The Constant Elasticity of Variance Option Pricing

The Constant Elasticity of Variance Option Pricing Model John Randal

A thesis submitted to the Victoria University of Wellington in partial fulfilment of the requirements for the degree of

Master of Science in Statistics and Operations Research April, 1998

2

Acknowledgements The author would like to thank his supervisors, Peter Thomson and Martin Lally, for their guidance and encouragement. Also, thanks to Credit Suisse First Boston NZ Limited for providing data, and to Edith Hodgen for valuable assistance with the preparation of this document.

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Abstract The Constant Elasticity of Variance (CEV) Model was first presented in 1976 by John Cox and Stephen Ross as an extension to the famous Black-Scholes European Call Option Pricing Model of 1972/3. Unlike the Black-Scholes model, which is accessible to anyone with a pocket calculator and tables of the standard normal distribution, the CEV solution consists of a pair of infinite summations of gamma density and survivor functions. Its derivation rested on the risk-neutral pricing theory and the results of Feller. Moreover, descriptions of this model in journal and text-book literature frequently contained errors. One difficulty in implementation of the Black-Scholes model is that one of its arguments is an unobserved parameter σ, the share price volatility. Much research has concentrated on estimating this parameter with a general conclusion that it is better to imply this volatility from observed option prices, than to estimate it from stock price data. In the case of the CEV model there are two unobserved parameters, δ 2 , with a relationship to the Black-Scholes parameter, and β, which defines the relationship between share price level and the variance of the instantaneous rate of return on the share. Attempts made to estimate this second parameter in the early 1980’s were not altogether satisfactory, perhaps condemning the CEV model to obscurity. A breakthrough was made in 1989 with a paper by Mark Schroder, who devised a method of evaluating the CEV option prices using the non-central Chi-Squared probability distribution, hence facilitating significantly simpler computation of the prices for those with suitable statistical software 1 . 1

I use the statistical package SPLUS extensively in this thesis.

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iv This thesis attempts to summarise the development of the CEV model, with comparisons made to the industry standard, the Black-Scholes model. The elegance of Schroder’s method is also made clear. Joint estimation of the two parameters of the CEV model, δ and β, is attempted using both simulated data, and a sample of stocks traded on the Australian Stock Exchange. In this section, it appears that significant improvements can be made to earlier estimation methods. Finally, I would like to note that this thesis is primarily a statistical analysis of a financial topic. As a consequence of this, the focus of my analysis differs to that of articles in the financial journal literature, and that of finance texts. Furthermore, the literature on the CEV model is sparse, and in general theorems therein are stated without proof. Some theorems found in this thesis reflect the statistical nature of the analysis and are hence absent from the financial literature which I have surveyed and referenced. Throughout the thesis, I have attempted to make it clear when an idea or proof follows previously established material. Unattributed material is generally that which I have worked on with my supervisors’ guidance, but which is not found in the references I have used.

Contents 1 Introduction to Options 1.1 1.2

1

The Call Option . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.1

An example . . . . . . . . . . . . . . . . . . . . . . . .

2

The Value of Call Options . . . . . . . . . . . . . . . . . . . .

2

2 GBM and the Black-Scholes Model

9

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2

Share Price Evolution . . . . . . . . . . . . . . . . . . . . . . . 10

2.3

2.4

2.5

2.2.1

Simulation of Geometric Brownian Motion . . . . . . . 15

2.2.2

The Future Share Price ST . . . . . . . . . . . . . . . . 18

Properties of CT - the Exercise Payoff . . . . . . . . . . . . . . 20 2.3.1

Mean and Variance of CT given St

. . . . . . . . . . . 23

2.3.2

Simulation of CT . . . . . . . . . . . . . . . . . . . . . 25

The Black-Scholes Formula . . . . . . . . . . . . . . . . . . . . 25 2.4.1

Properties of Call Price Prior to Maturity . . . . . . . 32

2.4.2

Best Predictor of a Future Black-Scholes Price . . . . . 34

2.4.3

Graphical Examination of Ct . . . . . . . . . . . . . . . 37

Use of the Black-Scholes Model . . . . . . . . . . . . . . . . . 42

3 The CEV Model

47

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2

The CEV Share Price Solution . . . . . . . . . . . . . . . . . . 50 3.2.1

Solution of the Forward Kolmogorov Equation . . . . . 54 v

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CONTENTS 3.2.2

Simulation of CEV Share Prices . . . . . . . . . . . . . 62

3.2.3

CEV Share Price Series . . . . . . . . . . . . . . . . . . 66

3.2.4

Graphical Examination of ST

. . . . . . . . . . . . . . 71

3.3

Properties of CT - the Exercise Payoff . . . . . . . . . . . . . . 77

3.4

The CEV Option Pricing Formula . . . . . . . . . . . . . . . . 78

3.5

3.6

3.4.1

The CEV Solution . . . . . . . . . . . . . . . . . . . . 81

3.4.2

Reconciling Various Forms of the CEV Solution . . . . 84

Computing the Option Price . . . . . . . . . . . . . . . . . . . 85 3.5.1

The Absolute CEV Model . . . . . . . . . . . . . . . . 85

3.5.2

Computing the General Model . . . . . . . . . . . . . . 86

3.5.3

CEV Option Prices . . . . . . . . . . . . . . . . . . . . 91

3.5.4

Behaviour of CEV Prices . . . . . . . . . . . . . . . . . 96

Use of the CEV Model . . . . . . . . . . . . . . . . . . . . . . 98

4 Data Analysis 4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.1.1

4.2

105

Summary of Alternative Methods . . . . . . . . . . . . 107

Estimating β from a share price series . . . . . . . . . . . . . . 113 4.2.1 4.2.2

β Estimation Strategy . . . . . . . . . . . . . . . . . . 114 The Variance of βˆ . . . . . . . . . . . . . . . . . . . . . 120

4.3

Appraisal of the Estimation Technique . . . . . . . . . . . . . 122

4.4

Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5 Conclusions

141

A Definitions

143

B Proofs for Selected Results

147

B.1 Result 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 B.2 Result 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 B.3 Result 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 B.4 Result 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

CONTENTS

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C Complete List of Shares

155

D SPLUS Code

157

D.1 GBM Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 157 D.2 Inversion of the Black-Scholes Formula . . . . . . . . . . . . . 158 D.3 CEV Share Price Simulation . . . . . . . . . . . . . . . . . . . 160 D.4 Estimation of β . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Bibliography

164

viii

CONTENTS

List of Figures 1.1

The lower bounds for call option value, and a possible form for the call price. . . . . . . . . . . . . . . . . . . . . . . . . .

2.1

5

A realisation of GBM with initial value St = $5, and parameters µ = 0.1, σ = 0.3 and 250 subintervals. . . . . . . . . . . . 16

2.2

The daily returns for the series shown in Figure 2.1 with estimated and actual mean, and the estimated mean ± 2 standard deviation limits. . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3

5000 realisations of ST , a future geometric Brownian motion price, with initial value St = $5, and parameters τ = 1, µ = 0.1 and σ = 0.3, and the theoretical lognormal distribution for ST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4

The unbroken series is a realisation of GBM with initial value St = $5 and parameters µ = 0.1, σ = 0.3, and 250 subintervals; the second series is the Black-Scholes prices for this share series added to the present value of the exercise price, with K = $5, and r = 0.06 and maturity at T = 1 year; finally the smooth broken line represents the present value of the exercise price. . . . . . . . . . . . . . . . . . . . . . . . . . 39 ix

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LIST OF FIGURES 2.5

The unbroken series is a realisation of GBM with initial value St = $5 and parameters µ = 0.1, σ = 0.3, and 250 subintervals; the second series is the Black-Scholes prices for this share series added to the present value of the exercise price, with K = $5, and r = 0.06 and maturity at T = 1 year; finally the smooth broken line represents the present value of the exercise price. . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.6

The distribution of call prices obtained using the Black-Scholes formula on a sample of share prices St+s , where s = 1 year, St = $5, µ = 0.1, σ = 0.1, K = $5, r = 0.06 and time to maturity τ − s of 5 months. Also shown is the theoretical density function. . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.7

The distribution of call prices obtained using the Black-Scholes formula on a sample of share prices St+s , where s = 1 year, St = $5, µ = 0.1, σ = 0.1, K = $5, r = 0.06 and time to maturity τ − s ranges from 2 years to half a month. Also the theoretical density function for these prices. . . . . . . . . . . 45

2.8

The distribution of call prices obtained using the Black-Scholes formula on a sample of share prices St+s , where s = 1 year, St = $5, µ = 0.1, σ = 0.1, K = $5, r = 0.06 and time to maturity τ − s ranges from 2 to 75 years. Also the (solid) theoretical density function for these prices, and the (dotted) lognormal density function of St+s . . . . . . . . . . . . . . . . 46

3.1

The relationship between β and P (ST = 0), with St = $5, β/2−1

τ = 1 year, µ = 0.10, σ = δSt

= 0.3 in the solid curve and

σ = 0.28 in the broken curve. . . . . . . . . . . . . . . . . . . 59

LIST OF FIGURES 3.2

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Firstly, a typical realisation of GBM, with St = $5, τ = 1 year, µ = 0.10, σ = 0.3 and n = 250 subintervals; secondly, the daily returns for this series with estimated mean, and mean ±

2 standard deviation series; thirdly, a plot of the share price

level against the estimated standard deviation of the daily returns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.3

Firstly, a typical realisation of share price, with St = $5, τ = 1 year, µ = 0.10, σ = 0.3 and n = 250 subintervals and CEV parameter β = −1; secondly, the daily returns for this series

with estimated mean, and mean ± 2 standard deviation series; thirdly, a plot of the share price level against the estimated

standard deviation of the daily returns. . . . . . . . . . . . . . 70 3.4

5000 realisations of ST , a future CEV price with β = −1, with St = $5, τ = 1 year, µ = 0.1 and σ = 0.3, the (solid)

theoretical density for these prices, and the lognormal density function with the same parameters. . . . . . . . . . . . . . . . 74 3.5

The empirical cumulative distribution function of the 5000 realisations of ST shown in Figure 3.4, a future CEV price with β = −1, with parameters St = $5, τ = 1, µ = 0.1 and

σ = 0.3, and the theoretical distribution function for these

prices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.6

The standard deviation of a future CEV share price ST , with St = $5, τ = 1 year, µ = 0.1 and σ = 0.3, for −2 ≤ β ≤ 2. . . . 76

3.7

A realisation of a CEV share price with initial value St = $5, and parameters β = −1, τ = 1, µ = 0.1, σ = 0.3, and 250

subintervals; also the CEV option prices for this share series

added to the present value of the exercise price, with K = $5, and r = 0.06 and time to maturity indicated on the horizontal scale; finally the present value of the exercise price itself. . . . 92

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LIST OF FIGURES 3.8

Black-Scholes implied volatilities for Absolute CEV option prices with St = $5, τ = 1 year, σ = 0.3, r = 0.06 and $4 ≤ K ≤ $6. . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.9

Out-of-the-money CEV option prices, with β between -2 and 6, σ between 0.4 and 2, and additional parameters St = $50, K = $55, τ = 0.5 years, and r = 0.06.

. . . . . . . . . . . . . 101

3.10 At-the-money CEV option prices, with β between -2 and 6, σ between 0.4 and 2, and additional parameters St = $50, K = $50, τ = 0.5 years, and r = 0.06.

. . . . . . . . . . . . . 102

3.11 In-the-money CEV option prices, with β between -2 and 6, σ between 0.4 and 2, and additional parameters St = $50, K = $45, τ = 0.5 years, and r = 0.06. 4.1

. . . . . . . . . . . . . 103

The log-likelihood surface, ¯l(β, δ), for a simulated series with St = $5, τ = 3 years, µ = 0.1, σ = 0.3, n = 250 subintervals per year, and β = 0. . . . . . . . . . . . . . . . . . . . . . . . 119

4.2

The cross-section of the log-likelihood surface in Figure 4.1, ¯l(β, δ) ˆ for a simulated series with St = $5, τ = 3 years, µ = 0.1, σ = 0.3, n = 250 subintervals per year, and β = 0. In addition, the line β = βˆ which identifies the maximum. . . . . . . . . . 119

4.3

en , given by Equation (4.14), for the simulated series examined previously with St = $5, τ = 3 years, µ = 0.1, σ = 0.3, n = 250 subintervals per year, and β = 0. . . . . . . . . . . . . 123

4.4

Estimates of β, resulting from CEV share price simulation, used for the figures in Table 4.2. Superimposed on these are ¯ 1) random variable over the the density function of an N (β, range of the estimates, where β¯ is the sample average of the β

4.5

estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 ˆ the standard deviation of β, ˆ from Table 4.2 Estimates of s(β), against the true β. . . . . . . . . . . . . . . . . . . . . . . . . 128

LIST OF FIGURES 4.6

Sample values en , defined in Equation (4.14), for the share price of AMC, with superimposed N (0, 1) density function.

4.7

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. 133

The BHP share series; in addition, the daily returns for this series with moving mean, and mean ± 2 standard deviation series; thirdly, a plot of the share price level against standard

deviation of the daily returns. . . . . . . . . . . . . . . . . . . 137 4.8

The MIM share series; in addition, the daily returns for this series with moving mean, and mean ± 2 standard deviation series; thirdly, a plot of the share price level against standard

deviation of the daily returns. . . . . . . . . . . . . . . . . . . 138 4.9

The BOR share series; in addition, the daily returns for this series with moving mean, and mean ± 2 standard deviation series; thirdly, a plot of the share price level against standard

deviation of the daily returns. . . . . . . . . . . . . . . . . . . 139

xiv

LIST OF FIGURES

List of Tables 3.1

A section of MacBeth and Merville’s Table 1, of CEV option prices, calculated for the parameters shown, with additional parameters: St = $50 and r = 0.06. . . . . . . . . . . . . . . . 93

3.2

Beckers’ δ values (rounded to 4 d.p.) for the Square Root CEV process, found using Equation (3.45), with additional parameters St = $40, and µ = log(1.05). . . . . . . . . . . . . 94

3.3

A section of Beckers’ Table II, showing Square Root CEV prices, with additional parameters St = $40, and r = log(1.05). 95

3.4

Beckers’ δ values (rounded to 4 d.p.) for the Absolute CEV process, with additional parameters St = $40, and µ = log(1.05). 96

4.1

MacBeth and Merville’s β estimates for six stocks. . . . . . . . 110

4.2

Summary of the β estimates for simulated series, obtained using the maximum likelihood procedure described above, with St = $5, µ = 0.1, σ = 0.3 and where 3 years’ data is used. . . . 126

4.3

p-values for hypothesis test H0 : β = i, i = −2, −1, . . . , 6

using the simulated data summarised in Table 4.2. . . . . . . . 127

4.4

β estimates for the 44 ASX share series, with estimates of ˆ the standard deviation of β, ˆ and the resulting confidence s(β), intervals obtained by simulation. . . . . . . . . . . . . . . . . . 130

4.5

p-values for the test of normality of en for the 44 ASX share series, where en is given by Equation (4.14). . . . . . . . . . . 132

C.1 ASX Company Codes . . . . . . . . . . . . . . . . . . . . . . . 155

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LIST OF TABLES

Chapter 1 Introduction to Options

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1.1

CHAPTER 1. INTRODUCTION TO OPTIONS

The Call Option

A call option gives the holder the opportunity, but not the obligation, to purchase a unit of its underlying asset some time in the future, at a price decided now. If the holder decides to buy the unit of underlying asset, they will exercise the option. The price they pay is called the exercise price. If the option is European the holder may exercise the option only on the exercise or maturity date of the option. American options may be exercised at any time up to and including the maturity date. An option will be either American or European, and it will have the fundamental characteristics: underlying asset, exercise price and maturity date.

1.1.1

An example

In New Zealand, it is possible to buy call options on the shares of Brierley Investments Limited (BIL). These options are traded along with options on other underlying assets on the New Zealand Futures and Options Exchange (NZFOE), information about which can be found on the internet at http://www.nzfoe.co.nz/. For example, Investor X could purchase a BIL option traded on the NZFOE, maturing in August 1998, with an exercise price of $1.20. All share options on the NZFOE are American, and so this option allows Investor X to buy 1000 BIL shares for $1200 any time between now and August 1998. If Brierley shares trade above $1.20 between now and August, Investor X may choose to exercise the option and receive the thousand shares. Alternatively, X might exercise the option and immediately sell the shares at the current share price (which would be greater then $1.20) for a profit. If the BIL share price does not exceed $1.20 before August, Investor X can (and should) let the option lapse, at no further cost.

1.2

The Value of Call Options

Although many traded options are American, and upon exercise yield many shares, for the remainder of this project I will consider only European call

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1.2. THE VALUE OF CALL OPTIONS options, whose underlying asset is a single share.

A call option is a derivative asset, whose value is based on the value of another asset, namely the underlying stock. Since exercise of the call option in the future delivers this share, the price of the option will obviously reflect the present worth of this share and the chances of it being higher than the exercise price on the maturity date. Let me present the following definitions: Definition 1.1. Let St be the price at time t of a share paying no dividends over the time interval [t, T ]. Definition 1.2. Let Ct be the price at time t of a European option. Definition 1.3. Let K be the exercise price of a European option, payable on exercise, for a single share with price St at time t. Definition 1.4. Let T be the maturity date of the European option, and τ = T − t the time until maturity from time t. Definition 1.5. Let r be the risk-free rate, which is payable continuously on an asset whose future value is certain. Definition 1.6. A European option is “in-the-money” if the current share price St exceeds K, “at-the-money” if the current share price is equal to K, and “out-of-the-money” if the current share price is less than K. On the maturity date T , the European option either expires worthless, or delivers a single share, with value ST , in exchange for a cash payment K. The payoff of the option at maturity can be represented mathematically by the equation: Payoff = CT =

(

ST − K 0

ST > K ST ≤ K

or equivalently CT = max(0, ST − K) = (ST − K)+ .

(1.1)

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CHAPTER 1. INTRODUCTION TO OPTIONS Prior to maturity, at time t, the option will have value not less than zero,

since the option cannot yield a negative payoff, and also not less than the current share price St less the present value of the exercise price discounted at the risk-free rate Ke−rτ . This is established in the following theorem, whose proof is standard and can be found in Hull (1997). Theorem 1.1 (Lower Bounds for Call Option Value). Ct ≥ max(0, St − Ke−rτ ) Proof. Since the option yields a non-negative payoff in the future, the price paid now for the opportunity to receive those payoffs must not be less than zero, Hence Ct ≥ 0.

Consider now two portfolios:

• Portfolio A, consisting of a single option; • Portfolio B, consisting of a share and a liability of K, payable at T . At T , the values of portfolios A and B are: VTA = CT = max(0, ST − K) VTB = ST − K respectively, and so we see VTA ≥ VTB , and hence it follows that VtA ≥ VtB .

If the latter relationship were not true, arbitrage profits could be earned by selling portfolio B and investing in portfolio A. Now VtB is given by VtB = St − Ke−rτ and hence Ct ≥ St − Ke−rτ .

(1.2)

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1.2. THE VALUE OF CALL OPTIONS

Option prices given by any option pricing model should obey the bounds given in Equation (1.2), and will have the same general appearance as the function shown in Figure 1.1. Properties of option prices appear in Merton (1973), and in particular, he proves that call prices must be convex in the

Option Price

share price1 .

Ct

St − Ke−rτ

0 0

Ke−rτ

Share Price

Figure 1.1: The lower bounds for call option value, and a possible form for the call price.

From the relationship given in Theorem 1.1 it is immediately apparent that the option price must depend on at least four factors: • St , the current share price; • τ , the time to maturity of the option; • K, the exercise price of the option;

1

• and r, the continuously compounding risk-free rate. Merton (1973), Theorem 10, page 150.

6


In addition, the option price will depend on the stochastic properties of ST . The fact that the call price is not equal to the lower bound given in Theorem 1.1 is due to the random nature of the future share price ST , and in particular: • σ, the share price volatility. This parameter, which will be defined more formally later, represents the uncertainty of future share prices. As this parameter increases, the risk associated with a future share price increases. The option price may be thought of as the sum of the lower bound and a premium, where the premium is monotonically increasing in σ. Treating St as fixed, and considering each of τ , K and r in turn, ceteris paribus, we can anticipate the effect a change in each factor might have on the current option price, using the lower bound for the option price given in Theorem 1.1. As the time to maturity τ increases, the present value of the exercise payment at the risk-free rate r diminishes: lim Ke−rτ = 0,

τ →∞

and so the lower bound Ct = St − Ke−rτ in Figure 1.1 is translated to the left

as the y intercept term decreases, and so the option price Ct corresponding to any particular St must increase to compensate. As the exercise payment increases, there is an opposite effect on the lower

bound Ct = St − Ke−rτ . In this case the y intercept term will increase,

thus translating the lower bound to the right, and allowing Ct to decrease. Increasing the exercise price therefore decreases the value of a European option. As the risk-free rate r increases, the present value of the exercise payment Ke−rτ will decrease as it did when τ was increased, and hence the option value will increase. At exercise, the option delivers the payoff (ST − K)+ . ST is of course a

random variable, since at the current time t we do not know for sure what

7

1.2. THE VALUE OF CALL OPTIONS

the share price at T will be. In order to make any statements about the probabilistic properties of ST , assumptions must be made on how the share price evolves through time. The evolution of a general process Xt may be described by the following stochastic differential equation (SDE): dXt = µ(Xt , t)dt + σ(Xt , t)dBt

(1.3)

where dXt may be interpreted as the change in Xt over the period [t, t + dt], µ(Xt , t) and σ(Xt , t) are functions of Xt and t, and {Bt } is Brownian motion, with initial condition B0 = 0. This process is too general for our purposes, and I will restrict attention to two specific cases, geometric Brownian motion (GBM), and Constant Elasticity of Variance (CEV) evolution. Definition 1.7. A share price that follows geometric Brownian motion (GBM) is a solution to the following SDE: dSt = µSt dt + σSt dBt

(t > 0)

where µ and σ are constant, and B0 = 0. Definition 1.8. A share price that follows the Constant Elasticity of Variance (CEV) model is a solution to the following SDE: β

dSt = µSt dt + δSt2 dBt

(t > 0)

(1.4)

where µ, δ and β are constant, and B0 = 0. It is clear from Definitions 1.7 and 1.8 that GBM is a special case of the CEV model, and corresponds to the case when β = 2. The solution to Equation (1.4) has very different behaviour for the three cases β = 2, β < 2, and β > 2. In the first case, the solution to the SDE above is geometric Brownian motion, which is a well known and widely studied process. This process, and the price of a call option over a share price following GBM,

8


are described in Chapter 2. When β < 2, the share price is the original Constant Elasticity of Variance process, and is considered, again with its companion option prices, in Chapter 3. The third case, corresponding to β > 2 is mentioned briefly at the end of Chapter 3, and is applied in Chapter 4 along with both other cases.

Chapter 2 GBM and the Black-Scholes Model

9

10

CHAPTER 2. GBM AND THE BLACK-SCHOLES MODEL

2.1

Introduction

Analysis of share prices over time would suggest that they are discrete-time, discrete-variable processes. This means that they change at discrete time points, and may take on only discrete values. In practice, however, share prices are generally modelled using continuous-time, continuous-variable processes. The geometric Brownian motion (GBM) process examined in this chapter, and the Constant Elasticity of Variance (CEV) process considered in the following chapter are continuous-time, continuous-variable processes. In addition, both these models are Markovian, meaning that the only relevant information regarding the future of the process is the present value, and that the past is irrelevant. This can be expressed mathematically as follows: P (St+s < s|Su , 0 ≤ u ≤ t) = P (St+s < s|St ). This property is consistent with weak form efficiency in the share market, since it implies that all information in prices S0 , S1 , . . . , St−1 is encapsulated in the present price St . The Black-Scholes Model was first presented in an empirical paper by Black & Scholes (1972), which was quickly followed by its derivation in Black & Scholes (1973). This model is based on a number of restrictive assumptions, one of which is that the price of the underlying asset has a lognormal distribution at the end of any finite (forward) interval, conditional on its value at some initial starting point. It will be shown that if we assume that the share price follows GBM, then this condition will be met. In this chapter I will deal initially with properties of geometric Brownian motion, and then examine the Black-Scholes model.

2.2

Share Price Evolution

Derivation of the Black-Scholes option pricing equation requires the assumption that the future share price has a lognormal distribution. This condition

11

2.2. SHARE PRICE EVOLUTION

is met if share price follows GBM, a process which is defined in Definition 1.7, and is a solution to the following SDE: dSt = µSt dt + σSt dBt

(2.1)

with initial condition B0 = 0, and where St is the share price at time t, µ is the continuously compounding expected growth rate of St , σ is the standard deviation of the instantaneous return on St , and {Bt } is Brownian motion, with E(dBt ) = 0 and Var(dBt ) = dt. Note that both µ and σ are constants,

independent of time and the current share price, and that Equation (2.1) is a special case of Equation (1.4), with β = 2 and δ = σ. I show below that the solution of the SDE above has a lognormal distribution. To prove this it is necessary to use a result from stochastic calculus called Itô’s Differentiation Lemma. Result 2.1 (Itˆ o’s Differentiation Lemma). Suppose that f (x, t) and its partial derivatives fx , fxx and ft are continuous. If Xt is given by dXt = µ(Xt , t)dt + σ(Xt , t)dBt then Yt = f (Xt , t) has stochastic differential: dYt = (µ(Xt , t)fx + ft + 12 σ 2 (Xt , t)fxx )dt + σ(Xt , t)fx dBt = fx dXt + ft dt + 12 σ 2 (Xt , t)fxx dt. Hull (1997) gives a sketch proof of this result using a Taylor series expansion. Using Itô’s Lemma, we can now prove the following well known theorems. Theorem 2.1. Conditional on its value at time t, a share price ST that follows GBM will have a lognormal distribution at T > t, with parameters E(ln ST ) = ln St + (µ − 21 σ 2 )τ and Var(ln ST ) = σ 2 τ , where τ = T − t.

Moreover, the form for ST given St will be: 1

ST = St e(µ− 2 σ

2 )τ +σ √τ Z

where Z is a standard normal variable with mean zero and unit variance.

12


Proof. Consider the transformation Yt = ln St . Referring to the SDE (2.1), and to Result 2.1, we see that: Xt = S t µ(Xt , t) = µSt σ(Xt , t) = σSt f (Xt , t) = ln St and so applying Itô’s Lemma ∂ln St ∂ln St ∂ 2 ln St dt + dSt + 12 (σSt )2 dt ∂t ∂St ∂St 2 1 1 = 0 dt + (µSt dt + σSt dBt ) − 12 σ 2 St2 2 dt St St ¡ ¢ 1 2 = µ − 2 σ dt + σdBt

d ln St =

(2.2)

Thus, integrating both sides from t to T yields: ¡

1 2 σ 2

¢

Z

T

ln ST − ln St = µ − dBt (T − t) + σ t ¡ ¢ = µ − 12 σ 2 τ + σ(BT − Bt )

From the properties of Brownian motion, in particular its independent increments: BT − Bt ∼ BT −t − B0 = Bτ since B0 = 0, and where ∼ denotes “is distributed as”. Hence: ¢ ¡ ln ST ∼ ln St + µ − 21 σ 2 τ + σBτ .

(2.3)

Conditioned on St , it is clear that the only random variable in the RHS

of the above equation is the Brownian motion term Bτ . Using the fact that Bt ∼ N (0, t) for any t > 0, we obtain the mean and variance of ln ST given

St :

¡ ¢ E(ln ST |St ) = E(ln St + µ − 12 σ 2 τ + σBτ |St ) ¡ ¢ = ln St + µ − 12 σ 2 τ + σE(Bτ ) ¡ ¢ = ln St + µ − 12 σ 2 τ

13

2.2. SHARE PRICE EVOLUTION ¡ ¢ Var(ln ST |St ) = Var(ln St + µ − 12 σ 2 τ + σBτ |St ) = σ 2 Var(Bτ )

= σ2τ In addition, if X is a normal random variable, then the linear combination a + bX, where a and b are both constant, is also normal with mean aE(X) and variance b2 Var(X). Combining this with the mean and variance above it is clear that ¡ ¡ ¢ ¢ ln ST |St ∼ N ln St + µ − 12 σ 2 τ, σ 2 τ (2.4) ¢ ¡ i.e. ST is lognormal with parameters ln St + µ − 12 σ 2 τ and σ 2 τ . Moreover,

since Z =

Bτ √ τ

∼ N (0, 1), it follows directly from Equation (2.3) that 1


2 )τ +σ √τ Z

(2.5)

as required. Theorem 2.2 (Moments of a Share Price following GBM). The mean and variance of the share price ST conditional on an earlier share price St , 0 ≤ t < T are St eµτ and St2 e2µτ (eσ

2τ

− 1) respectively.

Proof. Note first that the moment generating function of an N (0, 1) variable 1 2

is E(esZ ) = e 2 s . This can be used to find the moments of the random variable of interest ST |St .

¯ ¶ µ √ ¯ (µ− 12 σ 2 )τ +σ τ Z ¯ E(ST |St ) = E St e ¯ St ³ √ ´ (µ− 21 σ 2 )τ E eσ τ Z = St e 1

= St e(µ− 2 σ

2 )τ

1

e2σ

2τ

= St eµτ ¯ ¶ µ √ ¯ 2 2(µ− 12 σ 2 )τ +2σ τ Z ¯ 2 E(ST |St ) = E St e ¯ St ³ √ ´ 1 2 = St2 e2(µ− 2 σ )τ E e2σ τ Z 1

= St2 e2(µ− 2 σ

= St2 e(2µ+σ

2 )τ

2 )τ

e2σ

2τ

(2.6)

14


Forming the variance using the relationship Var(X) = E(X 2 ) − E(X)2 , we obtain

2

Var(ST |St ) = St2 e2µτ (eσ − 1)

(2.7)

as required. Note that if share price evolution is a deterministic process, i.e. σ = 0, then ST equals the mean of the stochastic process: St eµτ . Thus using either the expected value, or the deterministic case, the parameter µ can be thought of as the continuously compounding expected rate of return on the share per unit time, and may be modelled for a particular stock using the Capital Asset Pricing Model (CAPM), which features both a market risk premium, and a measure of the “riskiness” of the particular firm compared to the “market”. The CAPM describes the expected return on a particular asset: µj = r + MRP

Cov(Rj , Rm ) 2 σm

where MRP is the market risk premium, Rj is the return on asset j, µj is the expected value of that return, Rm is the return on the market portfolio, 2 and σm is the variance of the market return. Aggregate investor attitudes

towards risk affect the size of the market risk premium, which has a direct effect on the size of µj . The size of this effect is determined by the measure of systematic risk: Cov(Rj , Rm ) 2 σm which compares the risk associated with the particular asset to the risk associated with the market as a whole. The parameter µ is the only component of the GBM model that reflects investor risk attitudes. The return on an asset j can be modelled using the standard univariate linear regression model: Rj = α j + b j Rm + ² j

15


where Rj and Rm are as above, αj and bj are the regression model parameters, and ²j is a stochastic error term, with variance σ²2j . Applying the variance operator to all terms in the equation above yields the relationship: 2 σj2 = b2j σm + σ²2j . 2 measures the systematic risk in the The variance of the market return, σm

system, whereas σ²2j measures the non-systematic risk. Hence the parameter σ reflects a combination of these, but not investor risk attitudes.

2.2.1

Simulation of Geometric Brownian Motion

Simulation of GBM series, or prices at a particular time in the future is a useful way of analysing the properties of GBM, and later, analysing the properties of Black-Scholes option prices. Because the solution to the SDE (2.1) is known, there is no need to use a numerical method to approximate the solution, but rather the solution can be simulated directly. The solution to the SDE was given in Equation (2.5) and is: 1


2 )τ +σ √τ Z

where Z ∼ N (0, 1). Hence in order to simulate a single price at T , a single

realisation of Z can be obtained, and the share price computed directly.

In order to simulate an entire GBM series, we can divide the interval of interest [t, T ] into n subintervals defined by the times: t = t0 < t1 < · · · < ti < · · · < tn−1 < tn = T where the intervals are not necessarily of equal length. Equation (2.5) can also be written to give the share price at time ti conditional on the share price at ti−1 : 1

Sti = Sti−1 e(µ− 2 σ

2 )(t

i −ti−1 )+σ

√

ti −ti−1 Z

16


hence simulating n realisations of Z, the series can be constructed as follows: i Y St j St i = S t S j=1 tj−1

= St

i Y

1

e(µ− 2 σ

2 )(t

j −tj−1 )+σ

√

tj −tj−1 Zj

j=1

= St exp = St e

Ã

i X j=1

(µ −

(µ− 12 σ 2 )(ti −t)

1 2 σ )(tj 2

Ã

p − tj−1 ) + σ tj − tj−1 Zj

i X p exp σ tj − tj−1 Zj j=1

!

!

.

(2.8)

A program which simulates GBM using Equation (2.8) is given in Appendix

4.5

5.0

Share Price

5.5

6.0

D.1.

0.0

0.2

0.4

0.6

0.8

1.0

Time - years

Figure 2.1: A realisation of GBM with initial value St = $5, and parameters µ = 0.1, σ = 0.3 and 250 subintervals.

Figure 2.1 shows a single realisation of geometric Brownian motion with St = $5, τ = 1 year, µ = 0.1 and σ = 0.3. This particular realisation has


17

ST = $5.12, which is smaller than its expected value E(ST |St ) = $5e0.1 = √ $5.53 but well within a single standard deviation s(ST |St ) = St eµ eσ2 − 1 = $1.696 of it.

The daily returns for the series {St } are defined as: Rt = ln St+∆t − ln St where ∆t =

1 250

years is approximately one trading day. Whilst the series

{St } is clearly not a stationary process, Equation (2.2) indicates that the

daily returns should be stationary, with a mean (µ − 12 σ 2 )∆t and variance σ 2 ∆t. A plot of the daily returns can be seen in Figure 2.2, with an estimate

of the mean level and standard deviation shown1 . These have been estimated using a Lowess filter with a smoothing window of 30 observations. The Lowess filter is a robust, centred moving average filter, and was derived by Cleveland (1979). This filter was designed to smooth scatterplots, but has application to the equi-spaced observations of a time series. It estimates the average value at ti using weights from the bisquare function: ( (1 − x2 )2 |x| < 1 B(x) = 0 |x| ≥ 1 where x depends on the time ti and the smoothing window, and by making adjustments for outlying values. In this case I have used a smoothing window of 30 days, so that values outside this window are given zero weight, and hence do not affect the estimate. The estimates produced at the ends of the series are unreliable due to back- and forecasting of share price values. These estimates, obtained using estimated share price data, are not shown in the graph. Also shown in Figure 2.2 is the true mean for the daily returns of (µ −

1 2 σ )∆t 2

which is negligible for the parameters chosen2 . We see that the es-

timated mean does indeed oscillate about the actual mean level, and that 1

The series actually shown are the estimated mean, and the estimated mean ± two estimated standard deviations, from which the estimated standard deviation itself can be recovered. 2 Nevertheless, over a year it compounds to a significant amount.

18

0.02 0.0 -0.04

-0.02

Log-returns

0.04

0.06


0.0

0.2

0.4

0.6

0.8

1.0

Time - years

Figure 2.2: The daily returns for the series shown in Figure 2.1 with estimated and actual mean, and the estimated mean ± 2 standard deviation limits. the estimated standard deviation appears approximately constant. In fact, an autocorrelation plot for the daily return series shows no significant autocorrelation for non-zero lags. This is not surprising, since the series was generated in the first place using realisations of the standard normal variable that should indeed be independent of one another, and hence uncorrelated.

2.2.2

The Future Share Price ST

As discussed in Chapter 1, the value of a European option at t will certainly depend on the share price at the future exercise time T . Since we are able to simulate geometric Brownian motion, we can also simulate the distribution of future share prices by generating many realisations of the same process. The value of this is largely illustrative, since many of the properties of GBM are well known. Suppose a particular share price truly follows GBM, with known µ and σ, then N realisations of the share price at T can be obtained


19

using the second program seen in Appendix D.1. Simulation of the quantity ST is a useful way of deciding what properties the share price will have at T , and hence what payoff (if any) the option is likely to deliver. In the case of GBM, the theoretical distribution is known, and in this case can be compared to the results of the simulation to appraise the simulation procedure, and help guide the eye. Figure 2.3 shows the result of a simulation of 5000 share prices τ = 1 year in the future, with additional parameters St = $5, µ = 0.1 and σ = 0.3. The final value of the time series in Figure 2.1 has the same properties as each of the observations shown in the histogram. Superimposed on the observed distribution is the theoretical lognormal density curve with parameters E(ln ST |St ) = ln 5 + (0.1 + 12 0.32 ) and Var(ln ST |St ) = 0.32 . It is clear from the graph that the fit is very good,

particularly when the bars are small. This is expected since the height of fre-

quency histogram bars is a Poisson random variable with mean and variance equal to the expected height of the bars. Hence, the smaller bars’ heights will have a small standard deviation and should be closer to the curve than the higher bars. The heights of the equi-width bars in the relative frequency histogram shown in Figure 2.3 are proportional to the Poisson heights in a frequency histogram, and so the variability comments hold. A useful means of gauging the success of the simulation procedure is to estimate the mean and variance of a sample of share prices, and compare these to the theoretical values given by Equations (2.6) and (2.7) respectively. These estimates are given by the sample mean and sample variance of 5.5397 and 2.8756 respectively, compared to theoretical values of 5.5259 and 2.8756. Note that while the means differ slightly, the variance estimate is accurate to within 4 decimal places, again testimony to the accuracy with which GBM can be simulated. Note that these estimates do not correspond to the parameters of the lognormal distribution, which are the mean and variance of the log share prices. The maximum likelihood method could be used to fit the “best” lognormal distribution to the sample values but this has not been done here.

20

0.15 0.10 0.0

0.05

Relative Frequency

0.20

0.25


2

4

6

8

10

12

14

Share Price

Figure 2.3: 5000 realisations of ST , a future geometric Brownian motion price, with initial value St = $5, and parameters τ = 1, µ = 0.1 and σ = 0.3, and the theoretical lognormal distribution for ST .

2.3

Properties of CT - the Exercise Payoff

The properties of the call value at maturity are intimately linked to those of the share price by the equation: CT = max(ST − K, 0) = (ST − K)+

(2.9)

where K is the exercise price of the call option. It is this payoff that investors are interested in valuing. Such a future cash flow could be valued using the equation Pt = e−λτ E(CT |St )

(2.10)

where Pt would be the price paid now for the payoff CT , which would be received at time T . This equation features two rates particular to the risk

2.3. PROPERTIES OF CT - THE EXERCISE PAYOFF

21

preferences of investors in aggregate: µ, the continuously compounded expected rate of return which features in the expectation, and λ, the discount rate for future cash flows. It will be shown later that both µ and λ can be treated as if they were the risk-free rate r, for valuing this particular future cash flow. Consider the simple transformation YT = ST − K. It is clear YT has mean

E(ST )−K, variance Var(ST ), and YT +K has a lognormal distribution. Note that YT itself does not have a lognormal distribution, since the lognormal

distribution is defined on the range [0, ∞) whereas YT is defined on [−K, ∞).

However, the shape of the distribution of YT will be identical to that of ST ,

since we have the relationship P (YT < y) = P (ST − K < y) = P (ST < y + K). Next consider CT = YT+ = max(0, YT ). It is clear that P (CT = 0) = P (ST ≤ K) and so CT will not have a continuous distribution function. Theorem 2.3 (Distribution of CT given St ). Given St , CT = (ST −K)+

has a mixed distribution

P (CT ≤ c|St ) =

(

0 FST |St (K + c, τ )

c 0: FCt+s |St (c, τ ) = P (Ct+s < c|St ) = P (BS(St+s ) < c|St ) = P (St+s < BSS−1 (c; t + s)|St ) = P (ln St+s < ln BSS−1 (c; t + s)|St ) µ ¶ ln BSS−1 (c; t + s) − ln St − (µ − 12 σ 2 )s √ =Φ σ s since ln St+s ∼ N (ln St + (µ − 21 σ 2 )s, σ 2 s). This distribution function features

the inverse of the Black-Scholes function with respect to St+s . Whilst the form for this inverse cannot be written directly, the Black-Scholes model is a monotonic increasing function of St , and so computation of the single inverse value can be achieved using a numerical method such as the Newton-Raphson algorithm. This method is particularly useful here since the first derivative of the inverse with respect to c does have an explicit form which is easily evaluated. Differentiating this cumulative distribution function on the range c > 0 gives the form for the density function on the same range. Noting that if y = BSS−1 (c; t + s), then

∂y ∂c

can be found by using the chain rule for

differentiation: c = BS(y) ∂BS(y) ∂y 1= ∂y ∂c ∂y 1 = ³ ln y−ln K+(r+ 1 σ2 )(τ −s) ´ ∂c √ 2 Φ σ τ −s

since

∂C ∂S

= Φ(ht ). So, noting

1 1 ∂BSS−1 (c; t + s) ´= = ³ −1 1 2 ln BS (c;t+s)−ln K+(r+ σ )(τ −s) ∂c Φ(h∗t+s ) S 2 √ Φ σ τ −s

34


the derivative of FCt+s |St (c, τ ) with respect to c, with c > 0 is: ∂ FC |S (c) ∂c t+s t µ ¶ ln BSS−1 (c; t + s) − ln St − (µ − 12 σ 2 )s 1 1 √ = √ φ −1 Φ(h∗t+s ) σ s BSS (c; t + s)σ s µ ¶ ln BSS−1 (c; t + s) − ln St − (µ − 12 σ 2 )s 1 √ φ = √ σ s BSS−1 (c; t + s)σ sΦ(h∗t+s )

fCt+s |St (c, τ ) =

giving finally the density function for Ct+s :  0 c≤0 ³ ´ −1 1 2 fCt+s |St (c, τ ) = (c;t+s)−ln S −(µ− ln BS σ )s t 1 S 2 √  BS −1 (c;t+s)σ √ φ c>0 σ s sΦ(h∗ ) S

t+s

(2.21)

Note that in order to evaluate this density function, the Black-Scholes function must be inverted with respect to St . This cannot be done analytically and the equation c = BS(S) must be solved for S numerically. This can be done using a numerical method such as the Newton-Raphson method. Details of this method, and its implementation in this case can be found in Appendix D.2.

2.4.2

Best Predictor of a Future Black-Scholes Price

Although the moments of the theoretical distribution look impossible to derive using the density function above, it is possible to derive the mean price at a time t + s where 0 < s < τ . It is known that the best predictor of this future price, in a mean square sense, is given by the expected value, conditional on the past and present values of the series. In order to calculate the expected value of CT , the fact that ST given St is a function of a standard normal random variable Z was utilised. The same procedure can be used here.

35

2.4. THE BLACK-SCHOLES FORMULA

Theorem 2.9. The expected value of Ct+s , given St , where s ∈ (0, τ ), is

given by the equation:

³ E(Ct+s |St ) = St eµs Φ ht +

(µ−r)s √ σ τ

³ √ − Ke−r(τ −s) Φ ht − σ τ +

´

(µ−r)s √ σ τ

´

where ht is as in the Black-Scholes formula, and is given in Theorem 2.7. Proof. The form of St+s , where 0 < s < τ , is given by Equation (2.5) 1

St+s = St e(µ− 2 σ

2 )s+σ √sZ

.

Moreover Ct+s is given by the Black-Scholes formula (2.20) and so: √ E(Ct+s |St ) = E(St+s Φ(ht+s )|St ) − Ke−rτ E(Φ(ht+s − σ τ − s)|St )

(2.22)

with ht+s a function of the random variable St+s . Noting then that the Black-Scholes price Ct+s is a function of St+s which is in turn a function of Z, we conclude that Ct+s is itself a function of Z and its expected value can be computed using the standard normal density function φ(z) rather than the very complicated fCt+s |St (c, τ ). I will evaluate the two expectations in the above equation separately. The first expectation of interest is Z E(St+s Φ(ht+s )|St ) =

∞

St+s Φ(ht+s )φ(z)dz

z=−∞

and can be written as two separate components, which I will simplify independently: 1

St+s φ(z) = St e(µ− 2 σ

2 )s+σ √sz

1 2 1 √ e− 2 z 2π

√ 2 1 1 = St eµs √ e− 2 (−z+σ s) 2π µs = St e φ(y)

√ where y = −z+σ s. Simplifying first ln St+s , using the same transformation: √ ln St+s = ln St + (µ − 21 σ 2 )s + σ sz √ = ln St + (µ + 21 σ 2 )s − σ sy

36


thus ht+s becomes: ln St+s − ln K + (r + 12 σ 2 )(τ − s) √ σ τ −s √ ln St + (µ + 12 σ 2 )s − σ sy − ln K + (r + 21 σ 2 )(τ − s) √ = σ τ −s r ¶ r µ ln St − ln K + (r + 12 σ 2 )τ (µ − r)s τ s √ √ = + −y τ −s τ σ τ σ τ r ¶ r µ (µ − r)s τ s √ = ht + −y τ −s τ σ τ

ht+s =

Substituting these expressions into the first expectation on the right hand √ side of Equation (2.22), and making the change of variables z = −y + σ s yields:

E(St+s Φ(ht+s )|St ) =

where

Z

∞ y=−∞



St eµs φ(y)Φ 

³ = St eµs Φ2 ∞, ht + Φ2 (x, y; ρ) =

Z

x

φ(z)Φ −∞

Ã

ht +

(µ−r)s √ ; σ τ

(µ−r)s √ σ τ

q

−y

τ −s τ

q ´

y − ρz p 1 − ρ2

s τ

!

ps  τ  dy

dz

is the cumulative distribution function for a bivariate normal random variable with zero means, unit variances, and correlation coefficient ρ. This distribution function is symmetric about x and y, and so the order of integration can be changed to give an alternative distribution function: ! Ã Z y x − ρz dz. Φ2 (x, y; ρ) = φ(z)Φ p 1 − ρ2 −∞ and in particular

Φ2 (∞, y; ρ) =

Z

y

φ(z)Φ(∞)dz = Φ(y) −∞

since Φ(∞) = 1. This leads to the simplification of the first expectation: µ ¶ (µ − r)s µs √ E(St+s Φ(ht+s )|St ) = St e Φ ht + . σ τ

37


Now consider the second expectation in Equation (2.22). I will again simplify the component of the expression separately, but without making the y transformation as done in the previous term: √ ln St+s − ln K + (r − 21 σ 2 )(τ − s) √ ht+s − σ τ − s = σ τ −s √ ln St + (µ − 12 σ 2 )s + σ sz − ln K + (r − 21 σ 2 )(τ − s) √ = σ τ −s r ¶ r µ ln St − ln K + (r − 12 σ 2 )τ τ (µ − r)s s √ √ = + +z τ −s τ σ τ σ τ r ¶ r µ √ (µ − r)s τ s √ = ht − σ τ + +z τ −s τ σ τ

Substituting this into the expectation and interchanging the order of integration as above gives: √ E(Φ(ht+s − σ τ − s|St ) =

Z

∞ z=−∞



φ(z)Φ 

√

ht − σ τ + q

(µ−r)s √ σ τ τ −s τ

q ´ ³ √ s √ ;− = Φ2 ∞, ht − σ τ + (µ−r)s τ σ τ ´ ³ √ √ . = Φ ht − σ τ + (µ−r)s σ τ

ps  +z τ  dz

Hence substituting the two expectations into Equation (2.22), we obtain the expected value of Ct+s : ³ E(Ct+s |St ) = St eµs Φ ht +

(µ−r)s √ σ τ

´

³ √ − Ke−r(τ −s) Φ ht − σ τ +

(µ−r)s √ σ τ

´

(2.23)

as required. This formula is intriguing, since it is so similar to the Black-Scholes formula itself, but with the first St+s replaced by E(St+s |St ), and the correction term

in the Φ terms.

2.4.3

Graphical Examination of Ct

Just as the series {St } can be simulated, so too can the series {Ct }. This is done using a share price time series and the Black-Scholes formula. In

38


addition to parameters selected for simulation of share price evolution, St , τ , µ and σ, parameters K and r must also be chosen. In this case I have treated the time period over which the share is simulated as the remaining life of the option. Figures 2.4 and 2.5 show three time series, the first {St }, and the second

{Ct + Ke−rτ } are given by the solid and dotted lines respectively. The third,

much smoother curve, is the series {Ke−rτ }. The reason for showing the second series instead of simply {Ct } is to place both the share and option price

series in the same region of the graph, and is facilitated by the relationship

given in Theorem 1.1 (Ct ≥ (St −Ke−rτ )+ ). The series {Ct } is still evident in

the graphs by taking the difference of the series {Ct + Ke−rτ } and {Ke−rτ }.

It is apparent that Ct is always at least as great as St − Ke−rτ as required.

It is also clear that the two series are highly correlated, as expected; all

movements in {St } are mirrored by movements in the call price. However this correlation is not linear, since the Black-Scholes formula is non-linear in St . Figure 2.4 features a share price series that has ST < K, and so the option matures out-of-the-money, and will not be exercised. This is reflected in the option price series, which converges to zero as the share price falls sharply in the latter part of the series. There is a period in the middle of the share series where St < Ke−rτ , and the option price becomes small, but with a reasonable amount of time remaining, it does not disappear altogether. At this stage there is still a good chance that the option will mature in-the-money, and will be exercised. Whilst we see similar share prices at 0.8 and 0.2 years to maturity, as the share price falls from this level, the option price decreases much more quickly in the latter case, since there is little opportunity left for the share price to rise above K. Figure 2.5 shows a share price series that generally increases with time. It is a realisation from the same process as the share price series in Figure 2.4 but there are clear differences between the two. In this case the option

39

4.5 3.0

3.5

4.0

Price

5.0

5.5


1.0

0.8

0.6

0.4

0.2

0.0

Time to Maturity - Years

Figure 2.4: The unbroken series is a realisation of GBM with initial value St = $5 and parameters µ = 0.1, σ = 0.3, and 250 subintervals; the second series is the Black-Scholes prices for this share series added to the present value of the exercise price, with K = $5, and r = 0.06 and maturity at T = 1 year; finally the smooth broken line represents the present value of the exercise price.

price plus the present value of the exercise price converges on the share price series, until with approximately 0.2 years to maturity the two series are indistinguishable. In this case the share is so deep-in-the-money (St À K),

the probability of the share price at maturity being below the exercise price

is so small that a portfolio of the option and a cash asset Ke−rτ compounding at the risk-free rate is equivalent to a portfolio containing only the share. Whilst the option price Ct is a deterministic function of St , for s > 0, Ct+s will depend on a random variable St+s and so Ct+s is itself a random variable, with density function given by Equation (2.21). It is interesting to examine this density function for a fixed value of s ∈ [0, τ ]. We have already seen that

when τ − s = 0 and the option is at maturity, the distribution of call prices is

40

5.0

5.5

Price

6.0

6.5


1.0

0.8

0.6

0.4

0.2

0.0


Figure 2.5: The unbroken series is a realisation of GBM with initial value St = $5 and parameters µ = 0.1, σ = 0.3, and 250 subintervals; the second series is the Black-Scholes prices for this share series added to the present value of the exercise price, with K = $5, and r = 0.06 and maturity at T = 1 year; finally the smooth broken line represents the present value of the exercise price.

a mixed distribution given by Equation (2.12). Alternatively, for a fixed s, as T → ∞ the value and distribution of a call option will converge to the value

and distribution of the share. This is financially sensible, since as the share

pays no dividends, the only difference between the option and the share is the exercise payment, and is consistent with putting τ − s = ∞ in both the

Black-Scholes formula at t + s and in the density function fCt+s |St (c, τ − s) given by Theorem 2.8. As the exercise date is moved further and further into the future, the present value of the exercise payment shrinks to zero. Hence, an option with infinite life has the same characteristics as the share. At time t + s, the option price distribution and its limiting properties

can be examined graphically. The first series of graphs in Figure 2.7 shows


41

the behaviour of fCt+s |St (c, τ ) as the remaining life of the option decreases to zero (s → τ ). The sequence of six graphs has time to maturity decreasing from two years to approximately two weeks. The histograms in the figure

represent the distribution of simulated values of Ct+s , with s = 1 year, and the superimposed curve, the theoretical distribution of such prices. As the time to maturity of the option decreases, the sequence of graphs shows the convergence of the density function to the mixed distribution of CT (where s = τ ). We see the number of simulated option prices near zero increase, and the density function approach the continuous part of the mixture density (the truncated, translated lognormal density) and a spike at c = 0 with infinite height, and area P (St+s < K). Again, there is good agreement between the histograms and the theoretical density functions, particularly when the value of fCt+s |St is small. The closeness of the observed and the expected distributions stems from our ability to simulate {St } with only sample

error. Hence, a good fit in the St+s distribution should result in a good fit to any derivative distribution. In this way, simulation provides a means of confirming that the density functions I have derived are correct, with a close fit indicating success. The density function, when τ −s is small, has an interesting characteristic

which is not always apparent, and does not feature for the choice of parameters used in Figure 2.7. By decreasing the volatility parameter σ, however, the effect is observed. Figure 2.6 shows the results of a new simulation of 5000 share prices, and the density function of its call prices with 5 months to maturity. The density function in this case has an interesting “saddle” effect, created as the probability of an option maturing out-of-the-money increases, but the probability of it expiring at-the-money is still moderate. Figure 2.8 shows a sequence of graphs with τ − s increasing from 2 years

to 75 years, with two curves superimposed, and can be used to demonstrate the behaviour of the call price distribution as time to maturity increases. The solid curve is the theoretical density function of the call prices, and the second

42

0.0

0.2

0.4

0.6

f(c)

0.8

1.0

1.2


0.0

0.5

1.0

1.5

2.0

2.5

3.0

c

Figure 2.6: The distribution of call prices obtained using the Black-Scholes formula on a sample of share prices St+s , where s = 1 year, St = $5, µ = 0.1, σ = 0.1, K = $5, r = 0.06 and time to maturity τ − s of 5 months. Also shown is the theoretical density function.

broken curve is the lognormal density function of St+s , and the limiting form of fCt+s |St (c, τ ). The convergence of the option prices to the call prices is apparent as the sample distribution and theoretical curve move along the c axis towards the density function of the share price, until at τ − s = 75 the

two curves are virtually indistinguishable.

2.5

Use of the Black-Scholes Model

Geometric Brownian motion is a process whose transition probability density is known, whose moments are easily determined, and which can be accurately simulated with minimal computing resources. In addition, the formula for a European call option over an asset whose value follows GBM is known, and is easily obtained, verified and evaluated. The Black-Scholes option price

2.5. USE OF THE BLACK-SCHOLES MODEL

43

depends on only five parameters: St , τ , K, r and σ, of which the first three are directly observable, and the fourth is relatively easy to estimate. The model prices also have properties which are easily derived. The above analysis gives the density function for call options both at and prior to maturity. The expected value of option prices at a time in the future, up to and including maturity can also be determined. Black-Scholes prices are also easily simulated. Why then is an alternative model needed, when the GBM and Black-Scholes framework provide so many benefits? The answer to this question is that unfortunately Black-Scholes model prices do not correspond to those observed in the market. Even before the theoretical derivation was published, Black & Scholes (1972) noted that the model tends to overprice options on high variance stocks and under price options on low variance stocks. They also raise questions regarding the estimation of the final parameter σ. A series of empirical studies followed, attempting to obtain estimates of σ, which, when used in the Black-Scholes formula will more closely approximate future market option prices. These studies include those by Latané & Rendleman (1976), Chiras & Manaster (1978), and Beckers (1981). These authors, and others, discover that market option prices yield better estimates of volatility experienced over the remaining life of the option than estimates obtained using time series data of the share price. The estimates obtained from market option prices are called implied volatilities, and provide further evidence that the GBM assumption of the Black-Scholes model is not appropriate. Implied volatilities are calculated using a numerical method in much the same way as BSS−1 (c; t + s) was calculated for use in Equation (2.21). The implied volatility is the solution to the equation Cmt = BS(σ; St , τ, K, r) where Cmt is the market option price at time t, and BS(σ; St , τ, K, r) is the Black-Scholes function with single argument σ. The implied volatility σ ˆ is

44


the value of σ that satisfies the equation above. It can be interpreted as an estimate of the forward average volatility discussed in Result 2.2. If the Black-Scholes GBM assumption is correct, options over the same underlying asset, with the same maturity date, should yield identical implied volatilities. In fact if share price volatility σ is indeed constant, any option over a particular share should yield the same implied volatility. In practice this is not observed, and a “smile” relationship is seen between strike price K and the Black-Scholes implied volatility, and a similar relationship between time to maturity τ and implied volatility. Based on these phenomena, it appears that an option pricing model which incorporates changing volatility is required. Such a model is considered in the following chapter.

45

2.5. USE OF THE BLACK-SCHOLES MODEL 18 Months to Maturity

0.3

f(c)

0.0

0.0

0.1

0.1

0.2

0.2

f(c)

0.3

0.4

0.4

0.5

24 Months to Maturity

2

4

6

8

10

0

2

4

6

8

c

c



10

1.0 0.5

f(c)

0.4

0.0

0.0

0.2

f(c)

0.6

1.5

0.8

0

2

4

6

8

10

0

2

4

6

c

c


0.5 Months to Maturity

8

10

3 2

f(c)

1.5

0

0.0

0.5

1

1.0

f(c)

2.0

2.5

4

0

0

2

4

6 c

8

10

0

2

4

6

8

10

c

Figure 2.7: The distribution of call prices obtained using the Black-Scholes formula on a sample of share prices St+s , where s = 1 year, St = $5, µ = 0.1, σ = 0.1, K = $5, r = 0.06 and time to maturity τ − s ranges from 2 years to half a month. Also the theoretical density function for these prices.

46

CHAPTER 2. GBM AND THE BLACK-SCHOLES MODEL 5 Years to Maturity

f(c) 0.0

0.0

0.1

0.1

0.2

f(c)

0.2

0.3

0.3

0.4

2 Years to Maturity

2

4

6

8

10

0

6

8

15 Years to Maturity


10

0.25

0.30

f(c)

0.10

0.15

0.20

0.25 0.20 0.15

0.0

0.05

0.05

0.10

f(c)

4 c

0.0

4

6

8

10

12

2

4

6

8

10

c

c



12

14

0.0

0.0

0.05

0.05

0.10

0.10

f(c)

0.15

0.15

0.20

0.20

0.25

0.25

2

f(c)

2

c

0.30

0

2

4

6

8 c

10

12

14

2

4

6

8

10

12

14

c

Figure 2.8: The distribution of call prices obtained using the Black-Scholes formula on a sample of share prices St+s , where s = 1 year, St = $5, µ = 0.1, σ = 0.1, K = $5, r = 0.06 and time to maturity τ − s ranges from 2 to 75 years. Also the (solid) theoretical density function for these prices, and the (dotted) lognormal density function of St+s .

Chapter 3 The CEV Model

47

48

CHAPTER 3. THE CEV MODEL

3.1

Introduction

Empirical analysis of observed option prices, and observed phenomena like the volatility smile, led researchers to formulate option pricing models that included non-constant volatility. In response to observations of an inverse relationship between share price and share price volatility, documented by Fischer Black (1975), the Constant Elasticity of Variance (CEV) model was derived. Cox (1996) suggests that the CEV model is the simplest way to describe such an inverse relationship. In fact, the CEV model was derived after a direct request from Fischer Black to John Cox, for a share price evolution model that includes an inverse dependence of volatility and the share price, as described in Cox (1996). The standard CEV model assumes that share price St evolves according to the stochastic differential equation: β

dSt = µSt dt + δSt2 dBt

(t > 0, β < 2)

(3.1)

where µ and δ are constants, and with initial condition B0 = 0. Unlike GBM, a solution to Equation (3.1) can become negative unless otherwise constrained. This is clearly inappropriate for an asset price, so an absorbing barrier must be imposed at zero, such that if St = 0 then Su = 0 for all u > t. Note that a reflecting barrier at zero is not sufficient, since once a firm’s share price falls to zero, the firm will be bankrupt, and the share price cannot become positive again. The absorbing barrier, however, overcomes this problem and is consistent with bankruptcy of the firm. As mentioned above, the stochastic process for St assumed by Black and Scholes has β = 2 in Equation (3.1). Emanuel & MacBeth (1982) consider the above process when β > 2. Hence, the option price over an underlying asset whose price is a solution to the general SDE introduced in Definition 1.8 is known for any value of β. The case when β > 2 is mentioned briefly in Result 3.5 and estimated in Chapter 4, but this chapter focuses primarily on the case β < 2.

49

3.1. INTRODUCTION

The variance of the stock’s instantaneous rate of return, given data to time t is: Var

µ

¯ ¶ dSt ¯¯ St = δ 2 Stβ−2 dt St ¯

(3.2)

which, in the Black-Scholes case, is constant over time. In general, when β < 2, the variance of the instantaneous rate of return decreases as stock price increases, thus loosely satisfying empirical observations. The rate of return on a firm’s equity, in the absence of debt, Ra , can be written in the presence of debt, as a weighted average of the rates of return on its debt and equity, Rd and Re respectively: Ra =

B S Re + Rd (1 − α) V V

where S is the present value of equity, B the present value of debt, V = S +B the present value of the firm, and α reflects the tax scenario1 . Rearranging the relationship for Re , and taking the variance of both sides, with random variables Re and Ra , and all other quantities assumed constant, yields: ¯ ¶ ¶2 µ µ dSt ¯¯ B . (3.3) St = Var(Ra ) 1 + Var(Re ) = Var St ¯ S This is the desired relationship between share price and the variance of the

stock’s rate of return, since as St increases relative to Bt , the variance of its rate of return (the constant volatility parameter in the Black-Scholes case) will decrease, and vice versa, provided Var(Ra ) remains fixed. Moreover, as St → 0, under the CEV model, the share price volatility will tend to

infinity, as Equation (3.3) dictates. However, as St → ∞, in such a way that

Bt St

→ 0, the CEV model indicates that share price volatility should

become zero, whilst financial theory says that it should converge to a positive

constant Var(Ra ), where this quantity remains fixed. This fact is reflected in 1

In a Modigliani and Miller tax world α = Tc , the corporate tax rate, but in a Miller world, α = 0. For further details, see Copeland & Weston (1988) and the references therein.

50


the compound option pricing model derived by Geske (1979) but not in the CEV model, since under the CEV model, as share price goes to infinity, the volatility of the share price will go to zero. Therefore whilst the CEV model is theoretically motivated by the relationship between the firm’s return on equity and the value of equity outlined above, it does not fully describe this relationship. The CEV model acquired its name from the fact that the elasticity of variance of the rate of return on St is constant. This means that the ratio of any proportional change in St and the resulting proportional change in the variance of the rate of return on St will be constant. The proportional change in St over a period dt will be simply

dSt St

and the

proportional change in the variance of the rate of return on St over the same period dt, conditional on St , will be dVar(dSt /St ) . Var(dSt /St ) Taking the ratio of these two quantities, and noting the variance of the rate of return on St given by Equation (3.2), yields the desired relationship: Á Á dVar(dSt /St ) dSt dVar(dSt /St ) Var(dSt /St ) = Var(dSt /St ) St dSt St Á 2 β−2 δ St dt = (β − 2)δ 2 Stβ−3 dt St = β − 2. where again, all variances are conditional on St . Hence, a 10% increase in St will result in an increase of 10%(β − 2) in the

variance of the rate of return on St . Note that since β < 2 this will actually be a decrease in the volatility as required.

3.2

The CEV Share Price Solution

Unlike the Black-Scholes case, for the general CEV model it is not possible to derive a form for ST similar to the one we saw in Equation (2.5). According

3.2. THE CEV SHARE PRICE SOLUTION

51

to Cox & Ross (1976), the CEV option pricing formula can be derived using the same technique that Black and Scholes used to obtain their formula. This involves taking a transformation of the process Ct , chosen so that its governing PDE and boundary condition have a known solution. Both the Black-Scholes PDE and the appropriate PDE for the CEV option price can be transformed into the famous heat equation of physics, and solved using standard mathematical techniques. Cox and Ross used the CEV and other models as examples to pioneer a pricing method called “risk-neutral pricing” alluded to in Section 2.4. In order to price a future payment, PT , whose solution satisfies a PDE independent of investor risk preferences, it may be assumed that the underlying process has instantaneous mean r, and all future payments may be discounted at r. Hence the value of PT now will be Pt = e−rτ E∗ (PT |Ft ) where the expectation E ∗ is a risk-neutral expectation, taken under the assumption that the underlying process has instantaneous mean r, and Ft is

available information up to and including t. In the case of an option price,

the future payment we are interested in is PT = (ST − K)+ , where ST is

the underlying stochastic process. The implications of Cox and Ross’ result are that if we can find a density function that describes the evolution of PT

between t and T , or in this particular case, of St to ST , then we may be able to evaluate the (risk-neutral) expected value above. The Kolmogorov Equations can be used in this context. They are a set of two partial differential equations which describe the “transition” probabilities of a Markov diffusion process. The lognormal density function given in Equation (2.14) is an example of a transition probability density for a process following geometric Brownian motion. Cox & Miller (1965) define the Kolmogorov equations on page 215, which describe properties of a continuous time, continuous variable stochastic process X(t). These equations are defined in the following result.

52


Result 3.1 (The Kolmogorov Equations). Suppose the continuous time, continuous variable process X(t) is a solution of the SDE: p dX(t) = β(x, t)dt + α(x, t)dB(t)

with initial condition X(t0 ) = x0 . Then X(t) has instantaneous mean β(x, t)

and instantaneous variance α(x, t). In addition, X(t) has a transition probability density p(x0 , t0 ; x, t) which satisfies the Kolmogorov equations, and whose interpretation is given by the following equation: Z b p(x0 , t0 ; x, t)dx. P (a < X(t) < b|X(t0 ) = x0 ) = a

The Kolmogorov equations are the forward equation: 1 ∂2 ∂ ∂p {α(x, t)p} − {β(x, t)p} = 2 2 ∂x ∂x ∂t

(3.4)

in which the backward variables x0 and t0 may be considered constant and enter by way of the boundary conditions, and the backward equation: 1 ∂2p ∂p ∂p α(x0 , t0 ) 2 + β(x0 , t0 ) =− 2 ∂x0 ∂x0 ∂t0

(3.5)

in which the forward variables x and t enter only through boundary conditions. In the CEV case we are again interested in the function fST |St (s, τ ), which we may use to examine the properties of the future share price ST , given St . It appears that this probability density function might be the solution to the forward and backward Kolmogorov equations. St is a continuous time, continuous variable process, which satisfies the equation β

dSt = µSt dt + δSt2 dBt with initial condition B0 = 0. The process St has instantaneous mean and variance E(dSt |St ) = µSt and Var(dSt |St ) = δ 2 Stβ respectively, and we intend

to find its transition probability density

fST |St (s, τ ) = p(st , t; s, T )

53

3.2. THE CEV SHARE PRICE SOLUTION where St = st is given.

One point of concern is the fact that the process St has an absorbing barrier at zero, imposed in order to ensure that the share price cannot become negative. Cox and Miller consider this case on page 219, and state that an additional boundary condition must be imposed when solving each of the Kolmogorov equations. In particular, for an absorbing barrier at a, the additional boundary condition for the forward equation is: p(x0 , t0 ; x, t) = 0 where x < a < x0 , yielding for s < 0: fST |St (s, τ ) = 0. It follows from Cox and Miller’s condition, that if x ≤ a: Z x p(x0 , t0 ; u, t)du = 0 P (X(t) < x|X(t0 ) = x0 ) = −∞

since for x < a < x0 , the density function p is identically zero. Thus in order to determine the density function for ST = s given St = st we must solve the backward and forward Kolmogorov equations given by: 1 ∂2 2 β ∂ ∂f {δ st f } − {µst f } − =0 2 2 ∂s ∂s ∂T

(3.6)

1 2 β ∂2f ∂f ∂f δ st 2 + µst =0 + 2 ∂st ∂st ∂t

(3.7)

and

respectively, subject to the boundary conditions: fST |St (s, τ ) = 0

if s < 0

fST |St (s, 0|St = st ) = δ(s − st ), where δ(x) is the Dirac delta function. The first of these conditions follows from the absorbing barrier, and the second from the fact that f is a transitional density function.

54


3.2.1

Solution of the Forward Kolmogorov Equation

Feller (1951) solves the class of partial differential equations of the form: ut = (axu)xx − ((bx + c)u)x

(3.8)

where u = u(t, x), and a, b, c are constants, and a > 0. This equation is equivalent to the Fokker-Planck, or forward Kolmogorov equation seen in Equation (3.4), with α(x, t) = 2ax and β(x, t) = bx + c. In order to apply Feller’s results to find the CEV solution, a transformation must be made to the process {St } to obtain a partial differential equation of the required form,

and the instantaneous mean and variance of this new process obtained.

The transformation made by Cox (1996), and by Emanuel & MacBeth (1982), is Y = S 2−β , and is applicable for all β 6= 2. We can use Itô’s Lemma

to obtain the instantaneous mean and variance of the new process Yt . By

Itô’s Lemma, since β

dSt = µSt dt + δSt2 dBt and Yt = f (St ) = St2−β , dYt =

∂2f 1 ∂f dSt + δ 2 Stβ 2 dt ∂S 2 ∂S β

= (2 − β)St1−β (µSt dt + δSt2 dBt ) + 21 δ 2 Stβ (2 − β)(1 − β)St−β dt p ¡ ¢ = µ(2 − β)Yt + 12 δ 2 (2 − β)(1 − β) dt + δ(2 − β) Yt dBt .

Thus, the process Yt has instantaneous mean:

β(Yt ) = µ(2 − β)Yt + 12 δ 2 (2 − β)(1 − β) and variance: α(Yt ) = δ 2 (2 − β)2 Yt . The transition density f = fYT |Yt (y, τ ) will satisfy the forward Kolmogorov equation: 1 α(Yt ) 2

∂f ∂f ∂2f =− 2 + β(Yt ) ∂Yt ∂t ∂Yt

(3.9)

55

3.2. THE CEV SHARE PRICE SOLUTION which is of the form specified by Feller when β < 2, with a = 21 δ 2 (2 − β)2

b = µ(2 − β)

c = 12 δ 2 (2 − β)(1 − β)

and can be used in his result to obtain the transition density function of Yt . This function is given by Feller in his Lemma 9, as: fYT |Yt (y, τ ) =

b a(ebτ − 1)

exp

µ

−b(y + Yt ebτ ) a(ebτ − 1)

¶µ

ye−bτ Yt

¶ c−a 2a

I1− ac

Ã

1

2b(e−bτ yYt ) 2 a(1 − e−bτ ) (3.10)

where a, b and c are given above, and Iν (z) is the modified Bessel function of the first kind of order ν. In order to obtain the transition probability density for ST conditional on St , we must again make a change of variables: FST |St (s, τ ) = P (ST < s|St ) = P (ST2−β < s2−β |St2−β ) = P (YT < s2−β |Yt ). Differentiating with respect to s we obtain the density function: ∂ FS |S (s, τ ) ∂s T t ¡ ¢ = (2 − β)s1−β fYT |Yt s2−β , τ .

fST |St (s, τ ) =

Substituting into this final equation the constants a, b and c, and making the replacements y = s2−β and Yt = St2−β in the density function given by Feller, we obtain the density function for ST given St : 1 1 1 xz˜1−2β ) 2 2−β e−˜x−˜z I fST |St (s, τ ) = (2 − β)k˜ 2−β (˜

1

1 2−β

(2(˜ xz˜) 2 )

(3.11)

where k˜ =

δ 2 (2

2µ − β)(eµ(2−β)τ − 1)

˜ t2−β eµ(2−β)τ x˜ = kS

˜ 2−β z˜ = ks

(3.12)

and s > 0. This density function of ST conditional on St , is a function of St and the time elapsed, τ , between t and T , but also depends on CEV

!

56


parameters µ, δ and β. It describes the statistical properties at T , of a general process satisfying Equation (3.1), where β < 2, given its value at t. Abramowitz & Stegun (1968) give the power series expansion of the modified Bessel function seen in the density function above in their Equation (9.6.10) as: Iν (z) =

( 12 z)ν

∞ X n=0

( 41 z 2 )n . n!Γ(ν + n + 1)

(3.13)

Using this identity, the Bessel function in the density function above may be rewritten: I

1 2−β

1 2

(2(˜ xz˜) ) =

³

1 1 (2(˜ xz˜) 2 ) 2

= (˜ xz˜)

1 1 2 2−β

1 ∞ 1 X ´ 2−β xz˜) 2 )2 )n ( 41 (2(˜ 1 n!Γ(n + 1 + 2−β ) n=0

∞ X n=0

(˜ xz˜)n n!Γ(n + 1 +

1 ) 2−β

giving the alternative form for the transition density function: 1 1 fST |St (s, τ ) = (2 − β)k˜ 2−β (˜ xz˜1−β ) 2−β e−˜x−˜z

1 = (2 − β)k˜ 2−β e−˜x−˜z

∞ X n=0

(˜ xz˜)n n!Γ(n + 1 +

1 ) 2−β

1−β

1 ∞ X xñ+ 2−β zñ+ 2−β 1 ) n!Γ(n + 1 + 2−β n=0

(3.14)

Probability of Bankruptcy The density function given above is valid only for ST > 0. The absorbing barrier imposed at Su = 0 where t < u ≤ T prevents the share price from

becoming negative, and results in a positive probability of bankruptcy under the CEV model. The probability of bankruptcy given below is well known, but is often stated with the unnecessary assumption µ = r. Result 3.2. G(x, ν − 1) = 1 −

∞ X

g(y, n + ν)

n=0

where G(y, ν) and g(y, ν) are the survivor and probability density functions at y for a gamma random variable with shape parameter ν, and unit scale parameter.

57

3.2. THE CEV SHARE PRICE SOLUTION Proof. A proof of this result is given in Appendix B.2.

Theorem 3.1 (Probability of Bankruptcy for CEV Stocks). The prob1 ability of bankruptcy under the CEV model, with β < 2, will be G(˜ x, 2−β )

where G(y, ν) is the survivor function at y for a gamma random variable with shape parameter ν and unit scale parameter, and ˜ tβ−2 eµ(2−β)τ x˜ = kS

k˜ =

and

2µ δ 2 (2 − β)(eµ(2−β)τ − 1)

are as previously defined. ˜ 2−β , we find: Proof. First evaluate fST |St (s, τ )ds. Noting that z˜ = ks µ ¶1/(2−β) z˜ s= k˜ 1 ds = k˜−1/(2−β) z˜−1+1/(2−β) d˜ z. 2−β Making this transformation, and combining with the alternate form for the density function in Equation (3.14) yields: 1−β

fST |St (s, τ )ds = (2 − β)k˜ =e

−˜ x−˜ z

1 2−β

∞ X n=0

1 ∞ X xñ+ 2−β zñ+ 2−β −˜ x−˜ z e 1 ) n!Γ(n + 1 + 2−β n=0 1

Ã

! 1 − 2−β − 1−β ˜ 2−β k z˜ d˜ z 2−β

xñ+ 2−β zñ d˜ z 1 n!Γ(n + 1 + 2−β )

(3.15)

In addition since β < 2, z˜ is a 1-1, increasing function of ST and when ST = 0, z˜ = 0. Hence: P (ST = 0) = 1 −

Z

∞

s=0 Z ∞

fST |St (s, τ )ds 1

∞ X

xñ+ 2−β zñ d˜ z =1− e 1 n!Γ(n + 1 + ) z˜=0 2−β n=0 1 Z ∞ −˜z n ∞ −˜ x n+ 2−β X e x˜ e z˜ =1− d˜ z 1 n! Γ(n + 1 + ) 0 2−β n=0 =1−

∞ X n=0

−˜ x−˜ z

g(˜ x, n + 1 +

1 ) 2−β

58


since g(˜ z , ν) =

e−˜z z˜ν−1 Γ(ν)

is the gamma probability density function on z˜ ∈ [0, ∞).

Finally, using Result 3.2 we can determine the probability of bankruptcy: P (ST = 0) = 1 − =

∞ X

g(˜ x, n + 1 +

n=0 1 ) G(˜ x, 2−β

where G(y, ν) =

Z

∞ y

1 ) 2−β

(3.16)

ex xν−1 dx Γ(ν)

is the survivor function for the gamma random variable with shape parameter ν and unit scale parameter . Hence, the probability of bankruptcy between now and T > t, for a firm whose share price is a solution to the SDE (3.1), depends on the parameters µ, δ and β, the initial share price St , and on the length of the time period [t, T ] in question. Analysis of the relationship between β and the probability of bankruptcy indicates that for a fixed T , P (ST = 0) is insensitive to changes in share price at t, but sensitive to changes in the size of µ and δ. As we would expect, an increase in µ decreases the probability of bankruptcy for any β while an increase in volatility (via the parameter δ) increases the probability of bankruptcy. Figure 3.1 shows the relationship between β and P (ST = 0), β/2−1

with St = $5, τ = 1 year, µ = 0.10, σ = δSt

= 0.3 in the solid curve

and σ = 0.28 in the broken curve. Making small changes in δ result in very large movements in the curve, as seen in the graph. Here a 6.67% decrease in σ (from 0.3 to 0.28), via the parameter δ, results in a 29.64% decrease in P (ST = 0) when β = −2, and larger percentage decreases for larger values

of β, with a 97.10% decrease in P (ST = 0) when β = 1. Note that since k˜ = ∞ when β = 2, the probability of bankruptcy given

by Equation (3.16) generalises to the Black-Scholes case where P (ST = 0) =

59

0.015 0.010 0.0

0.005

P(bankruptcy)

0.020

0.025


-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

beta

Figure 3.1: The relationship between β and P (ST = 0), with St = $5, τ = 1 β/2−1 year, µ = 0.10, σ = δSt = 0.3 in the solid curve and σ = 0.28 in the broken curve.

0. This result is given by Markov’s Inequality: P (X ≥ x) ≤

E(X) . x

Letting X be a gamma random variable, with shape parameter scale parameter, then: 1 G(˜ x, 2−β ) = P (X ≥ x˜)

E(X) x˜ Á 1 2µSt2−β = 2 − β δ 2 (2 − β)(eµ(2−β)τ − 1) δ 2 (eµ(2−β)τ − 1) = St2−β =0

≤

1 ) = 0 as required. when β = 2, and so limβ→2− G(˜ x, 2−β

1 , 2−β

and unit

60

CHAPTER 3. THE CEV MODEL Since the probability of bankruptcy is positive for β < 2, the mixed

distribution of ST given St can be written as a linear combination of a step function, and a continuous distribution function:

FST |St (s, τ ) =

 ³ G x˜,

1 2−β

0

´

³ ³ ´´ R s 1 + 1 − G x˜, 2−β 0

fST |St (u,τ )du 1 1−G(˜ x, 2−β )

s≥0 s St , the volatility will be less than σ. Differences between the daily return series for the CEV process in Figure 3.3 and the same series for the GBM process in Figure 3.2 are easier to see than differences between the raw share price series. In particular the series are very different near the end of the graph, which, on close inspection, is one region of the share price series where a difference is apparent. The third graph in each respective figure does show clearly the difference between the two processes. In the case of the geometric Brownian motion, there does not seem to be any clear relationship between share price level and the standard deviation of the daily returns. The standard deviation should be constant, a fact which is not entirely supported by the graph. This is not overly alarming since we are dealing with a single realisation of a process, and examining a property that holds in continuous time using non-parametric smoothing technology. Comparison of this scatterplot to the third graph of the CEV process shows highly evident differences. Here we see a decreasing relationship between share price level, and the standard deviation of its daily returns. The relationship does not appear to be linear, but it is certainly decreasing, in confirmation of what we know to be true for a CEV process with β < 2.

69

4.5 3.5

4.0

Share Price

5.0

5.5


0.0

0.2

0.4

0.6

0.8

1.0

0.6

0.8

1.0

0.0 -0.02 -0.08

-0.06

-0.04

Log-returns

0.02

0.04

Time - years

0.0

0.2

0.4

0.014

Time - years

0.013

•

0.012

•

• •

0.011

• • •

•

•

•

• 0.010

Log-return Standard Deviation

•

• • • • • •

• • • • • •

3.5

•

•

• ••• •

•• • • ••

• • •

• •

•

•

•

•

• • • • • • • • • • • • • •• • • • • • • • • ••• • ••• •• • • ••• • • • •• • •• •• •• •• • • • • •• • • •• • • •• • • • • • • • • • • • •• • • • • • • • • • • • •• • • ••• • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • •• • • • • 4.0

• •

•

•

•

•

4.5

•

• •

• • •

••

• • •

•

• • ••

• • • •

• •

•

• •

•• • • 5.0

5.5

Share Price

Figure 3.2: Firstly, a typical realisation of GBM, with St = $5, τ = 1 year, µ = 0.10, σ = 0.3 and n = 250 subintervals; secondly, the daily returns for this series with estimated mean, and mean ± 2 standard deviation series; thirdly, a plot of the share price level against the estimated standard deviation of the daily returns.

70

4.5 4.0 2.5

3.0

3.5

Share Price

5.0

5.5


0.0

0.2

0.4

0.6

0.8

1.0

0.6

0.8

1.0

0.0 -0.08

-0.04

Log-returns

0.02

0.04

0.06

Time - years

0.0

0.2

0.4

0.020

Time - years

•

0.018

• • •

0.016

•

•

0.014

• •

•

••

• • • • •• •••••• • • ••

3.0

3.5

•

• • •

•

• •

•

•• • • •

•

•

•

•

•

• • •• • • • • • • • • • • •• ••• ••• • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ••• • •• • • • • • • • • •• • • • • • • • • •• • • • ••• • • • •• • • • • • • • • • • • • • • •• • • •• • •• • • • • •• • • •• •• • • • • • • • • •• •• •• • • • • • •

• •

0.012

• •

•

•

0.010


• •

• •

• •• • •

•

4.0

4.5

5.0

•

5.5

Share Price

Figure 3.3: Firstly, a typical realisation of share price, with St = $5, τ = 1 year, µ = 0.10, σ = 0.3 and n = 250 subintervals and CEV parameter β = −1; secondly, the daily returns for this series with estimated mean, and mean ± 2 standard deviation series; thirdly, a plot of the share price level against the estimated standard deviation of the daily returns.

71


3.2.4

Graphical Examination of ST

The theoretical distribution for CEV share prices at T , given an initial starting value at t, is given by Equation (3.11), with an alternative form in Equation (3.14). Both these (improper) density functions present problems in terms of computation. In the first case, the Bessel function Iν (z) is not internally computed in SPLUS, and in the second, there is an infinite summation3 . An alternative is due to Schroder (1989), who discusses computation of the CEV option price. Whilst not the focus of his paper, Schroder represents the transition survivor function of ST given St in terms of a non-central Chisquared random variable. Johnson, Kotz & Balakrishnan (1995) introduce the non-central Chi-squared distribution on page 433 as in the following result. Result 3.3 (The Non-central χ2 Distribution). If U1 , U2 , . . . , Uν are independent unit normal random variables, and δ1 , δ2 , . . . , δν are constants then: ν X

(Uj + δj )2

(3.21)

j=1

has a non-central χ2 distribution, with ν degrees of freedom, and non-centrality parameter: λ=

ν X

δj2 .

j=1

The density function of such a variable is given as: ∞

1

exp(− 12 (y + λ)) X (y) 2 ν+j−1 λj . p(y; ν, λ) = 1 Γ( 12 ν + j)22j j! 22ν j=0 3

(3.22)

It does appear that the terms in the summation should converge very quickly to zero, due to the presence of both n! and Γ(ν + n + 1) in the denominator. Together these terms are like n!2 and will quickly dominate the term in the numerator, suggesting that the summation should converge after few terms.

72

CHAPTER 3. THE CEV MODEL Given the definition of Johnson et al., it appears ν ∈ Z+ is required, how-

ever the moment generating function for a non-central Chi-squared variable is defined for all ν > 0. This moment generating function: µ ¶ λt − 12 ν MX (t) = (1 − 2t) exp 1 − 2t

uniquely defines the non-central Chi-squared variable, and so non-integer degrees of freedom are permitted. This is noted by Johnson et al. Note also that λ must be positive. Using the series expansion of the modified Bessel function shown in Equation (3.13) the density function p(y; ν, λ) can be manipulated to give: p © ª 1 p(y; ν, λ) = 21 (y/λ) 4 (ν−2) exp − 21 (λ + y) I 1 (ν−2) ( λy) 2

(3.23)

which is the same as that given by Johnson et al. in their Equation (29.4). Beginning with Equation (3.23), the equivalence of the two forms of the density function is demonstrated below. p 1 1 p(y; ν, λ) = 12 (y/λ) 4 (ν−2) e− 2 (λ+y) I 1 (ν−2) ( λy) 2

=

1 1 1 (y/λ) 4 (ν−2) e− 2 (λ+y) ( 21 2

∞ X p 1 (ν−2) λy) 2 j=0

1

1

1

= ( 12 ) 2 ν y 2 (ν−2) e− 2 (λ+y)

∞ X j=0

√ ( 41 ( λy)2 )j j!Γ( 12 (ν − 2) + j + 1)

λj y j 22j j!Γ( 21 ν + j)

The last equation is of course the same as Equation (3.22). Theorem 3.3 (The CEV Share Price Distribution Function). The transition distribution function for a CEV share price ST given St , FST |St (s, τ ) = P (ST < s|St ) is given by the equation:   s 0

73


˜ 2−β , k˜ and x˜ are defined in Equation (3.12), and Q(x; ν, λ) where y˜ = ks is the survivor function at x for a non-central Chi-squared variable with ν degrees of freedom and non-centrality parameter λ. Proof. Note firstly that ST ≥ 0, so that for s = 0, P (ST < s|St ) = 0.

1 Secondly, P (ST = 0|St ) is given by Theorem 3.1, and is G(˜ x, 2−β ) as required.

I will defer the proof for the final case, where s > 0, until Section 3.5.2, since the proof is made considerably easier by results which follow. Thanks to Schroder’s result, and the fact that the non-central Chi-squared cumulative probability function is an internal function in SPLUS, we can again compare the theoretical distribution for share price at a future time T , to an empirical distribution of simulated values. Using the Euler method, I have simulated 5000 share prices τ = 1 year into the future, with β = −1, and

additional parameters St = $5, µ = 0.1, σ = 0.3 and n = 250 subintervals. Of these simulated series, 38 had final values ST = 0, giving an estimate of P (ST = 0) of

38 5000

= 0.0076. This is less than the true probability of 0.00896,

corresponding to an expected number of bankruptcies of 44.8. A histogram of the remaining 4962 values is shown in Figure 3.4 along with the density function for a GBM process with the same parameters (plotted as a broken line), and the true density function (plotted as a solid line) conditional on ST > 0, calculated using an approximation to Equation (3.14): 1−β

1 fST |St (s, τ ) = (2 − β)k˜ 2−β e−˜x−˜z

1 100 X xñ+ 2−β zñ+ 2−β 1 n!Γ(n + 1 + 2−β ) n=0

where the upper limit of the summation has been replaced by 100. In fact, remarkable accuracy can be achieved using only the first 11 terms of the summation, with a maximum difference of only 0.00005555 between the two approximations, where both are calculated at 500 values of ST in the range [0, 10.4]. The theoretical curve is a true density function, which is formed by scaling the function fST |St (s, τ ) above as seen in Equation (3.17): fST |St (s, τ |ST > 0) =

fST |St (s, τ ) . 1 1 − G(˜ x, 2−β )

74

0.20 0.15 0.10 0.0

0.05

Relative Frequency

0.25

0.30


0

2

4

6

8

10

Share Price

Figure 3.4: 5000 realisations of ST , a future CEV price with β = −1, with St = $5, τ = 1 year, µ = 0.1 and σ = 0.3, the (solid) theoretical density for these prices, and the lognormal density function with the same parameters.

In Figure 3.4, we see that again there is good agreement between the simulated share prices and theory. Unlike the GBM equivalent seen in Figure 2.3, in this case there are two sources of error, due to sampling error and use of the Euler approximation for the solution to the SDE (3.1). The close fit of the density function to the histogram suggests that aggregate errors in the Euler method of simulating share price under the CEV model have been small. We can also compare the empirical cumulative distribution function to the theoretical one, which, thanks to Schroder, can be computed directly. These functions are shown in Figure 3.5 where the solid line represents the theoretical curve and the broken line the empirical results from the sample shown in Figure 3.4,with St = $5, τ = 1, µ = 0.1, σ = 0.3 and β = −1. Even using fairly coarse subdivisions for the estimation of the empirical

75

0.6 0.4 0.0

0.2

P(Share Price < s)

0.8

1.0


0

2

4

6

8

10

s

Figure 3.5: The empirical cumulative distribution function of the 5000 realisations of ST shown in Figure 3.4, a future CEV price with β = −1, with parameters St = $5, τ = 1, µ = 0.1 and σ = 0.3, and the theoretical distribution function for these prices.

curve, the empirical and theoretical distributions are virtually indistinguishable. Estimating the sample mean and standard deviation for all 5000 realisations yields estimates S¯T = 5.5096, and s(ST ) = 1.5516, which are both below the theoretical values of 5.5259 and 1.5782, despite the fact that fewer bankruptcies were observed than expected. These differences are not significant at the 5% level4 . In Theorem 3.2 I derived a form for the variance of a future CEV share price ST , given St and β. The variance is given in Equation (3.18) and is used to calculate the theoretical standard deviation for the sample above. 4

A two-sided test of H0 : µ = 5e0.1 yields a test statistic of -0.7293, and a p-value of 0.4658. Similarly a two-sided test of H0 : σ 2 = 1.5782 yields a test statistic of 4831.91 and a p-value of 0.0923.

76


Using SPLUS, it is possible to calculate the standard deviation of ST given St by restricting the infinite summation to a finite number of terms. This computation fails when β is close to two, since the quantile at which the gamma density function must be evaluated grows very large. Increasing the shape parameter, and the upper limit of the summation does not solve the problem since both Γ(n + 1 +

i ) 2−β

terms, where i = 1, 2, become too large,

even though their ratio is likely to remain moderate in comparison. Figure 3.6 shows the standard deviation function for −2 ≤ β ≤ 2, and the additional

parameters St = $5, τ = 1 year, µ = 0.1 and σ = 0.3. Due to computation

problems the standard deviation for 1.5 < β < 2 has been estimated by linear interpolation using the standard deviation when β = 1.5 found using

1.66 1.64 1.62 1.60 1.58

Standard Deviation of Share Price

1.68

1.70

the approximation, and that for β = 2, given by Equation (2.7).

-2

-1

0

1

2

beta

Figure 3.6: The standard deviation of a future CEV share price ST , with St = $5, τ = 1 year, µ = 0.1 and σ = 0.3, for −2 ≤ β ≤ 2. The relationship shown in the graph is not monotonic, with the variance decreasing as β decreases from 2 to approximately -0.50, after which the

3.3. PROPERTIES OF CT - THE EXERCISE PAYOFF

77

variance begins to increase.

3.3

Properties of CT - the Exercise Payoff

Many results proved for the GBM case apply to the CEV exercise payoff also. In particular the form for the probability density function of C T given St , derived in Theorem 2.3 holds, except of course the functions FST |St (s, τ ) and fST |St (s, τ ) are different. In addition we saw that E(CT |St ) is very sim-

ilar to the price of the option at t, but in future value terms, and with µ

present in the solution instead of r. In the case of CEV exercise payoffs I will restrict myself to noting the form of the cumulative distribution function of the payoff. A graphical representation of both the theoretical and simulated exercise payoffs follows from Figure 3.5 by setting FST |St (s, τ ) = 0 for s < K, and repositioning the curve, moving the origin to s = K. The shape of both curves will be identical for s > K. This result is stated and proved in the following Theorem. Theorem 3.4 (The Distribution of CT given St ). At exercise, the call option will deliver CT =

(

0 ST − K

ST ≤ K ST > K

the cumulative distribution function of which is, given St :   c 0 where x˜ and k˜ are defined in Equation (3.12).

Proof. For K > 0: P (CT = 0|St ) = P (ST < K|St ) 2 ˜ 2−β ) = Q(2˜ x; 2−β , 2kK

(3.24)

(3.25)

78


from Theorem 3.3. Likewise, for c > 0 P (CT < c|St ) = P (ST − K < c|St ) = P (ST < K + c|St ) ³ ´ 2 ˜ = Q 2˜ x; 2−β , 2k(K + c)2−β again using the distribution function of ST given in Theorem 3.3.

3.4

The CEV Option Pricing Formula

The price of a European option over a stock whose share price evolves according to Equation (3.1), was derived by John Cox and circulated as an unpublished note5 , which was summarised in Cox (1996). In addition, two special cases of the CEV model, the Absolute, and Square Root CEV models, were described in Cox & Ross (1976). The derivation of these models used the risk-neutral pricing method discussed in Section 2.4, as an alternative to the classical method used by Black and Scholes. The risk-neutral method is applicable because again the option price may be described using an equation which is independent of investor risk preferences. The assumptions of the CEV model are: • the instantaneous interest rate is known, and constant; • share price is a solution to the SDE (3.1), which implies: – the parameters δ and β are constant, and known; – no dividends are paid on the share over the life of the option; • there are no transaction costs, differential taxes, or short-selling restrictions, and it is possible to trade any fraction of the stock or option.

5

John Cox (1975), ’Notes on option pricing I: constant elasticity of variance diffusions’, unpublished note, Stanford University, Graduate School of Business.

3.4. THE CEV OPTION PRICING FORMULA

79

The only difference between these assumptions and those of Black and Scholes are those regarding the way share price evolves over time. All other assumptions are the same. Theorem 3.5 (The CEV Option Price PDE). If St is a solution to the SDE (3.1), then the price of a European call option Ct , with time τ to maturity, must satisfy the partial differential equation (PDE): 1 2 β∂ δ S 2

2

C ∂C ∂C + rS + − rCt = 0 ∂S 2 ∂S ∂t

(3.26)

subject to the boundary condition CT = (ST − K)+ where r is the continuously compounding risk-free rate. Proof. The CEV option price PDE follows from Theorem 2.5, with β

σ(St , t) = δSt2

−1

as does the boundary condition. As in the Black-Scholes case, the PDE (3.26) which the option value must satisfy, is independent of µ and is hence independent of investor preferences. Cox and Ross thus infer that the option may be valued assuming any investor preferences, and in particular, assuming risk-neutrality. Why this should be so can be seen by applying the Kolmogorov Equations introduced in Result 3.1 to their solution. Theorem 3.6 (The Risk-Neutral CEV Solution). The correct solution Ct , of the PDE (3.26) subject to the boundary condition CT = (ST − K)+ is

given by the expectation equation:

Ct = C(St , τ ) = e−rτ E∗ {(ST − K)+ |St } where E∗ is an expectation using a risk-neutral transition density function for share price with the replacement µ = r, and r is the continuously compounding risk-free rate.

80


Proof. Considering C in the form C(St , τ ) = e−rτ P (St , τ ) will yield the partial derivatives 2 ∂2C −rτ ∂ P = e ∂S 2 ∂S 2

∂C ∂P = e−rτ ∂S ∂S

∂C ∂P = −re−rτ P + e−rτ ∂τ ∂τ

which can be substituted in Equation (3.26) to give the similar PDE for the function P : 1 2 β∂ δ St 2

2

P ∂P ∂P + rSt − =0 2 ∂S ∂S ∂τ

(3.27)

and boundary condition CT = PT = (ST − K)+ . From the relationship

between P and C, the form of P (St , t) is known, and its partial derivatives can be evaluated: ∗

+

P (St , t) = E {(ST − K) |St } =

Z

∞ s=K

(s − K)fS∗T |St (s, τ )ds

and so: Z ∞ ∂P ∂f ∗ = (s − K) ds ∂S ∂S K Z ∞ ∂2P ∂2f ∗ = ds (s − K) ∂S 2 ∂S 2 K Z ∞ ∂P ∂f ∗ = ds. (s − K) ∂τ ∂τ K Using the above partial derivatives, the left hand side of Equation (3.27) can be evaluated to give: ½ ¾ Z ∞ ∂f ∗ ∂f ∗ 1 2 β ∂2f ∗ (s − K) δ St + rSt − ds 2 ∂S 2 ∂S ∂τ K

(3.28)

in which a risk-neutral transition density function, f ∗ , is being differentiated with respect to the backward variables St = S and t. The backward Kolmogorov Equation (3.7) states that the transition density for a general CEV process will satisfy the PDE: 1 2 β ∂2f ∂f ∂f δ St + = 0. 2 + µSt 2 ∂St ∂t ∂St

81


Thus, the transition density for a process with µ = r will satisfy the backward Kolmogorov Equation with µ replaced by r, and for such a process C(St , t) = e−rτ E∗ {(ST − K)+ |St }

(3.29)

is a solution of (3.26). We know the transition density function for the general CEV process, in which we can replace µ by r, and form the risk-neutral density function for ST∗ given St , where ST∗ is a random variable that is a solution to the SDE: ∗β

dSt∗ = rSt∗ dt + δSt 2 dBt

(3.30)

subject to the initial condition St∗ = St . I will denote the density function for this variable, conditional on St : fS∗T |St (s, τ ).

Note also that Equation (3.29) can be rewritten:

C(St , t) = e−rτ E∗ {(ST − K)+ |St } Z ∞ −rτ =e (s − K)fS∗T |St (s, τ )ds ZK∞ Z −rτ −rτ ∗ =e sfST |St (s, τ )ds − Ke K

∞

K

fS∗T |St (s, τ )ds

= e−rτ E∗ (ST |St , ST > K)P ∗ (ST > K|St ) − Ke−rτ P ∗ (ST > K|St ) (3.31)

where P ∗ (.) is a risk-neutral probability whose definition follows from the final equations above, and since, given ST > K, the correct risk-neutral density function for ST given St is fS∗T |St ,ST >K (s, τ )

3.4.1

=

fS∗T |St (s, τ )

P ∗ (ST > K|St )

.

The CEV Solution

The solution to the CEV partial differential equation has been outlined by various people, with conflicting results. Most of the disagreement can be

82


attributed to typographical errors, but I believe one error in particular is genuine. In order to shed more light on the different versions of the CEV solution published, I will attempt to employ consistent notation, and identify and correct inconsistencies. Cox (1996) presents the CEV solution option pricing formula, and the risk-neutral density function from which it was derived. Comparison of this solution with that in Cox & Ross (1976) for the special case β = 1 (the Square Root CEV model) indicates that the density function given in Cox (1996) is incomplete. Ignoring dividends, Cox’s risk-neutral density function should read, for s > 0: 1

1

1

fS∗T |St (s, τ ) = (2 − β)k 2−β (xz 1−2β ) 2 2−β e−x−z I

1

1 2−β

(2(xz) 2 )

(β < 2) (3.32)

where: k=

δ 2 (2

2r − β)(er(2−β)τ − 1)

x = kSt2−β er(2−β)τ

z = ks2−β

(3.33)

with s = ST , and the missing term e−x−z is inserted. As we saw earlier, using the power series expansion of the modified Bessel function yields an alternative form for the risk-neutral density function: 1−β

fS∗T |St (s, τ )

= (2 − β)k

1 2−β

e

−x−z

1 ∞ X xn+ 2−β z n+ 2−β . 1 ) n!Γ(n + 1 + 2−β n=0

The risk-neutral density function shown in Equation (3.32) can be used to obtain a formula for the option price given by Cox. Theorem 3.7 (The CEV Option Pricing Formula). The solution to Equation (3.26) subject to the boundary condition CT = (ST − K)+ is given by

the formula:

Ct = St

∞ e−x xn G X n=0

³ kK 2−β , n + 1 + Γ(n + 1)

1 2−β

´

¡ 2−β ¢ 1 ∞ −x n+ 2−β X e x G kK , n + 1 − Ke−rτ 1 ) Γ(n + 1 + 2−β n=0

83


where g(x, ν) and G(y, ν) are the gamma probability density and survivor functions respectively, with shape parameter ν and unit scale parameter. Proof. From Theorem 3.5 and the results of Cox & Ross (1976) we can use the risk-neutrality argument to give Cox’s option pricing formula. Thus, the solution can be obtained by evaluating: Z ∞ −rτ C(St , t) = e (s − K)fS∗T |St (s, τ )ds. K

Note that since z = ks

2−β

, when s = ST = K then z = kK 2−β . Again

drawing parallels to previous results, and in particular Equation (3.15), we can make the change of variable to z: fS∗T |St (s, τ )ds

=e

−x−z

∞ X n=0

1

xn+ 2−β z n dz. 1 n!Γ(n + 1 + 2−β )

Noting x = kSt2−β er(2−β)τ , the solution takes the form: 1 ¶ µ³ ´ 1 Z ∞ ∞ X xn+ 2−β z n z 2−β −x−z −rτ −K e dz C(St , t) = e 1 k ) n!Γ(n + 1 + 2−β z=kK 2−β n=0 1 1 1 Z ∞ ∞ X xn (x/k) 2−β z n+ 2−β − Kxn+ 2−β z n −x−z −rτ e dz =e 1 ) n!Γ(n + 1 + 2−β kK 2−β n=0 1 1 Z ∞ ∞ X xn St erτ z n+ 2−β − Kxn+ 2−β z n −x−z −rτ dz e =e 1 ) n!Γ(n + 1 + 2−β kK 2−β n=0 1 Z ∞ X e−z z n+ 2−β e−x xn ∞ = St dz 1 n! kK 2−β Γ(n + 1 + 2−β ) n=0 1 Z ∞ ∞ X e−x xn+ 2−β e−z z n −rτ − Ke dz (3.34) 1 Γ(n + 1 + 2−β ) kK 2−β n! n=0

which is the required solution, and so consistent with the conditional density function for ST given St in Equation (3.32) above. It is not a trivial matter to show directly that this solution satisfies the partial differential equation above, nor that the density function satisfies the forward equation. Despite this, the CEV option prices exhibit limiting qualities that

instill confidence in the solution, as does the earlier comparison of CEV share prices to their theoretical distribution.

84


3.4.2

Reconciling Various Forms of the CEV Solution

Cox’s option pricing formula for the CEV model with β < 2, as given in Equation (3.34), is consistently reported in papers by Beckers (1980), Emanuel & MacBeth (1982), MacBeth & Merville (1980) and Schroder (1989). Furthermore the solution that appears in Jarrow & Rudd (1983) on page 154 is also consistent with that of Cox, however, this solution cannot be obtained by integration of the risk-neutral density function also given in this text. The risk-neutral density function given by Jarrow and Rudd is: fS∗T |St (s, τ )

= −φe

−rτ

∞ ³ ´ X g λSt−φ , n + 1 − φ1 g(λ(se−rτ )−φ , n + 1) (3.35) n=0

where: 2r

λ=

δ 2 φ(eφrτ

− 1)

φ = β − 2.

To identify the function that should appear in Jarrow and Rudd’s notes, the risk-neutral density given in Equation (3.32) can be written in Jarrow and Rudd’s notation. Comparison of the two density functions, and noting the definitions of k, x, and z, yields immediately the relationships: x = λSt−φ

k = λeφrτ

z = λ(se−rτ )−φ .

These can be substituted into Cox’s risk-neutral density function to give: 1−β

fS∗T |St (s, τ )

= (2 − β)k = −φ(λe = −φλe

1 2−β

e

−x−z

1 ∞ X xn+ 2−β z n+ 2−β 1 n!Γ(n + 1 + 2−β ) n=0

1 φrτ − φ −x−z

)

e

1 1 ∞ X (λSt−φ )n− φ (λ(se−rτ )−φ )n+1+ φ

n=0

1 φrτ −1− φ

s

1

= −φλeφrτ s−1− φ

n!Γ(n + 1 − φ1 )

1 −φ −rτ −φ ∞ X (λSt−φ )n− φ eλSt (λ(se−rτ )−φ )n eλ(se )

n=0 ∞ X n=0

Γ(n + 1 − φ1 )

Γ(n + 1)

g(λSt−φ , n + 1 − φ1 )g(λ(se−rτ )−φ , n + 1)

85

3.5. COMPUTING THE OPTION PRICE

This final equation is clearly different to the density given by Jarrow and Rudd, who have a constant −φeφrτ multiplying the sum. Since the riskneutral density given in Equation (3.32) yields the option pricing formula

given by Cox, computing e−rτ E∗ ((ST −K)+ |St ) using the risk neutral density

function given by Jarrow and Rudd cannot lead to the required formula.

3.5

Computing the Option Price

The option pricing formula given by Cox appears formidable, especially compared to the industry standard Black-Scholes formula. Empirical papers that consider the analysis of simulated CEV prices include Beckers (1980) and MacBeth & Merville (1980). These authors produce prices by limiting the infinite summations to a finite number of terms: Ct ≈ St

n2 X

n=n1

³ g(x, n + 1)G kK 2−β , n + 1 + − Ke−rτ

n2 ³ X g x, n + 1 +

1 2−β

1 2−β

n=n1

´

´

(3.36)

G(kK 2−β , n + 1)

where x and k are defined in Equation (3.33), and where n1 and n2 are chosen to ensure convergence of the solution6 . Alternatively, these authors and others, including Rubinstein (1985), focus their attention on special cases of the CEV class for which closed form solutions or efficient numerical approximations existed. In particular, there is a closed form solution for Absolute CEV prices, where β = 0, and a numerical approximation for the Square Root CEV prices, where β = 1.

3.5.1

The Absolute CEV Model

In the Absolute CEV model, share price evolves according to the SDE: dSt = µSt dt + δdBt 6

Beckers uses n1 = 1 and n2 = 995.

(3.37)

86


which is an Ornstein-Uhlenbeck process with an absorbing barrier at zero. Cox & Ross (1976) give a form for the risk-neutral transitional density function with β = 0: fS∗T |St (s, τ |β

1 = 0) = √ 2πWt

where Wt =

δ 2 2rτ (e 2r

¶ µ ¶¶ µ µ (s + St erτ )2 (s − St erτ )2 − exp − exp − 2Wt 2Wt (3.38)

− 1). Using the formula in Equation (3.29) and the

density function above yields the closed form solution for the option price when β = 0:

Ctabs = (St − Ke−rτ )Φ(dt ) + (St + Ke−rτ )Φ(yt ) + vt (φ(dt ) − φ(yt )) (3.39) where φ(x) is the unit normal probability density function at x, Φ(x) is the unit normal cumulative distribution function, and: vt = δ

µ

1 − e−2rτ 2r

¶ 21

dt =

St − Ke−rτ vt

yt =

−St − Ke−rτ . vt

It is relatively simple to obtain the solution from the risk-neutral density function in the usual way, and in this case it is also easy to show by evaluating partial derivatives, that the solution (3.39) is consistent with the PDE (3.26) with β = 0. This solution is as easy to compute as the Black-Scholes equation, and would be an ideal alternative, if the specification of the model were empirically appropriate.

3.5.2

Computing the General Model

Schroder (1989) derives a result that allows easy computation of option prices for any CEV process. His result would have allowed Beckers, MacBeth and Merville, Rubinstein and others to easily examine CEV prices other than those that prior to Schroder’s result were easy to compute. This work has been alluded to in Section 3.2.4 where it was used to obtain an expression for the distribution function of future share prices. The key to that result,

87


and this one, is to note that the integral of interest, in this case Equation (3.29), can be written in terms of the non-central Chi-squared cumulative distribution function. Theorem 3.8. The CEV option price may be written Z ∞ Z ∞ −rτ 2 2 p(2z; 2 + 2−β , 2x)dz − Ke 2 p(2x; 2 + Ct = St y

where y = kK

y

2−β

2 , 2z)dz 2−β

, and p(y; ν, λ) is the probability density function at y of

a Chi-squared random variable with ν degrees of freedom and non-centrality parameter λ, and x and k are defined in Equation (3.33). Proof. Recalling Equation (3.32) and noting from the proof to Theorem 3.1 ds = k −1/(2−β)

1 z −1+1/(2−β) dz 2−β

we obtain a further form for fS∗T |St (s, τ )ds:

fS∗T |St (s, τ )ds = x1/(4−2β) z −1/(4−2β) e−x−z I

1

1 2−β

(2(xz) 2 )dz

which we can use to examine the option price: Ct = e−rτ E∗ ((ST − K)+ |St ) Z ∞ −rτ (s − K)fS∗T |St (s, τ )ds =e ¶ µ³ ´ 1 ZK∞ 1 z 2−β −rτ − K x1/(4−2β) z −1/(4−2β) e−x−z I 1 (2(xz) 2 )dz =e 2−β k 2−β Z ∞z=kK √ e−x−z (z/x)1/(4−2β) I1/(2−β) (2 xz)dz = St y Z ∞ √ −rτ − Ke (3.40) e−x−z (x/z)1/(4−2β) I1/(2−β) (2 xz)dz y

since:

³ ´1/(2−β) e−rτ (x/k)1/(2−β) = e−rτ St2−β er(2−β)τ = St .

Schroder recognises that the integrands are both non-central Chi-squared density functions, of the form given by Johnson et al. (1995) seen in Result 3.3. Recall the form for p(y; ν, λ) given in Equation (3.23): p ª © 1 p(y; ν, λ) = 21 (y/λ) 4 (ν−2) exp − 21 (λ + y) I 1 (ν−2) ( λy). 2

88


The first integrand in Equation (3.40) is: √ e−x−z (z/x)1/(4−2β) I1/(2−β) (2 xz). Comparison of this with p(y; ν, λ) above suggests that the integrand is almost a non-central Chi-squared density function with y = 2z

ν =2+

2 2−β

λ = 2x

except for the factor 21 , which is absent. Hence: √ e−x−z (z/x)1/(4−2β) I1/(2−β) (2 xz) = 2 p(2z; 2 +

2 , 2x). 2−β

(3.41)

The second integrand in the expression for Ct above is: √ e−x−z (x/z)1/(4−2β) I1/(2−β) (2 xz). Again comparison with p(y; ν, λ) indicates that the integrand is almost a non-central Chi-squared density function with y = 2x

ν =2+

2 2−β

λ = 2z

except for the scale factor 21 . Hence: √ e−x−z (x/z)1/(4−2β) I1/(2−β) (2 xz) = 2 p(2z; 2 +

2 , 2x). 2−β

(3.42)

Combining Equations (3.41) and (3.42) yields the desired result: Z ∞ Z ∞ −rτ 2 2 Ct = St , 2z)dz 2 p(2z; 2 + 2−β , 2x)dz − Ke 2 p(2x; 2 + 2−β y

y

(3.43)

Note that as written, neither of the integrals in Equation (3.43) are noncentral Chi-squared survivor functions at y. The first integral has integration with respect to z rather than the required 2z, whilst the second has integration with respect to half the non-centrality parameter 2z. Before proving Schroder’s result, it is necessary to consider the following result.

89

3.5. COMPUTING THE OPTION PRICE Result 3.4. Z

∞ y

2 p(2x; 2ν + 2, 2z)dz = 1 −

Z

∞

2 p(2z; 2ν, 2y)dz

x

where p(y; ν, λ) is the probability density function at y of a non-central Chisquared random variable with ν degrees of freedom and non-centrality parameter λ. Proof. A proof of this result can be found in Appendix B.3. Theorem 3.9 (Schroder’s Option Pricing Formula). The form of the CEV option price given by Schroder (1989) is: Ct = St Q(2y; 2 +

2 , 2x) 2−β

2 − Ke−rτ (1 − Q(2x; 2−β , 2y)).

where y = kK 2−β , x and k are given in Equation (3.33), and Q(y; ν, λ) is the survivor function at y for a non-central Chi-squared random variable with ν degrees of freedom and non-centrality parameter λ. Proof. The proof of this result follows directly from Theorem 3.8 and Result 3.4. Note the form of Ct given by Equation (3.43): Z ∞ Z ∞ −rτ 2 Ct = St 2 p(2z; 2 + 2−β , 2x)dz − Ke 2 p(2x; 2 + y

y

2 , 2z)dz. 2−β

Making the substitution w = 2z in the first integral yields: Z ∞ Z ∞ 2 2 2 p(2z; 2 + 2−β , 2x)dz = p(w; 2 + 2−β , 2x)dw y

2y

= Q(2y; 2 +

2 , 2x). 2−β

Similarly, applying Result 3.4 to the second integral and again making the substitution w = 2z yields: Z ∞ Z ∞ 2 2 2 p(2x; 2 + 2−β , 2z)dz = 1 − 2 p(2z; 2−β , 2y)dz y x Z ∞ 2 p(w; 2−β , 2y)dw =1− 2x

2 = 1 − Q(2x; 2−β , 2y).

90


Hence the option pricing formula is: Ct = St Q(2y; 2 +

2 , 2x) 2−β

2 − Ke−rτ (1 − Q(2x; 2−β , 2y))

(3.44)

as required. Recall that Theorem 3.3 gave the form of the cumulative distribution function for the CEV share price ST given St . The proof for the form of this function in the case s > 0 utilises results now available. Proof of Theorem 3.3 Proof. We are required to prove the proposition that for s > 0, the cumulative distribution function of ST given St is given by the equation: 2 FST |St (s, τ ) = Q(2˜ x; 2−β , 2˜ y)

where k˜ =

δ 2 (2

2µ − β)(eµ(2−β)τ − 1)

˜ t2−β eµ(2−β)τ x˜ = kS

˜ 2−β y˜ = ks

and Q(x; ν, λ) is the survivor function at x of a non-central Chi-squared variable with ν degrees of freedom and non-centrality parameter λ. Note that from Equation (3.31), the option price at t may be written: Ct = e−rτ E∗ (ST |St , ST > K)P ∗ (ST > K|St ) − Ke−rτ P ∗ (ST > K|St ) whilst Schroder gives the option price as: Ct = St Q(2y; 2 +

2 , 2x) 2−β

2 − Ke−rτ (1 − Q(2x; 2−β , 2y)).

Comparing the two formulae it is clear that: 2 P ∗ (ST > K|St ) = 1 − Q(2x; 2−β , 2y)

where y = kK 2−β , and so replacing r by µ in x and y, we obtain: 2 ˜ 2−β ) P (ST > K|St ) = 1 − Q(2˜ x; 2−β , 2kK


91

which yields the desired result: 2 ˜ 2−β )) P (ST < s|St ) = 1 − (1 − Q(2˜ x; 2−β , 2ks 2 , 2˜ y) = Q(2˜ x; 2−β

˜ 2−β . where y˜ = ks

3.5.3

CEV Option Prices

In the same way that a call price series can be computed for geometric Brownian motion using the Black-Scholes formula, the call price series that corresponds to a simulated CEV process can be calculated using Schroder’s formula. A difference is that SPLUS cannot evaluate Schroder’s formula when τ is close to zero, because both the quantile at which the non-central Chi-squared survivor function must be evaluated, and the non-centrality parameter become infinite. It appears then that both the Q terms will be zero, yielding an option price of 0, not the same as the required (ST −K)+ , however a limiting argument will undoubtedly yield the correct result. Computation

of Schroder’s formula becomes increasingly slow as the time to maturity nears zero (as would computation of the infinite sums, since n2 must be increasingly large to compensate for growth in k). Figure 3.7 displays a realisation of {St }, a CEV process with β = −1,

St = $5, τ = 1, µ = 0.1, σ = 0.3, and 250 subintervals. In addition, series

corresponding to {Ct + Ke−rτ } and {Ke−rτ } are shown. The difference be-

tween the two latter curves is of course the option price series {Ct }. The graph exhibits the same general qualities shown in the corresponding graphs for the GBM realisations seen in Figures 2.4 and 2.5. A high degree of correlation is seen between the series {St } and {Ct }, where every movement in the share price is mirrored by a movement in the option price. In the particular realisation in Figure 3.7, the option matures in-the-money, and so, near exercise when the discounting effect on the exercise price is minimal, the option price plus (discounted) exercise price converges to the share

92 6.0


5.0 4.0

4.5

Price

5.5

Share price Option price + pv(exercise price) pv(exercise price)

1.0

0.8

0.6

0.4

0.2

0.0


Figure 3.7: A realisation of a CEV share price with initial value St = $5, and parameters β = −1, τ = 1, µ = 0.1, σ = 0.3, and 250 subintervals; also the CEV option prices for this share series added to the present value of the exercise price, with K = $5, and r = 0.06 and time to maturity indicated on the horizontal scale; finally the present value of the exercise price itself.

price, or equivalently, the option price converges to the share price less the (discounted) exercise price. Both MacBeth & Merville (1980) and Beckers (1980) have computed and analysed CEV prices. In particular they compare option prices for various CEV classes with Black-Scholes option prices. MacBeth and Merville choose CEV processes with β = 0, -2, and -4, aligning the processes to GBM using the relationship 1− β2

δ = σSt

where σ is equivalent to the Black-Scholes volatility parameter, as described earlier. I have chosen to reproduce part of MacBeth and Merville’s Table 1, using SPLUS and Schroder’s pricing formula seen in Equation (3.44). These figures

93


are shown in Table 3.1 and correspond to those of MacBeth and Merville except in two cases, shown with an asterisk. These values are given as 0.89 and 0.94 by MacBeth and Merville, instead of 0.88 and 0.95 respectively. The correspondence between the two sets of results indicates that Schroder’s formula is indeed reproducing Ct , correctly and without the need to make decisions regarding the upper and lower limits of summation in Cox’s formula. The fact that two values differ in the second decimal place, indicates that at least twice, the values of n1 and n2 chosen by MacBeth and Merville have not ensured convergence of the option price seen in Equation (3.36). In pricing a large bundle of options, such differences could amount to significant pricing errors. τ 30 365

90 365

180 365

270 365

K 40 50 60 40 50 60 40 50 60 40 50 60

σ = 0.2 β = 0 β = −2 10·20 10·20 1·27 1·27 0·00 0·00 10·62 10·64 2·36 2·36 0·06 0·03 11·32 11·42 3·55 3·56 0·39 0·28 12·05 12·21 4·55 4·57 ∗ 0·69 0.88

σ = 0.4 β = 0 β = −2 10·28 10·34 2·41 2·41 0·11 0·06 11·30 11·58 4·31 4·33 ∗ 0·73 0.95 12·81 13·35 6·28 6·34 2·35 1·97 14·15 14·89 7·85 7·97 3·65 3·18

Table 3.1: A section of MacBeth and Merville’s Table 1, of CEV option prices, calculated for the parameters shown, with additional parameters: St = $50 and r = 0.06. Beckers compares Square Root and Absolute CEV prices with BlackScholes prices. In addition to using Equation (3.36) with n1 = 0 and n2 = 995 to obtain the CEV prices, Beckers uses an approximation (attributed to John Cox, but not referenced) for the Square Root CEV case, and the exact formula for the Absolute CEV case, given in Equation (3.39).

94

CHAPTER 3. THE CEV MODEL Earlier, I described the alignment process used by Beckers, to ensure his

CEV prices were comparable to Black-Scholes prices. This involved solving the equation 2

Var(ST |St ) = St2 e2µτ (eσ − 1) where the random variable ST follows a CEV process with β < 2, for δ [which features in Var(ST |St )], so that all three share price processes considered have the same variance at T . In contrast, MacBeth and Merville’s alignment

procedure ensured all processes have the same variance over the interval [t, t + dt]. Using the form for Var(ST |St , β = 1) given in Theorem 3.2, Beckers shows

that for a Square Root CEV process, δ is given by the equation: 2

2

µSt eµτ +σ τ (eσ − 1) δ = eµτ − 1 2

(3.45)

where σ is the Black-Scholes volatility parameter7 . I have used this relationship to give the values seen in Table 3.2.

σ 0·2 0·3 0·4

Months to maturity 1 4 7 1·2694 1·2828 1·2964 1·9100 1·9485 1·9877 2·5579 2·6439 2·7331

Table 3.2: Beckers’ δ values (rounded to 4 d.p.) for the Square Root CEV process, found using Equation (3.45), with additional parameters St = $40, and µ = log(1.05). Table 3.3 reproduces a section of Beckers’ Table II, for the Square Root CEV case with St = $40 and r = log(1.05). In all but one case, the values given by Schroder’s formula correspond exactly to those given by Beckers, 4 with the single exception when (σ, K, τ ) = (0.2, 45, 12 ), which reads 0.478 in

Beckers’ table.

95


σ

0.2

0.3

0.4

K 30 35 40 45 50 30 35 40 45 50 30 35 40 45 50

Months to Maturity 1 4 7 10·122 10·495 10·908 5·150 5·798 6·478 1·006 2·193 3·063 0·019 ∗ 0.486 1·093 0·000 0·059 0·287 10·123 10·639 11·307 5·235 6·363 7·393 1·471 3·147 4·357 0·149 1·248 2·298 0·004 0·393 1·082 10·136 11·010 12·067 5·427 7·121 8·557 1·941 4·145 5·755 0·397 2·156 3·670 0·043 1·001 2·221

Table 3.3: A section of Beckers’ Table II, showing Square Root CEV prices, with additional parameters St = $40, and r = log(1.05). I have been unable to confirm the Absolute CEV prices given by Beckers, and have not attempted to confirm the values of δ he gives in his Appendix B. These values are obtained by solving the equation: 2

Var(ST |St , β = 0) = St2 e2µτ (eσ − 1) for δ, and are reproduced in Table 3.4. These values are of quite a different scale to δ values given by MacBeth and Merville’s alignment procedure, which are 8, 12, and 16 corresponding to σ values 0.2, 0.3, and 0.4 respectively, indicating the values used by Beckers may have been reported incorrectly. Note that prices in both Tables 3.1 and 3.3 exhibit the following properties: 7

MacBeth and Merville’s method yields δ values of 1.2649, 1.8974, and 2.5398, corresponding to σ = 0.2, 0.3, and 0.4 respectively. These values are similar to those shown in Table 3.2, however use of these values, instead of Beckers’, results in option prices up to 15% less, with the difference increasing with time to maturity and σ.

96


σ 0·2 0·3 0·4

Months to maturity 1 4 7 0·1769 0·1818 0·1868 0·4006 0·4194 0·4394 0·7185 0·7727 0·8504

Table 3.4: Beckers’ δ values (rounded to 4 d.p.) for the Absolute CEV process, with additional parameters St = $40, and µ = log(1.05). • As K increases, Ct decreases; • As τ increases, Ct increases; • As σ increases, Ct increases. These general trends, observed for the particular values given above, correspond to the expected behaviour of call option prices as discussed in Chapter 1.

3.5.4

Behaviour of CEV Prices

As mentioned above, CEV prices appear to have similar properties to BlackScholes prices, with respect to changes in exercise price, time to maturity, or the volatility parameter σ. Of interest is the behaviour of CEV prices when β changes, and in particular whether CEV prices converge to the Black-Scholes price as β nears two. If this convergence were not evident, then this would indicate a deficiency in the CEV option pricing formula. Cox’s formula for CEV option prices, and Schroder’s method of computing these prices enable us to examine convergence as β nears two from below. At present we cannot examine convergence from above, however a result derived by Emanuel & MacBeth (1982) completes the CEV class, and gives us a pricing formula for the case β > 2. Result 3.5 (The CEV Option Price for β > 2). In the case β > 2, the risk-neutral transition density function, describing the evolution of the share

97

3.5. COMPUTING THE OPTION PRICE price St to ST is: √ 1 1 fS∗T |St (s, τ |β > 2) = (β − 2)k 1/(2−β) (xz 1−2β ) 2 2−β e−x−z I1/(β−2) (2 xz) where all terms are defined earlier8 . The option price is given by: 2 , 2y) − Ke−rτ (1 − Q(2y; 2 + Ct = St Q(2x; β−2

2 , 2x)) β−2

(β > 2). (3.46)

Proof. A proof of the option price formula can be found in Appendix B.4. Emanuel and MacBeth cite theoretical support for this model9 , however its use obviously conflicts with the motivation behind the original CEV model, including empirical observations regarding share price evolution. Regardless of the motivation behind each model, any option price over a share whose price evolves according to Equation (3.1) can now be computed using Equation (3.44) for β < 2, the Black-Scholes formula when β = 2, and Equation (3.46) when β > 2. Using these formulae, I have produced the graphs in Figures 3.9 to 3.11. These figures give call option prices for β in the range −2 ≤ β ≤ 6, for various σ and K, and where St = $50 and r = 0.06.

These graphs give the relationship between Ct and β for out-of-the-money (K = $55), at-the-money (K = $50), and in-the-money (K = $45) calls as σ changes. Both MacBeth and Merville, and Beckers note the relationship between call price and β. MacBeth and Merville restrict their comments to the parameter choices they consider, but Beckers makes a more general comment: “As a general rule, it can be inferred that for in-the-money and at-the-money 8

Note that the risk-neutral density that applies in the case β > 2 is very similar to that when β < 2, with differences only in the scale factor (in this case β − 2 instead of 2 − β), and the degrees of freedom of the modified Bessel function. In fact for β 6= 2, the general risk-neutral density function could be written: √ fS∗T |St (s, τ ) = |2 − β| k 1/(2−β) (xz 1−2β )1/(4−2β) e−x−z I1/|2−β| (2 xz) 9

Rubinstein, Mark ((1981),‘Displaced Diffusion Option Pricing’, Manuscript, University of California, Berkeley.

98


options the model price increases as the characteristic exponent [β] decreases, whereas exactly the opposite is true for out-of-the-money options.”10 Whilst Beckers uses a different alignment procedure to MacBeth and Merville, and I have produced the graphs in Figures 3.9 to 3.11 using the latter authors’ alignment procedure, it appears that the relationship Beckers describes does not hold in general. Even if Beckers’ comments were restricted to the case 0 ≤ β < 2, when σ is very large (perhaps too large given empirical observa-

tions), even in this range Ct is not a monotonic function of β as shown in the graphs. A positive aspect of the functions shown in Figures 3.9 to 3.11 is that the CEV prices appear to converge to the Black-Scholes price, from above and below, when β approaches 2, for the particular combination of parameters used. This observation lends support to the use of the CEV option pricing formulae as alternatives to the Black-Scholes formula.

3.6

Use of the CEV Model

In Section 2.5, I noted the volatility smile phenomenon, which provides evidence against the use of the Black-Scholes formula to price share options. Figure 3.8 shows the Black-Scholes implied volatilities11 for Absolute CEV option prices. Parameters used to obtain the CEV option prices were St = $5, τ = 1 year, σ = 0.3, r = 0.06 and $4 ≤ K ≤ $6. These prices were then used

as market option prices, to yield the Black-Scholes implied volatilities shown in the graph. In particular I solved the equation: Cmt = BS(σ)

for σ, where the Cmt are Absolute CEV prices, and BS(σ) is the BlackScholes formula treated as a function with a single parameter σ. The values of σ that satisfy the relationship are shown in the graph. 10

Beckers (1980), page 667. The implied volatility is the value of σ that equates the Black-Scholes model price, to the observed market price of the option, given other (observable) parameters S t , τ , K and r. 11

99

0.31 0.30 0.28

0.29

Implied Volatility

0.32

0.33

3.6. USE OF THE CEV MODEL

4.0

4.5

5.0

5.5

6.0

Strike Price

Figure 3.8: Black-Scholes implied volatilities for Absolute CEV option prices with St = $5, τ = 1 year, σ = 0.3, r = 0.06 and $4 ≤ K ≤ $6. The curve in Figure 3.8 does not look dissimilar to the implied volatility curve for options on the S&P 500 index on May 5, 1993, given by Hull (1997) in his Figure 19.412 . Suppose a share price was truly following a CEV process, and the market was correctly pricing options over that share. Using the Black-Scholes formula to obtain an estimate for σ, results in the volatility smile relationship shown in the graph. The fact that the relationship shown is similar to those obtained empirically lends indirect support for the use of the CEV model in practice. Empirical research already alluded to has endeavoured to determine whether the CEV model is an appropriate one. In the next Chapter, I attempt to estimate the coefficient β for real share price series, but am not able to use option price information in the analysis. Rubinstein (1985) uses non-parametric methods to compare option pricing models. Included in the models he con12

Hull (1997), page 504.

100


siders is the Absolute CEV model, whose option price formula is given in Equation (3.39). In order to identify pricing models consistent with his vast database of option prices, Rubinstein considers pairs of market option prices with all characteristics in common except for either the strike price, or the time to maturity. He then calculates implied volatilities for the pair, and compares the pattern of relative values to those theoretically observed for the various models he considers. The Absolute CEV model is consistent with some of the biases he observes, but like the other models considered, it does not explain significantly all the biases observed.

101

3.6. USE OF THE CEV MODEL sigma = 0.6

sigma = 0.8

6.5

3.8

9.5

4.0

7.0

10.0

10.5

Call price

7.5

Call price

4.4 4.2

Call price

4.6

11.0

8.0

4.8

sigma = 0.4

2

4

6

-2

2

4

6

-2

sigma = 1.2

sigma = 1.4

6

4

6

4

6

18.5 18.0 16.5

14.5 4

6

17.0

15.0

15.5

Call price

Call price

16.0

14.0 13.5

2

4

19.0

sigma = 1

13.0

-2

0

2

4

6

-2

0

2

beta

beta

beta

sigma = 1.6

sigma = 1.8

sigma = 2

Call price

23

24

25

23 22

18

21

20

22

19

21

Call price

20

Call price

21

26

0

2 beta

12.0 -2

0

beta

12.5

Call price

0

beta

17.5

0

16.5

-2

-2

0

2 beta

4

6

-2

0

2 beta

4

6

-2

0

2 beta

Figure 3.9: Out-of-the-money CEV option prices, with β between -2 and 6, σ between 0.4 and 2, and additional parameters St = $50, K = $55, τ = 0.5 years, and r = 0.06.

102

-2

-2

0

0

0

sigma = 0.4

2 beta

sigma = 1

2 beta

sigma = 1.6

2 beta

4

4

4

6

6

6

-2

-2

-2

0

0

0

4

4

4

6

6

6

-2

-2

-2

0

0

0

sigma = 0.8

2 beta

sigma = 1.4

2 beta

sigma = 2

2 beta

4

4

4

6

6

6

CHAPTER 3. THE CEV MODEL sigma = 0.6

2 beta

sigma = 1.2

2 beta

sigma = 1.8

2 beta

12.1 12.0 11.9 11.8 11.7 19.6

9.25 9.20 9.15 9.10 9.05 9.00 17.2

6.34 6.32 6.30 6.28 6.26 14.7

Figure 3.10: At-the-money CEV option prices, with β between -2 and 6, σ between 0.4 and 2, and additional parameters St = $50, K = $50, τ = 0.5 years, and r = 0.06.

-2

Call price Call price Call price

19.4 19.2 19.0 26.5 26.0 25.5 25.0 24.5 24.0 23.5


17.1 17.0 16.9 24.0 23.5 23.0 22.5


14.6 14.5 14.4 14.3 22.0 21.8 21.6 21.4 21.2 21.0 20.8 20.6

103

3.6. USE OF THE CEV MODEL sigma = 0.6

sigma = 0.8

Call price

14.0

Call price

11.5

9.0 8.6

11.0

13.5

8.8

Call price

9.2

14.5

12.0

9.4

15.0

9.6

12.5

sigma = 0.4

2

4

6

-2

0

2

4

6

-2

beta

sigma = 1

sigma = 1.2

sigma = 1.4

4

6

6

4

6

22.0

4

20.0 19.5 -2

0

2

4

6

-2

0

2

beta

beta

sigma = 1.6

sigma = 1.8

sigma = 2

26

28

beta

21

24

23

25

26

Call price

Call price

24

22

Call price

23

25

27

24

6

21.0

Call price

19.0

Call price

18.5 18.0 17.5

2

4

21.5

19.5

17.5 17.0 Call price

16.5 16.0

0

2

beta

15.5 -2

0

beta

20.5

0

20.0

-2

-2

0

2 beta

4

6

-2

0

2 beta

4

6

-2

0

2 beta

Figure 3.11: In-the-money CEV option prices, with β between -2 and 6, σ between 0.4 and 2, and additional parameters St = $50, K = $45, τ = 0.5 years, and r = 0.06.

104


Chapter 4 Data Analysis

105

106

CHAPTER 4. DATA ANALYSIS

4.1

Introduction

In order to apply the Black-Scholes model to determine option prices on a stock, a single parameter, σ, must be estimated. A significant part of the empirical literature on option pricing, post Black & Scholes (1972) has tried to identify the “best” way to estimate this parameter. These studies have shown that whilst the Black-Scholes model does not seem appropriate due to its restrictive set of conditions, the Black-Scholes implied volatility is a better predictor of future share price volatility than an estimate based on past and present share prices. In order to use the CEV model to estimate option prices, not only must δ be estimated, but also the CEV parameter β. This parameter characterises the strength of the inverse relationship between share price level, and its volatility. Estimation of β values in real data that were significantly different from β = 2 would lend support to the use of the CEV model for option pricing, and provide evidence that the Black-Scholes assumptions regarding share price evolution and its volatility are not satisfied. In particular, I focus on the estimation of β, with an aim to identify whether or not the CEV model seems empirically appropriate. Some empirical papers do use particular CEV models to describe underlying asset evolution but do not attempt to chose the “best” model for their data, either by implying parameters for option and underlying asset price data or by direct estimation from the underlying asset series alone. These papers include Rubinstein (1985), whose work has been discussed in Chapter 3, and Lauterbach & Schultz (1990), who test the Square Root CEV model against the Black-Scholes and other models for the pricing of warrants1 . Other researchers do endeavour to determine the most appropriate CEV model, by estimating β from time series data. Previously, attempts 1

After adjustment for warrant pricing, this empirical research finds that the Square Root CEV formula describes observed warrant prices significantly better than the BlackScholes model.

107

4.1. INTRODUCTION

have been made by Beckers (1980), MacBeth & Merville (1980), Emanuel & MacBeth (1982), Marsh & Rosenfeld (1983), Tucker, Peterson & Scott (1988) and Melino & Turnbull (1991) to estimate β, with varying degrees of success. Whilst their estimation methods vary, a recurrent theme in the results is that the CEV model indeed appears to be an appropriate model for share price evolution. Since modelling the share price has direct implications on modelling option prices, this finding provides evidence for the use of the CEV option pricing model in the marketplace. Before introducing the estimation procedure I have used, I will briefly summarise the methodologies and the findings of the aforementioned papers.

4.1.1

Summary of Alternative Methods

Beckers (1980) Beckers estimates β values for 47 stocks, using almost five years’ data for each. The SDE (3.1) yields Var

µ

¯ ¶ dSt ¯¯ St = δ 2 Stβ−2 dt. ¯ St

With daily data, we can set dt = 1, and so rewrite the above relationship in the form: ¯ ¶¶ µ µ β−2 St+1 ¯¯ St ln St = ln δ + ln s ¯ St 2

where s(.) is the standard deviation operator. This is effectively the regression equation for β used by Beckers. Estimation of the regression parameters will yield estimates for both δ and β. Application of this technique requires some refinement however, as the dependent variable in the regression is not observable. Using the fact that for daily data µ is very small, giving ¶ µ St+1 = eµ ≈ 0. E St Beckers approximates the standard deviation on the LHS by |ln(St+1 /St )|.

This is appropriate if X = ln(St+1 /St ) is approximately normal, in which

108


case, the mean value of X will be proportional to its standard deviation. This result is proved by Beckers. Thus, using the single realisation of X instead of the standard deviation of interest, Beckers estimates β using the regression

where β = 2b + 2.

¯ ¯ ¯ St+1 ¯ ¯ = a + b ln St + wt ln ¯¯ln St ¯

(4.1)

He reports low R2 and Durbin-Watson statistics which lead him to conclude that his regression is incomplete. However, using this technique, Beckers finds that 38 of his 47 cases had β estimates significantly less than two, with three stocks yielding β significantly greater than two, leaving only six stocks consistent with GBM. Also, on the basis of his estimates, Beckers finds evidence to reject the hypothesis that the true β values for the stocks are the same, and concludes that in general, different stocks will have different values of β, just as their volatilities and other characteristics differ. Throughout his analysis Beckers restricts the CEV model to β : 0 ≤ β
2 respectively. Given a realisation of a such a share price process, the aim of this section is to obtain estimates of β and δ, for use in option pricing. Recall that Cox’s and Emanuel and MacBeth’s analyses give us the true transition density for the solution to the above SDE for the case β 6= 2: 1

1

1

xz˜1−2β ) 2 2−β e−˜x−˜z I fSt+1 |St (st+1 , 1) = |2 − β| k˜ 2−β (˜

1

1 |2−β|

(2(˜ xz˜) 2 )

where again: k˜ =

δ 2 (2

2µ − β)(eµ(2−β)τ − 1)

˜ t2−β eµ(2−β)τ x˜ = kS

˜ 2−β z˜ = ks

apply with τ = 1 day. Theoretically, the likelihood function for a realisation from this process could be formed, and maximum likelihood techniques employed to jointly obtain estimates for β and δ, βˆ and δˆ respectively . Problems arise not only due to the presence of the modified Bessel function but also the complicated nature of the density as a function of β. In order to numerically solve a system of equations like ¯ ∂f ¯¯ =0 ∂β ¯β=β,δ= ˆ δˆ ¯ ∂f ¯¯ =0 ∂δ ¯ ˆ ˆ β=β,δ=δ

the derivatives, which are likely to be of the form g(β, δ)f (β, δ) where f (β, δ) is the density function above treated as a function of β and δ only, and g(β, δ) is some other function, must be evaluated over a finely divided range of β,

114


and for each evaluation, approximation of the modified Bessel function must be made. Whilst the modified Bessel function appears to converge quickly, the maximisation process will still be computationally expensive. In addition, the likelihood function should be evaluated for each of the three cases: β < 2, β = 2 and β > 2, and the value of βˆ that gives the largest value chosen.

4.2.1

β Estimation Strategy

The β estimation method which I introduce in this section uses maximum likelihood technology, but is an approximate method which is not based on the true likelihood function for a series of CEV share prices. Using the true likelihood is possible, but this alternative method is both easier to implement, and to understand. Although the method is approximate, the estimates which it gives will hopefully be close to those which would be obtained using the true likelihood. At the very least, it is hoped that this approximate method will provide estimates which can be used as starting values in an estimation process that does use the CEV transitional density. In order to estimate β, I will use the fact that the Brownian motion increment dBt , which appears in the SDE (3.1), is a normal random variable with zero mean and variance dt. This will allow use of the normal density function to construct a likelihood function that is simpler than one using the true transition density for the CEV share price process. Simplification of the estimation process is made by considering the transformation Yt = ln St . Using Itô’s Lemma, ´ ³ β−2 d ln St = µ − 21 δ 2 Stβ−2 dt + δSt 2 dBt

and so integrating both sides from n to n + 1: Z Z n+1 β−2 1 2 St dt + δ ln Sn+1 − ln Sn = µ − 2 δ n

β−2

≈ µ − 12 δ 2 Snβ−2 + δSn 2 ²n

n+1

(4.2)

β−2 2

St n

dBt

4.2. ESTIMATING β FROM A SHARE PRICE SERIES

115

where ²n is a forward N (0, 1) increment that is independent of Sn (and hence Yn ), and it is assumed that the integrands are sufficiently smooth to allow the integral approximations to be made, or alternatively, that the time unit of n is sufficiently small. Letting n be measured in days, the rates µ and δ must now also be measured in days. Since Yn = ln Sn , we can write the discrete time equivalent of the SDE (3.1), and so conditional on Yn : β−2

Yn+1 = Yn + µ(Yn ) + δe 2 Yn ²n ¢ ¡ ∼ N Yn + µ(Yn ), δ 2 e(β−2)Yn

(4.3)

where µ(Yn ) = µ − 21 δ 2 e(β−2)Yn is the expected value of the daily returns Yn+1 − Yn , given Yn . Thus, an approximate transition density function for

Yn+1 conditional on Yn = yn can be written: ( µ ¶2 ) 1 1 yn+1 − yn − µ(yn ) fYn+1 |Yn (yn+1 , 1) = √ exp − β−2 β−2 2 2πδe 2 yn δe 2 yn

(4.4)

and so, for a sample of share prices of size N + 1: S0 , S1 , . . . , SN , the conditional likelihood for the corresponding series {Yn : n = 0, . . . , N } can be

formed. Use of the approximation directly above has an advantage over the

use of the true transition density functions for Sn+1 given Sn , in that it can be used for any CEV process. The true likelihood function would be a product of density functions that have different forms for β < 2, β = 2 and β > 2, and so given an unknown parameter β, there might be difficulty constructing the correct likelihood function, especially when β is close to two. Note that: L(β, δ| ∼s ) = fY0 (y0 )

N −1 Y n=0

fYn+1 |Yn (yn+1 , 1)

l(β, δ| ∼s ) = ln L(β, δ| ∼s ) = ln fY0 (y0 ) +

N −1 X n=0

ln fYn+1 |Yn (yn+1 , 1)

116


where L(β, δ| ∼s ) is the likelihood function for the parameters β and δ, given a realisation, ∼s , of a share price series. Conditional on the share price sample s , in order to maximise the likelihood function L(β, δ) to find an estimate

∼

for β, it is appropriate to maximise the function 1 l(β, δ) N

=

1 N

ln fY0 (y0 ) +

1 N

N −1 X n=0

ln fYn+1 |Yn (yn+1 , 1)

(4.5)

which, for large N , is approximately: Ã ( µ ¶2 )! N −1 X y − y − µ(y ) 1 1 1 n+1 n n ¯l(β, δ) = ln √ exp − β−2 β−2 y y n n N n=0 2 2πδe 2 δe 2 ( ¶2 ) µ N −1 β−2 1 1 1 X r¯n − ln δ − yn − ln 2π − = yn N n=0 2 2 2 δe β−2 2 = − ln δ −

N −1 N −1 r¯n2 1 β−2 X 1 X yn − ln 2π − 2N n=0 2 2N n=0 δ 2 e(β−2)yn

(4.6)

¯n = where r¯n = yn+1 − yn − µ(yn ) are realisations of the random variable R

Yn+1 − Yn − µ(Yn ).

This log-likelihood function is very similar to that derived by Christie

(1982), and implemented by Tucker et al. (1988), however there are a few differences. Christie produces the log-likelihood function: l(β, δ) = −

N β−2X 1 X 2 −(β−2) N ln(2π) − ln δ 2 − ln S − 2 rS S (4.7) 2 2 2 2δ

where N is sample size, and rS =

∆S S

is the percentage daily return on the

share price S. In the model above, I am dealing with mean corrected daily returns, rather than the percentage returns Christie uses. In addition Christie sees no need for mean-correction, since the parameter µ is of a much smaller order than the variance of returns, but by making the log transformation, I introduce the correction term seen in Equation (4.2). Finally, Christie does not explicitly take into account the unconditional density function for Y0 . The log-likelihood in Equation (4.6) can be constructed from share price data by first computing the natural logarithm of the share prices, then taking


117

the first difference of the logs to form a series {rn }. Once again I have chosen

to estimate the mean of this series, E(Yn+1 −Yn ) = µ(Yn ), non-parametrically

using the Lowess filter. This estimated mean level can then be subtracted

from the daily returns to give the series {¯ rn }, which can be combined with

{yn } to calculate l(β, δ).

Given the equivalence of maximisation of L and ¯l, as discussed above,

Maximum Likelihood estimates for β and δ could be found by solving the simultaneous equations: ¯ ∂ ¯l ¯¯ =0 ∂β ¯β=β,δ= ˆ δˆ ¯ ∂ ¯l ¯¯ = 0. ∂δ ¯ ˆ ˆ β=β,δ=δ

I do not do this, but rather find an estimate for δˆ from the second equation directly above, and substitute this back in to Equation (4.6), which I then maximise with respect to β. This second maximisation is done numerically, using SPLUS, the details of which follow. The first partial derivative of ¯l(β, δ) with respect to δ is: N −1 1 1 X r¯n2 ∂ ¯l . =− + ∂δ δ N n=0 δ 3 e(β−2)yn

(4.8)

Setting Equation (4.8) equal to zero as described above, and solving for δˆ2 (β) yields: N −1 1 X r¯n2 δˆ2 (β) = N n=0 e(β−2)yn

(4.9)

which may substituted into Equation (4.6). The function obtained by making this substitution is: ! Ã N −1 N −1 X r¯2 β−2 X 1 1 1 1 n ¯l(β, δ) ˆ = − ln − yn − ln 2π − (β−2)y n 2 N n=0 e 2N n=0 2 2

(4.10)

118


and maximisation of this with respect to β is equivalent to minimisation of the function: g(β) = ln

Ã

N −1 1 X r¯n2 N n=0 e(β−2)yn

!

N −1 β−2 X + yn N n=0

(4.11)

ˆ by scaling and removal of additive constants. This obtained from ¯l(β, δ) function can be minimised numerically in SPLUS using the function nlmin as shown in Appendix D.4. An example of a typical likelihood surface is shown in Figure 4.1, and can be seen to have a ridge-like surface. This means that the parameters β and δ are highly interdependent, and that the estimation procedure is probably not all that well defined for finding βˆ and δˆ jointly. Ideally, the likelihood surface would have a distinct peak, which would enable the maximisation ˆ The process to better pin-point the maximum likelihood estimates βˆ and δ. scale of δ 2 in the graph has been altered in the estimation process, by a scale factor dt. Hence an estimate of σ 2 is given by 1 × Stβ−2 δˆ2 × dt where dt is measured in years. The share price used to construct the log-likelihood surface shown in Figure 4.1 yields the estimates: βˆ = −0.3176

δˆ = 0.02504

σ ˆ = 0.06132.

Choosing to minimise g(β), as outlined above, restricts the maximisation process to a cross-section of the log-likelihood surface, obtained by setting δ 2 = δˆ2 (β) as given in Equation (4.9). The log-likelihood function is shown again in Figure 4.2, along the cross section δ 2 = δˆ2 (β). The vertical line ˆ and clearly cuts the β = βˆ has been superimposed on the graph of ¯l(β, δ), curve at its maximum. Hence it appears that the maximisation process is correctly identifying the maximum likelihood estimate of β.


119

0.00066

4.38

0.00064 0.00062

4.384 4.38406 4.384

4.383

0.00058

0.00060

(scaled) delta squared

4.383

4.38

-0.34

-0.32

-0.30

-0.28

beta

log-likelihood

4.3825

4.3830

4.3835

4.3840

Figure 4.1: The log-likelihood surface, ¯l(β, δ), for a simulated series with St = $5, τ = 3 years, µ = 0.1, σ = 0.3, n = 250 subintervals per year, and β = 0.

-0.34

-0.32

-0.30

-0.28

beta

ˆ Figure 4.2: The cross-section of the log-likelihood surface in Figure 4.1, ¯l(β, δ) for a simulated series with St = $5, τ = 3 years, µ = 0.1, σ = 0.3, n = 250 subintervals per year, and β = 0. In addition, the line β = βˆ which identifies the maximum.

120

4.2.2


The Variance of βˆ

Maximum likelihood theory indicates that the estimates of β and δ have desirable asymptotic properties. As sample size increases, they will be unbiased and normal, with variances obtained via the information matrix: I(β, δ) = −E

Ã

∂2l ∂β 2 ∂2l ∂δ∂β

∂2l ∂δ∂β ∂2l ∂δ 2

!

.

(4.12)

Approximating N −1 l by ¯l as before, and recalling the form of ¯l(β, δ) in Equation (4.6) the partial derivatives can be evaluated and the information matrix found. The first partial derivatives of l are: N −1 N −1 ¯ n2 Yn ∂l 1X R ∂ ¯l 1X Yn + =N =− ∂β ∂β 2 n=0 2 n=0 δ 2 e(β−2)Yn N −1 ¯ n2 ∂ ¯l N X R ∂l =N =− + ∂δ ∂δ δ δ 3 e(β−2)Yn n=0

yielding second partial derivatives: N −1 ¯ n2 Yn2 ∂2l 1X R = − ∂β 2 2 n=0 δ 2 e(β−2)Yn

=−

N −1 1X 2 2 ² Y 2 n=0 n n

N −1 X ¯ n2 Yn ∂2l R =− ∂δ∂β δ 3 e(β−2)Yn n=0

=−

N −1 2 X ²n Y n n=0

δ

N −1 X ¯ n2 N ∂2l R = − 3 ∂δ 2 δ2 δ 4 e(β−2)Yn n=0 N −1 2 X N ²n = 2 −3 δ δ2 n=0


121

where the ²n are approximately N (0, 1), with E (²2n ) = Var(²n ) = 1. Hence, by the independence of ²n and Yn , elements of the information matrix are: −E

µ

∂2l ∂β 2

¶

N −1 1X E(Yn2 ) = 2 n=0

¶ N −1 X ∂2l E(Yn ) −E = ∂δ∂β δ µ 2 ¶ n=0 ∂ l 2N = −E ∂δ 2 δ2 µ

which give the information matrix: Ã X

E(Yn2 ) − 1δ P E(Yn ) − 1δ 1 2

I(β, δ) =

This has the determinant:

P

E(Yn )

2N δ2

!

.

(4.13)

P N −1 ( E(Yn ))2 2N 1 X 2 E(Yn ) − |I(β, δ)| = 2 δ 2 n=0 δ2 µ ´2 ¶ 1 ³X N X 2 . E(YN ) = 2 E(YN ) − δ N

The information matrix yields the variances and covariances of the estimates through the relationship {I(β, δ)}

−1

=

µ

ˆ ˆ β) ˆ Var(δ) Cov(δ, ˆ β) ˆ ˆ Cov(δ, Var(β)

¶

These quantities will all feature the terms in the determinant, in particular E(ln Sn ) and E{(ln Sn )2 }, quantities which do not appear easy to calculate given the density function for Sn given St , where n > t.

An alternative to calculating the theoretical variance of the estimates of β and δ is simulation. Share price series with known β and δ can be simulated using the SPLUS function described in Appendix D.3. The likelihood function can then be constructed for these series, and the maximum likelihood estimates obtained using the method described in Appendix D.4. This is done in the following section.

122


4.3

Appraisal of the Estimation Technique

The estimation of β outlined above relies on the assumption √

¯n R δ 2 S β−2

∼ N (0, 1)

holding at least approximately. The maximum likelihood analysis that follows this assumption yields estimates for both β and δ. These estimates may be used to construct a sample which should be approximately unit normal. The values en = q

¯n R ˆ δˆ2 Snβ−2

(4.14)

can be compared graphically to a unit normal probability density function. If the two distributions are “close”, it is possible that the estimate of β will also be good. This analysis is not as important for simulated series as it is for real data series, since the simulated series are constructed directly from normal Brownian motion increments. Examination of the distribution of the quantity given in Equation (4.14) for selected real series will be done in the following section. The distribution of the en values for the simulated series, whose log-likelihood function was examined earlier, is shown in Figure 4.3. This histogram has the standard normal density function φ(z) superimposed over the range of the realised en values. The fit seems good, but there is an outlying value at z = −4.7067. This outlier does not pose a problem however,

and a Chi-squared goodness of fit test for the sample yields a very small test statistic, with a p-value of 96.74% for a test of the hypothesis that the distribution comes from a standard normal distribution. This is not at all surprising because of the simulation method for the share series. If the sample of en values does appear to have a standard normal distribution, the maximum likelihood procedure should indeed be appropriate. If so, the estimates should be asymptotically unbiased, and have variances

123

0.2 0.0

0.1

Relative Frequency

0.3

0.4

4.3. APPRAISAL OF THE ESTIMATION TECHNIQUE

-4

-2

0

2

Figure 4.3: en , given by Equation (4.14), for the simulated series examined previously with St = $5, τ = 3 years, µ = 0.1, σ = 0.3, n = 250 subintervals per year, and β = 0.

obtained via the information matrix. As noted above, the theoretical variances look difficult to compute, so I have used simulation to estimate the standard error of estimates of β. For given parameter values, it is possible to generate CEV share series, and then given these series, it is possible to estimate the parameter β. If all parameter values are fixed, and the share price simulation repeated a large number of times, a sample of βˆ values will form, from which the mean, standard deviation, and approximate distribution can be estimated. Maximum likelihood theory tells us that for long series, with N large, the estimate of β should be approximately normal, and so the standardised βˆ values: βˆ − β ˆ s(β)

ˆ is an estimate of the stanshould be approximately unit normal, where s(β) ˆ given by the sample standard deviation of the βˆ values. dard deviation of β,

-2

-2

0

2

beta = 1

0

beta = -2

-2

2

beta = 4

0

2

4

4

4

Relative Frequency

-6

-4

-4

-2

0

beta = 2

0

2

2

4

4

0.2

-4

0

beta = 5

-2

2

-4

-4

-2

-2

0

beta = 0

-2

2

beta = 3

0

beta = 6

0

2

4

2

4

6

CHAPTER 4. DATA ANALYSIS beta = -1

-2

4

0.5 0.4 0.1 Relative Frequency

Figure 4.4: Estimates of β, resulting from CEV share price simulation, used for the figures in Table 4.2. Superimposed on these are the density function ¯ 1) random variable over the range of the estimates, where β¯ is the of an N (β, sample average of the β estimates.

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I have simulated share series for the nine integer values of β in the range −2 ≤ β ≤ 6, with St = $5, µ = 0.1, and σ = 0.3. For each of these β

values I have simulated 1000 series 850 days long, and have used the last 750 observations of each series to estimate β. This is intended to reduce the impact of the fact that all share series begin in the same place, if indeed these initial conditions have an effect, and replicate the conditions under which β will be estimated for the real series. After 100 days, the share price will be a random variable whose properties have been discussed in Chapter 3. Recalling that the CEV model allows bankruptcy, it is possible that the 1000 share price series will result in fewer than 1000 estimates of β. In fact, given the initial conditions, the expected number of bankruptcies for some β is large, as is reflected in the observed numbers. This analysis is meant to estimate β values for series similar to those for the real data examined in the following section. Since these series are all approximately three years long, I have estimated β using only the complete simulated series. Estimating β from the shorter series, where bankruptcy is observed, should not improve the estimates, since fewer observations are available than in the full length series, nor should it affect them, since the bankrupted series contain no information not theoretically present in complete series. This same simulation procedure could be used to test any β estimation procedure. Figure 4.4 shows the distribution of the standardised βˆ values obtained from the simulation. The normal density curves that have been superimposed are drawn for the range of standardised β estimates, and are for normal ¯ and variance 1, where β¯ is the sample mean of the variables with mean β, βˆ values for each case. The normal curves are clearly not a good fit to the histograms. There are many outlying values in the estimates, and many of the histograms appear to have a sharper peak than the normal density functions, indicating positive kurtosis6 . The poor fit is confirmed by Chi-squared goodness of fit tests, with both the mean and standard deviation estimated from the data, where the null 6

Kurtosis is proportional to the fourth central moment of a random variable X, and is

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hypothesis that the sample values are normal is rejected in every case, with very high significance. Note also that many of the normal density functions are not centred on zero. The biasedness of the estimates is examined in a table which follows. Further results of the simulation are shown in Table 4.2. The true β value, used for simulating the share price series, is given, with the sample ˆ respectively, and mean and standard deviation of the βˆ values, β¯ and s(β) the number of series that did not result in bankruptcy, n. β -2 -1 0 1 2 3 4 5 6

β¯ −2·0265 −1·0259 −0·0154 1·0189 2·0218 3·0458 4·0598 5·0948 6·1384

ˆ s(β) 0·4308 0·4260 0·3925 0·3799 0·3850 0·3896 0·4200 0·4539 0·4922

n 909 931 975 1000 1000 1000 994 984 960

Table 4.2: Summary of the β estimates for simulated series, obtained using the maximum likelihood procedure described above, with St = $5, µ = 0.1, σ = 0.3 and where 3 years’ data is used. Comparison of the β¯ and β values in Table 4.2 appears to indicate that the estimates are biased. The normal density curves superimposed on the βˆ distributions were in fact centred on the respective β¯ − β values, which appear to differ from zero. For each of the above values of β it is possible to given by E{(X − µ)4 } −3 σ4 where µ = E(X) and σ 2 = Var(X). The kurtosis is zero for a normal random variable, and so positive (negative) values indicate that a density is more (less) peaked around its centre than a normal density function.


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test the hypotheses: H0 :β = i HA :β 6= i where i = −2, −1, . . . 6. The Central Limit Theorem provides the result: β¯ − β ˆ s(β) √ n

∼ N (0, 1).

Testing these hypotheses for the nine cases of β results in the test statistics and p-values seen in Table 4.3. The data in the table shows that there is True β -2 -1 0 1 2 3 4 5 6

Test statistic −1·8512 −1·8555 −1·2220 1·5718 1·7872 3·7171 4·4919 6·5487 8·7152

p-value 0·0641 0·0635 0·2217 0·1160 0·0739 0·0002 0 0 0

Table 4.3: p-values for hypothesis test H0 : β = i, i = −2, −1, . . . , 6 using the simulated data summarised in Table 4.2. a significant bias present in the estimation procedure when β > 2. The estimates for Cox’s CEV class have an average which is not significantly different from the true β at the 5% level, but at a higher significance level, the differences will become statistically significant. The figures in Table 4.3 do indicate that the method of estimating β is failing for some CEV processes, and in particular estimates for β that are greater than two may need adjustment to reflect this bias. The sample standard deviation of the β estimates is also given in Table 4.2, and these are shown in Figure 4.5. There does not appear to be much

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variation in this quantity, but in general it increases as β deviates from 2, particularly for β > 2. Both the bias noted for this latter case and the large standard deviation do not appear to be due to a preponderance of large values, with the histograms in Figure 4.4 showing symmetry in the βˆ values. In the following section, where I estimate β for 44 stocks, confidence intervals are formed for the true β (given that the model is appropriate and there really is a true β) using the standard deviations of βˆ listed in Table 4.2. This is an alternative to deriving the theoretical standard deviation using the information matrix, and estimating it for the particular share price series. Linear interpolation of adjacent standard deviations is used for non-integer estimates of β. It certainly appears that the range 0.35-0.50 should be adequate for the true standard error, which when compared to approximate standard errors of MacBeth and Merville of between 1.5 and 2.5 (assuming confidence interval width of around four standard errors) is very small. Tucker et al.

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(1988) also give the standard deviation of β estimates. For the range of β considered here, their standard errors are between 0.30 and 2.27. They do not state how these standard errors are obtained, although they differ for similar values of βˆ indicating a dependence on the realised time series data. It is also important that the method is producing sensible estimates. We know that the variance of daily returns should decrease as share price increases for β < 2, and vice versa for β > 2. This should be evident in a plot of the standard deviation of the daily returns against the share price level. Such a graph was seen in Figure 3.3 for a simulated series, and further plots will be produced for the real share series in the following section.

4.4

Data Analysis

This section describes an exploratory study of some Australian share price data, provided by Credit Suisse First Boston NZ Limited. The data set contains share price, trade, and dividend information for stocks traded on the Australian Stock Exchange (ASX) which have exchange traded options. Of this data, there are 44 stocks with full information over the period 2 January 1995 to 17 November 1997. A list of these stocks can be found in Appendix C. The time period spanned by the data contains a period of extraordinary turbulence in the world markets, experienced in October 1997. This period has been omitted from all analysis, as it appears influences were exogenous to the Australian market. Thus the data analysed was for a period 2 January 1995 to 26 September 1997. In particular, the analysis focusses on the estimation of β for each of the series, and on the appropriateness of the technique outlined above. The estimated β values for the 44 dividend corrected share series are ˆ obtained by linear interpolation shown in Table 4.4, with estimates of s(β) of the function shown in Figure 4.5. Confidence intervals are formed using these standard deviations, with “Type” indicating what sort of CEV process

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has been estimated. Type 1 corresponds to the CEV class of Cox, with β < 2, Type 2 corresponds to GBM, and Type 3 corresponds to the final class, with β > 2. The estimates have not been corrected for the biases identified in the previous section, but this does not have an effect on the classification of the estimates into the respective “Type” categories. The bias is significant for β > 2, however none of the lower confidence limits given in Table 4.4 are close enough to two for the bias to have an effect. The closest is that of ANZ, at 2.2263, which is well above two plus the estimated bias when βˆ ≈ 3 of 0.0458, given in Table 4.3.

Table 4.4: β estimates for the 44 ASX share series, with ˆ the standard deviation of β, ˆ and the estimates of s(β), resulting confidence intervals obtained by simulation. Stock Code AAA AMC ANI ANZ BHP BIL BOR BPC BRY CBA CCL CML CSR FBG FXJ GIO GMF ICI LLC MAY MIM NBH

βˆ 0·8188 2·2049 1·2969 3·0058 −1·2231 2·6350 6·6570 −0·1267 0·4202 3·1292 3·1093 3·9192 6·3625 2·3904 1·8815 2·0314 2·6529 2·3450 4·0313 1·5924 1·3895 1·8186

ˆ s(β) 0·3822 0·3860 0·3815 0·3898 0·4271 0·3879 0·5532 0·3967 0·3872 0·3935 0·3929 0·4175 0·5258 0·3868 0·3844 0·3852 0·3880 0·3866 0·4210 0·3830 0·3819 0·3841

ˆ βˆ − 2s(β) 0·0544 1·4330 0·5340 2·2263 −2·0772 1·8591 5·5507 −0·9201 −0·3542 2·3422 2·3235 3·0841 5·3108 1·6168 1·1126 1·2611 1·8769 1·5718 3·1892 0·8265 0·6256 1·0504

ˆ βˆ + 2s(β) 1·5833 2·9768 2·0598 3·7853 −0·3689 3·4108 7·7634 0·6667 1·1946 3·9162 3·8951 4·7543 7·4141 3·1640 2·6504 2·8018 3·4289 3·1183 4·8734 2·3583 2·1533 2·5868

Type 0 1 1 2 0 1 2 0 0 2 2 2 2 1 1 1 1 1 2 1 1 1

continued on following page

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4.4. DATA ANALYSIS continued from previous page

Stock Code NCM NCP NDY OSH PAS PBL PDP PLU PNI QNI QRL RIO RSG SEV SGB SRP STO TAH WBC WMC WOW WPL

βˆ 0·6649 0·8809 0·9427 2·7619 0·8373 2·2324 −0·0709 1·0940 3·3607 0·0424 0·2894 1·2305 0·0053 1·8940 3·2647 1·8503 3·2129 1·1087 3·5586 −0·1209 3·8585 2·3913

ˆ s(β) 0·3841 0·3814 0·3807 0·3885 0·3820 0·3861 0·3948 0·3804 0·4005 0·3919 0·3888 0·3811 0·3924 0·3845 0·3976 0·3843 0·3961 0·3805 0·4066 0·3965 0·4157 0·3868

ˆ βˆ − 2s(β) −0·1034 0·1180 0·1814 1·9849 0·0734 1·4602 −0·8606 0·3331 2·5596 −0·7415 −0·4883 0·4683 −0·7795 1·1250 2·4694 1·0817 2·4208 0·3477 2·7454 −0·9139 3·0271 1·6176

ˆ βˆ + 2s(β) 1·4331 1·6437 1·7040 3·5389 1·6013 3·0046 0·7188 1·8548 4·1618 0·8262 1·0670 1·9928 0·7901 2·6630 4·0599 2·6189 4·0050 1·8697 4·3717 0·6721 4·6898 3·1649

Type 0 0 0 1 0 1 0 0 2 0 0 0 0 1 2 1 2 0 2 0 2 1

The estimates of β range from -1.2231 for BHP, to 6.6570 for BOR7 . The table clearly shows that there are many stocks in the sample whose share price does not appear to follow GBM. In fact, at an approximate 95% level, 16 stocks have βˆ significantly less than 2, and 12 have βˆ significantly greater than 2, leaving 16 cases for which GBM cannot be ruled out as a plausible evolution model. The estimation of β depends on assumptions alluded to in the previous section, in particular, the quantity en defined in Equation (4.14) is required to be normal. For each of the share series listed in Table 4.4, I have computed the en values using the share price and daily return data. For each 7

To construct the confidence interval for the estimates of BOR and CSR, a further ˆ = 0.5850, and n = 924. simulation was run for β = 7, with β¯ = 7.1622, s(β)

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of the stocks, I have tested the hypothesis that the sample of en values has a standard normal distribution. Using the Chi-squared goodness of fit test, with 27 degrees of freedom, I obtain the p-values shown in Table 4.5. Stock Code AAA ANI BHP BOR BRY CCL CSR FXJ GMF LLC MIM NCM NDY PAS PDP PNI QRL RSG SGB STO WBC WOW

p-value 0·0040 0 0·2022 0·5043 0 0 0·0084 0·0010 0 0·0016 0·1158 0·0003 0·0011 0·0011 0 0·1122 0 0·0001 0·0008 0·0014 0·0070 0·0034

Stock Code AMC ANZ BIL BPC CBA CML FBG GIO ICI MAY NBH NCP OSH PBL PLU QNI RIO SEV SRP TAH WMC WPL

p-value 0·0032 0·3701 0·0004 0 0·1232 0·0149 0·0001 0·0002 0·0082 0·1913 0·1940 0·0085 0 0 0·0003 0·1590 0·1310 0·0005 0 0·0005 0·5169 0·2363

Table 4.5: p-values for the test of normality of en for the 44 ASX share series, where en is given by Equation (4.14). These p-values range from 0 to 0.5169, with 12 being larger than 10%, indicating that for these shares at least, the estimation approximation may be appropriate. It is interesting to note that the p-values are either very small, indicating clear rejection of the hypothesis that the en are normal, or quite large, where the null hypothesis cannot be rejected at any reasonable significance level.

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4.4. DATA ANALYSIS

Figure 4.6 shows the sample values en for the stock AMC, which has been randomly selected from the sample of stocks for which the test for normality was rejected. The p-value for this particular stock was 0.0032, and the graph helps to show why the null hypothesis of normality was rejected. The mean and variance of the sample are 0.0106 and 1.0014, which are very close to the required values of 0 and 1 respectively. The histogram shows some large positive values, and again, a sharper peak than the normal density function has (positive kurtosis). Removal of the outliers does not improve the Chi-squared statistic, indicating that the main problems are in the centre of the distribution. In particular the shape of the empirical distribution cannot

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The share series that do appear to have the desired normality properties belong to stocks: ANZ, BHP, BOR, CBA, MAY, MIM, NBH, PNI, QNI, RIO, WMC, and WPL. Of these twelve stocks, four belong to each of the

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three types, with estimates significantly less than 2, greater than 2, or not significantly different. Of these I have chosen BHP, BOR, and MIM for further analysis. Figures 4.7 to 4.9 are of the type seen before, with three graphs featuring in each. In particular these graphs show the share price series, the daily return series with an estimated mean level, and a plot of the estimated standard deviation of the daily returns against the share price level. This final graph is intended only to show the general relationship between share price level and volatility. As in Figures 3.2 and 3.3, superimposed on the daily returns are estimates of the mean level, and lines two estimated standard deviations from this mean. As before, both these averages were determined using the Lowess filter with a smoothing window of 30 days. Figure 4.7 gives the graphs for the BHP share price series. BHP has βˆ = −1.2231, and so we should see that the share price is more volatile when

the share price is low, and less volatile when it is high. The share price series shows a lot of movement in the share price, with periods of increasing

value and decreasing value. The daily returns seem to have a mean that is oscillating slowly about zero, and a standard deviation that is changing over time. In the scatterplot, we see that there appears to be a negative relationship between the share price level and the standard deviation of the daily returns. Figure 4.8 gives the same graphs for the MIM share price series. MIM has βˆ = 1.3895, a value that is not significantly different from 2, and hence is a process that could be GBM. The share price series appears similar to that of BHP, with periods of increase and decrease. Once again the daily returns have a mean level that appears to oscillate about zero, and again have a changing standard deviation. The difference between this series and that of BHP, is that the periods of low volatility do not correspond consistently to periods of high share price level, nor do the periods of high volatility correspond to periods of low share price level. This phenomenon is evident

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by comparing the first and second graphs for the series, but is best seen in the third graph. Here, the points seem to be scattered fairly randomly about the graph, with nothing to suggest an increasing or decreasing relationship, consistent with the estimation of β not significantly different from two. The final Figure 4.9 shows the three graphs for the BOR time series. BOR has βˆ = 6.6570 which indicates that periods of high volatility should correspond to high share price level, and that periods of low volatility should correspond to periods of low share price level. The share series in the first graph does not show the same oscillation as the other two series, but has a period of oscillation to begin with, and then a period of growth in the latter part of the series. Yet again, the daily returns have a mean that appears to oscillate about zero, and a changing standard deviation. The standard deviation of the daily returns is highest at the end of the series, consistent ˆ since this is where the share price is largest. Lower with the size of β, volatility is seen at other times. The third graph indicates that there is a positive relationship between share price level and the standard deviation of the daily returns. This and the relationship seen for the other two series, is probably not linear though, as the specification of the CEV model would suggest. The three share series BHP, MIM and BOR have properties that are consistent with the βˆ values estimated from them. In addition, the assumptions of the estimation procedure, in particular normality of the en are satisfied for these series. Thus, it appears that the estimation procedure is giving sensible estimates for these series at least. Further analysis of the ASX share data examined above could focus on the following issues: • Can the β estimation procedure be refined to: – incorporate the CEV density function and thus construct the exact likelihood function for a CEV process;

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CHAPTER 4. DATA ANALYSIS – identify the source of bias, and either eliminate it, or correct for it; – determine the asymptotic properties of the estimates?

• Does βˆ change significantly over the length of the series? • Using the estimates of β and δ, does the CEV model adequately describe market option prices?

• Does the CEV model provide better forecasts of future share prices than alternative models?

• Can implied estimates of β and δ be obtained from market option and share prices, in such a way that forecasts of future share price are improved?

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Figure 4.8: The MIM share series; in addition, the daily returns for this series with moving mean, and mean ± 2 standard deviation series; thirdly, a plot of the share price level against standard deviation of the daily returns.

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Figure 4.9: The BOR share series; in addition, the daily returns for this series with moving mean, and mean ± 2 standard deviation series; thirdly, a plot of the share price level against standard deviation of the daily returns.

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Chapter 5 Conclusions

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142

CHAPTER 5. CONCLUSIONS The CEV model has not been used extensively in the option pricing lit-

erature to describe observed option prices. Reasons for this may be that the option price solution is not an attractive one, and that there is difficulty in estimating the two parameters β and δ. In researching this thesis, I have attempted to gain an understanding of the CEV model, and examine its application to real share series. Furthermore, I have attempted to use this understanding to provide a more detailed analysis of the model than is found in the journal literature. I have described some of the statistical properties of CEV model share and option prices, and for the special case, the GBM and Black-Scholes models. The estimation of β using the method I describe is not altogether satisfactory, since the procedure appears to yield estimates that are biased. However, it does appear to give estimates that are sensible and conform to empirical observations. Moreover, share series are found to be consistent with each of the three CEV classes, corresponding to β < 2, β = 2, and β > 2. This thesis has raised many questions that remain to be answered, but has provided a platform for further research. I look forward to working further on the estimation of β, and analysing market option prices using the CEV model. Specific opportunities for extensions to this analysis are listed at the end of the previous chapter.

Appendix A Definitions Throughout this thesis, I have used the following notation consistently. Shown below is a brief description of the symbol, and a page reference to an appropriate definition. Other symbols used in the body of this document are redefined where used. Symbol β βˆ β¯ Bt Ct δ δˆ dBt dSt E ∗ (.) φ(q) Φ(q) Φ2 (x, y; ρ) f ∗ (.)

Description The CEV parameter; an estimate of the CEV parameter β; the sample mean of n estimates of β; the value of Brownian motion at time t; the value of a European call option (exercisable at T ) at t; CEV parameter; an estimate of the CEV parameter δ; Brownian motion increment over the period [t, t + dt]; share price increment over the period [t, t + dt]; expectation under risk-neutrality; the value of the standard normal probability density function at q; the value of the standard normal cumulative distribution function at q; the bivariate normal cumulative distribution function; a transition density function under risk-neutrality;

143

Page 48 113 125 11 3 48 113 7 7 51 22 22 36 81

144

APPENDIX A. DEFINITIONS

Symbol Description fXt+s |Yt (x, s) the (transition) density function for the random variable Xt+s given the value of another random variable Yt , evaluated at x, and with time interval s elapsed since t; g(x, ν) the gamma probability density function at x with shape parameter ν; G(x, ν) the gamma survivor function at x with shape parameter ν; gt a function that appears in the expected value of BlackScholes call prices; ht a function that appears in the Black-Scholes formula; ∗ ht a function that appears in the density function of a future Black-Scholes price; k˜ CEV variable, featuring in the general transition density function; k risk-neutral CEV variable, with µ = r; K the exercise price of a European call option; ) the likelihood function for unknown parameters θ L(θ | x ∼ ∼ ∼ ; given data x ∼

Page

58 58 23 30 32 55 82 3 116

) l(θ | x ∼

the log-likelihood function for unknown parameters θ ∼ given data x ; ∼

116

µ N (µ, σ 2 )

the mean rate of return on a stock; a normal random variable with mean µ and variance σ2; the density function of a non-central Chi-squared random variable, with ν degrees of freedom, and noncentrality parameter λ; the survivor function of a non-central Chi-squared random variable, with ν degrees of freedom, and noncentrality parameter λ; the continuously compounding risk-free rate; the variance of the rate of return on a stock following geometric Brownian motion; share price at time t; the standard deviation of the random variable X; the time until exercise of a European call option, T −t; present time, or the index of a time series; the exercise time of a European call option;

11

∼

p(x; ν, λ)

Q(x; ν, λ)

r σ2 St s(X) τ t T

71

73

3 11 3 3 3

145 Symbol x˜ x z˜ z Z

Description CEV variable, featuring in the general transition density function risk-neutral CEV variable, with µ = r; CEV variable, featuring in the general transition density function; risk-neutral CEV variable; the standard normal random variable, with zero mean, and unit variance.

Page 55 82 55 82 11

146

APPENDIX A. DEFINITIONS

Appendix B Proofs for Selected Results B.1

Result 2.2

Let the share price St be the solution to the SDE: dSt = µSt dt + σt St dBt where σt is a deterministic function of time. Then the price of a call option on a stock with share price St must satisfy the PDE: C¯ ∂ C¯ ∂ C¯ + rS + − rC¯t = 0 t ∂S 2 ∂S ∂t

1 2 2∂ σ S 2 t t

2

with boundary condition C¯T = (St − K)+ , and the call price C¯t is given by

Black-Scholes equation, with the substitution: Z 1 T 2 2 2 σ =σ ¯ ≡ σ du. τ t u

Proof. The form of the PDE follows from Equation (2.18), with σ(St , t) = σt . The Black-Scholes function, with 1 σ =σ ¯ ≡ τ 2

2

Z

T t

σu2 du

can be written: C¯t = C(St , τ, σ ¯) 147

(B.1)

148

APPENDIX B. PROOFS FOR SELECTED RESULTS

and this will satisfy the PDE: C¯ ∂ C¯ ∂ C¯ + rS + − rC¯t = 0. t ∂S 2 ∂S ∂t

1 2 2∂ σ S 2 t t

2

These partial derivatives are given by: ¯ ∂C ¯¯ ∂ C¯ = ∂S ∂S ¯σ=¯σ and

(B.2)

¯ ∂ 2 C ¯¯ ∂ 2 C¯ = ∂S 2 ∂S 2 ¯σ=¯σ

(B.3)

¯ ¯ ∂C ∂ σ ∂C ¯¯ ¯ ¯¯ ∂ C¯ + = ∂t ∂t ¯σ=¯σ ∂ σ ¯ ∂t ¯σ=¯σ

where σ is given by Equation (B.1). Now ¯ ¯ 2 ¯ √ ¯ ∂ C ∂C 2 ¯ = St φ(ht ) τ ¯¯ = τ σSt ∂σ ¯ ∂S 2 ¯σ=¯σ σ=¯ σ

since

∂2C ∂S 2

=

φ(ht ) √ . Sσ τ

In addition:

1 ∂ ∂σ ¯ = ∂t 2¯ σ ∂t

ÃR T

σu2 du t T −t

!

=

σ ¯ 2 − σt2 2τ σ ¯

yielding ¯ ¯ 2 ¯ ∂ C¯ ∂C ¯¯ 2 2 2∂ C¯ 1 σ − σt )St = + (¯ ∂t ∂t ¯σ=¯σ 2 ∂S 2 ¯σ=¯σ

(B.4)

and so substituting the partial derivatives given in Equations (B.3) and (B.4) into the PDE (B.2) gives: ∂ 2 C¯ ∂ C¯ ∂ C¯ + rS + − rC¯t t ∂S 2 ∂S ∂t ∂2C ∂2C ∂C ∂C 1 2 + + 2 (¯ σ − σt2 )St2 2 − rCt = 12 σt2 St2 2 + rSt ∂S ∂S ∂t ∂S ¯ 2 ¯ ∂ C ∂C ∂C + − rCt ¯¯ = 21 σ 2 St2 2 + rSt ∂S ∂S ∂t σ=¯ σ =0

LHS = 21 σt2 St2

by Equation (2.19) as required.

¯ ¯ ¯ ¯

σ=¯ σ

149

B.2. RESULT 3.2

B.2

Result 3.2 G(y, ν − 1) = 1 −

∞ X

g(y, n + ν)

n=0

where G(y, ν) and g(y, ν) are the survivor and probability density functions at y for a gamma random variable with shape parameter ν and unit scale parameter. Proof. Using techniques employed by Schroder (1989), the gamma survivor function G(y, ν) may be written in terms of a recurrence relation, found using integration by parts: G(y, ν) =

Z

∞

g(x, ν)dx

y ∞

e−x xν−1 dx Γ(ν) y ¯∞ Z ∞ −x −e−x xν−1 ¯¯ e (ν − 1)xν−2 dx = + Γ(ν) ¯y (ν − 1)Γ(ν − 1) y Z ∞ −x ν−2 e−y y ν−1 e x = + dx Γ(ν) Γ(ν − 1) y =

Z

= g(y, ν) + G(y, ν − 1).

(B.5)

Applying this relationship n + 1 times to G(y, n + ν), we obtain: G(y, n + ν) = g(y, n + ν) + G(y, n − 1 + ν) = g(y, n + ν) + g(y, n − 1 + ν) + G(y, n − 2 + ν) .. . n X g(y, k + ν) + G(y, ν − 1). =

(B.6)

k=0

Hence, letting n → ∞, we obtain: ∞ X n=0

g(y, n + ν) + G(y, ν − 1) = lim G(y, n + ν) n→∞

= 1 − lim P (Yn+ν < y) n→∞

=1

(B.7)

150


where Yn+ν is a gamma random variable with parameter n+ν, and the result follows.

B.3

Result 3.4 Z

∞ y

2 p(2x; 2ν + 2, 2λ)dλ = 1 −

Z

∞

2 p(2z; 2ν, 2y)dz

x

where p(y; ν, λ) and Q(y; ν, λ) are the probability density and survivor functions at y for a non-central Chi-squared random variable with ν degrees of freedom and non-centrality parameter λ. Proof. This proof follows the method of Schroder (1989). Note firstly that there are two alternative formulae for call price Ct given in Equation (3.34): ³ ´ ∞ e−x xn G kK 2−β , n + 1 + 1 X 2−β Ct = St Γ(n + 1) n=0 ¡ ¢ 1 ∞ X e−x xn+ 2−β G kK 2−β , n + 1 −rτ − Ke 1 Γ(n + 1 + 2−β ) n=0 and Equation (3.43) Z ∞ 2 p(2z; 2 + Ct = St y

2 , 2x)dz 2−β

− Ke

−rτ

Z

∞

2 p(2x; 2 +

y

These equations allow us to note directly that: Z ∞ ∞ X 2 g(x, n + 1)G(y, n + 1 + 2 p(2z; 2 + 2−β , 2x)dz = y

Z

∞

y

2 p(2x; 2 +

2 , 2z)dz 2−β

=

n=0 ∞ X

g(x, n + 1 +

1 )G(y, n 2−β

2 , 2z)dz. 2−β

1 ) 2−β

(B.8)

+ 1).

(B.9)

n=0

A result from the proof in Appendix B.2 is: G(y, n + 1) =

n X

g(y, i + 1) =

i=0

which follows since G(y, 1) = g(y, 1).

n X i=1

g(y, i)

(B.10)

151

B.3. RESULT 3.4 Letting ν = 1 + Z

∞

1 , 2−β

Equation (B.8) yields:

2 p(2z; 2ν, 2y)dz =

x

∞ X

g(y, n + 1)G(x, n + ν)

n=0

and so developing the right hand side of this expression: RHS = = =

∞ X

n=0 ∞ X

n=0 ∞ X n=0

g(y, n + 1)G(x, n + ν) Ã

g(y, n + 1) G(x, ν − 1) + Ã

g(y, n + 1) 1 −

∞ X

n X

g(x, k + ν)

k=0

g(x, k + ν)

k=n+1

!

!

where the last two equalities follow from Equations (B.6) and (B.7) respectively. Noting that: ∞ X

g(y, i) =

i=1

∞ ∞ X X yi e−y y i = e−y = e−y ey = 1 Γ(i + 1) i! i=0 i=0

we can expand the final equation above, and change the order of summation to yield: Z

∞ x

2 p(2z; 2ν, 2y)dz = 1 − =1− =1−

∞ ∞ X X

g(y, n + 1) g(x, k + ν)

n=0 k=n+1

∞ X k−1 X

g(y, n + 1) g(x, k + ν)

k=1 n=0 ∞ X

g(x, k + ν + 1)

k X

g(y, n)

n=1

k=0

Now noting Equation (B.10), the summation in the final equation can be written: ∞ X k=0

g(x, k + ν + 1)

k X

g(y, n) =

n=1

=

∞ X

g(x, k + ν + 1)G(y, k + ν)

Zk=0∞ y

2 p(2x; 2 +

2 , 2z)dz 2−β

152


by Equation (B.9) and making the substitution ν = Z

∞ y

2 p(2x; 2ν + 2, 2λ)dλ = 1 −

Z

∞

1 . 2−β

Hence the result

2 p(2z; 2ν, 2y)dz

(B.11)

x

is proved.

B.4

Result 3.5

The option price when β > 2 is given by the formula: 2 , 2y) − Ke−rτ (1 − Q(2y; 2 + Ct = St Q(2x; β−2

2 , 2x)) β−2

where Q(y; ν, λ) is the survivor function at y, for a non-central Chi-squared variable with ν degrees of freedom, and non-centrality parameter λ. Proof. This proof follows the derivation of Schroder’s pricing formula for the standard CEV case, where β < 2. The risk-neutral density function given in the first part of this result is: √ 1 1 fS∗T |St (s, τ |β > 2) = (β − 2)k 1/(2−β) (xz 1−2β ) 2 2−β e−x−z I1/(β−2) (2 xz). Emanuel and MacBeth point out that this function has a singularity at s = ∞

and so any integration should be over a range excluding ∞. Hence, they write the call price at t as the share price at t, less the discounted exercise price,

plus the present value of the amount saved by letting the option lapse when ST < K, which follows since: ST − K + (K − ST )+ = ST − K +

(

= (ST − K)+

0 K − ST

K < ST K ≥ ST

as required. The present value of the saving (K − ST )+ is: e

−rτ

Z

K s=0

(K − s)fS∗T |St (s, τ )ds

153

B.4. RESULT 3.5 and so the option price is given by: Ct = St − Ke

−rτ

+e

−rτ

Z

K s=0

(K − s)fS∗T |St (s, τ )ds.

Note that z is unchanged from its previous definition, and so using ds = k −1/(2−β)

1 z −1+1/(2−β) dz 2−β

we can write: 1

1

fS∗T |St (s, τ )ds = −(z/x)− 2 2−β e−x−z I

1 β−2

√ (2 xz)dz.

Noting that when s = 0 then z = ∞, and when s = K then z = kK 2−β = y, we find in this case the transformation is a negative one, and so the option

price is: Ct = St − Ke

−rτ

−e

−rτ

Z

∞ y

¶³ ´ µ³ ´ 1 √ z −1/(4−2β) −x−z z 2−β −K e I 1 (2 xz)dz. β−2 k x

In the proof of Theorem 3.8, we saw e−rτ (x/k)1/(2−β) = St which leads to yet another representation of the formula for Ct : ¶ µ Z ∞ √ −x−z 1/(2β−4) Ct = St 1 − e (x/z) I1/(β−2) (2 xz)dz y µ ¶ Z ∞ √ −x−z 1/(2β−4) −rτ e (z/x) I1/(β−2) (2 xz)dz . 1− − Ke y

The form of the non-central Chi-squared probability density function, p(x; ν, λ) is given in Equation (3.23), and comparison of the above integrands with this function yields the formula: ¶ µ Z ∞ 2 Ct = St 1 − 2 p(2x; 2 + β−2 , 2z)dz y µ Z ∞ −rτ 1− 2 p(2z; 2 + − Ke y

2 , 2x)dz β−2

¶

.

154


Using Result 3.4, and making the simple transformations seen in the proof of Theorem 3.9, yields the option pricing formula given by Schroder: 2 Ct = St Q(2x; β−2 , 2y) − Ke−rτ (1 − Q(2y; 2 +

as required.

2 , 2x)) β−2

Appendix C Complete List of Shares Table C.1: ASX Company Codes Stock Code AAA AMC ANI ANZ BHP BIL BOR BPC BRY CBA CCL CML CSR FBG FXJ GIO GMF ICI LLC MAY MIM NBH NCM

Company Name Acacia Resources Limited Amcor Limited Australian National Industries Limited Australia & New Zealand Banking Group Limited The Broken Hill Proprietory Company Limited Brambles Industries Limited Boral Limited Burns, Philp & Company Limited Brierley Investments Limited Commonwealth Bank of Australia Coca-Cola Amatil Limited Coles Myer Limited CSR Limited Foster’s Brewing Group Limited Fairfax (John) Holdings Limited GIO Australia Holdings Limited Goodman Fielder Limited ICI Australia Limited Lend Lease Corporation Limited Mayne Nickless Limited M.I.M. Holdings Limited North Limited Newcrest Mining Limited continued on following page

155

156

APPENDIX C. COMPLETE LIST OF SHARES continued from previous page

Stock Code NCP NDY OSH PAS PBL PDP PLU PNI QNI QRL RIO RSG SEV SGB SRP STO TAH WBC WMC WOW WPL

Company Name The News Corporation Limited Normandy Mining Limited Oil Search Limited Pasminco Limited Publishing & Broadcasting Limited Pacific Dunlop Limited Plutonic Resources Limited Pioneer International Limited QNI Limited QCT Resources Limited Rio Tinto Limited Resolute Limited Seven Network Limited St. George Bank Limited Southcorp Limited Santos Limited Tabcorp Holdings Limited Westpac Banking Corporation WMC Limited Woolworths Limited Woodside Petroleum Limited

Appendix D SPLUS Code D.1

GBM Simulation

The following program is an appropriate way of simulating a geometric Brownian motion series in SPLUS. It follows from Equation (2.8), and takes arguments St , τ , µ, σ, and n the number of subintervals in [t, T ]. It returns share with n + 1 elements, which is a realisation of the GBM process over the time period [t, T ]. gbm.f

The Constant Elasticity of Variance Option Pricing

The Constant Elasticity of Variance Option Pricing

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