Merton-Style Option Pricing under Regime Switching - CiteSeerX

Merton-Style Option Pricing under Regime Switching John Driffill Birkbeck College, University of London Turalay Kenc Imperial College, University of London Martin Sola Birkbeck College, University of London, Universidad Torcuato Di Tella October 1, 2002

Merton-Style Option Pricing under Regime Switching

Abstract This paper develops a valuation framework for a perpetual American call option when the underlying asset return dynamic is modelled by a regime switching process. In particular, asset return dynamic is governed by a stochastic dividend process which randomly switches between two regimes that are characterized by different rates of both drift and volatility. This regime-switching characterization of dividend growth is supported by empirical works. We provide analytical results by solving the fundamental differential equation. The analysis reveals that the option value may differ between states. Our empirical application shows that these differences may be quantitatively very important. Keywords: Option pricing, regime-switching. JEL numbers: C51, G12, G13.

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Introduction

This paper develops a valuation framework for a perpetual American call option when the underlying asset return dynamic is modelled by a Markov regime switching process.

In our case, the stochastic process randomly

switches between two regimes which are characterized by different rates of both drift and volatility. This characterization then enables us to capture discrete shifts in the stock prices as well as continuous changes as in the standard contingent claims valuation model - the Black-Scholes (1973) model. This paper is motivated by the fact that existing option pricing models, surveyed by Bakshi, Cao and Chen (1997), do not empirically perform well. The reason for this centers on the unrealistic assumptions made about the underlying price process (the distribution assumption). For instance, the benchmark (Black-Scholes) model assumes a diffusion process whose drift and volatility parameters are constant. Since then there have been several improvements on the benchmark model. Early models [such as Merton (1976) and Heston (1993)] allowed for jump diffusion processes to capture discontinuous changes in the underlying asset price and/or stochastic volatility to relax the constancy of the volatility parameter. Recent models [such as Duan (1995)] adopted a more realistic assumption – dynamics for the price of the underlying asset follows a General Autoregressive Conditionally Heteroskedastic (GARCH) process. The realism of this assumption comes from 1

the evidence on high-frequency data which reveals that stock returns are characterized very often by periods of high and low volatility, with the well known occurrence of “volatility clustering”, the tendency of one large shock to be followed by more large shocks. However, time series data over long periods of time, like a century, typically sampled at lower frequency, tends to show not only periods of low and high volatility but also periods of slower and faster mean growth. For such data GARCH process proves to be inappropriate. Instead, a more successful formulation is provided by James Hamilton’s (1989) model of stochastic regime switching. When converted into a continuous-time, his model implies that the underlying asset price switches occasionally between two states as well as exhibits continuous changes in each state. The shifts are governed by a Markov (point) process typically with constant transition probabilities. On the other hand, continuous changes in each state are governed by diffusion processes, i.e., each state has its own drift and volatility. The underlying asset may, for example, switch between a high growth/low volatility state and a low growth/high volatility state – or some other combination of the possibilities. Note that when diffusion process is combined with Markov chain (point process) leads to a Hidden Markov model which is topical in mathematics [see Elliott, Aggoun, and Moore (1997)]. In the traded options literature this regime switching idea has already been modelled by Naik (1993); Bollen

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(1998); Bollen, Gray and Whalley (2000); Chourdakis and Tzavalis (2000); and Duan, Popova and Ritchken (1999). Unlike the above papers this paper explicitly model dividends as a regime switching process in response to the findings of Cecchetti, Lam, and Mark (1990), Bonomo and Garcia (1994) and Driffill and Sola (1998). They found that dividends are well modelled by a two-state Markov-switching model, and that this can go a long way towards explaining the persistent deviation of US stock prices from those predicted by models which do not take account of regime-switching. To value options written on assets whose prices are governed by regime-switching dividends we solve the fundamental (second-order) differential equation(s). Our valuation approach differs from the valuation methods adopted in the above papers which mainly use lattice approximation or Monte Carlo simulation in providing analytical results. However, it does not come without costs: the analysis has to be restricted to pricing Merton-style options [see Merton (1973)]. In particular, we consider a perpetual American call option. In this respect, our approach is more useful to pricing real options while it provides a benchmark case for traded options. To sum up the Markov regime-switching formulation allows the derivation of analytical results on option pricing, taking account of the fact that participants in financial markets not only either observe or infer the current state of the system but also make predictions about future regime switches.

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The layout of this paper is as follows. In section 2 we develop a model of dividends and stock prices that allows for regime-switching in dividends and therefore also in stock prices. In section 3 we pursue its implications for the pricing of a perpetual American call option and we derive a closed form solution for the price of this option. Section 4 applies the theory to roughly a century of annual data on the New York Stock Market composite index for which a model for dividends and stock prices is estimated, and the option is priced. Section 5 explores the implications of the model further by sensitivity analysis, looking at the effects of changes in key parameters on the option value and exercise price. Section 6 concludes the paper.

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Models of Stock Prices and Dividends

We assume that the flow of dividends D generated by a stock (or an index of stocks) moves unpredictably over time. It is customary in problems of this kind to assume that the flow of returns D is driven by a diffusion process with rate of drift µ and instantaneous variance σ 2 . We follow this practice, with the difference that it is assumed here that there are two states of the world, 0 and 1, between which it switches from time to time, governed by a Markov process. In state 0 the drift in the return process is µ0 and the variance is σ02 , while in state 1 the corresponding values are µ1 and σ12 . The

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probability of shifting from state 0 to state 1 in an interval of time of length dt is θdt, and the probability of switching from state 1 to state 0 is φdt. The stochastic process for D is thus represented by:

dD = µi Ddt + σi Ddw

f or i = 0, 1

(1)

S, the value of the stock (or the index, in the case that we a dealing with an index here), is the present discounted value of the stream of returns D using the constant discount rate ρ. Then S is calculated using ρSdt = Ddt + Et (dS) The value of the stock in each state of the world will be a function of the profit flow D at the time. In state 0 the value can be written as S0 (D) and in state 1 as S1 (D). If the world is in state 0 at time t, at time t + dt it will be in state 0 with probability (1 − θdt) and in state 1 with probability θdt. The expected present value of the stock at t + dt will then be

Et (St+dt ) = (1 − θdt)Et (S0 (Dt+dt )) + θdtEt (S1 (Dt+dt ))

(2)

Then the expected change at time t in the value of the stock between t and

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t + dt is Et (dS) = Et (St+dt − St ) = (1 − θdt)Et (S0 (Dt+dt )) + θdtEt (S1 (Dt+dt )) − Et (S0 (Dt )) = Et (S0 (Dt+dt ) − S0 (Dt )) + θdtEt (S1 (Dt+dt ) − S0 (Dt+dt )) (3) Applying Ito’s Lemma to this last expression, we obtain: µ ¶ σ02 2 Et (dS) = µ0 S0D D + S0DD D + θ[S0 (D) − S1 (D)] dt 2 Thus, conditional on being on state 0, the evolution of the value of the project over time satisfies the following differential equation: ρS0 (D) = D + µ0 S0D D +

σ02 S0DD D2 + θ[S1 (D) − S0 (D)] 2

Analogously, conditional on being in state 1: ρS1 (D) = D + µ1 S1D D +

σ12 S1DD D2 + φ[S0 (D) − S1 (D)] 2

The general solutions to these equations consist of two parts, the solution to the characteristic function, and the particular integral. The solutions to the characteristic function are identified with bubbles in the asset price, and have to be equal to zero if the asset price is to equal the present value of future returns. The particular integrals, which equal the present value of future returns, are

S0 = k0 D,

S1 = k1 D 6

for k0 and k1 satisfying

ρk0 = 1 + k0 µ0 + θ[k1 − k0 ]

ρk1 = 1 + k1 µ1 + φ[k0 − k1 ] i.e., k0 =

ρ+φ+θ−µ1 (ρ+θ−µ0 )(ρ+φ−µ1 )−φθ

and k1 =

ρ+φ+θ−µ0 . (ρ+θ−µ0 )(ρ+φ−µ1 )−φθ

From the foregoing analysis it is clear that the stock price S follows geometric Brownian motion with drift, with different drift and variance in each of the two states. W have in fact dS = µi Sdt + σi Sdw

f or i = 0, 1

but note that when there is a change of state, the stock price jumps from k0 D to k1 D or vice versa.

The dividend process is continuous, but stock

prices are discontinuous at times of regime-switch.

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Valuing a Perpetual American Call Option with Markov Switching Stock Prices.

Having derived the model for stock prices, we are now in a position to consider the pricing of a perpetual American call option on the stock. The call option

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has a strike price K. The value of the option may be written as a function of the stock price S in state i as

c = ci (Si )

(4)

which allows for the option value being a different function of the stock price in each state. The value of the option at time t + dt given that the economy was in state 0 at time t is

Et (ct+dt ) = (1 − θdt)Et ( c0 (S0t+dt )) + θdt Et (c1 (S1t+dt ))

(5)

while the value of the option at time t + dt given that the economy was in state 1 at time t is

Et (ct+dt ) = φdt Et (c0 (S0t+dt )) + (1 − φdt) Et (c1 (S1t+dt ))

(6)

Then the change in the value of the option given that the economy is in state 0 at time t is Et (dc) = Et (ct+dt − ct ) = (1 − θdt)Et ( c0 (S0t+dt )) + θdt Et (c1 (S1t+dt ) − c0 (S0t )) =

Et (c0 (S0t+dt ) − c0 (S0t )) + θdt Et ((c1 (S1t+dt ) − c0 (S0t+dt )))

Applying Ito’s Lemma in order to evaluate (c0 (S0t+dt ) − c0 (S0t )) yields

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the following expression under state 0 at time t µ dc =

¶ σ02 2 µ0 c0S S0 + c0SS S0 + θ[c1 (S1 ) − c0 (S0 )] dt + σ0 c0S S0 dz 2

in which c0S ≡ ∂c0 (S)/∂S, c0SS ≡ ∂ 2 c0 (S)/∂S 2 , and so on. The option is priced by the widely used method of constructing an artificial portfolio whose returns are riskless, and which therefore has to earn the riskless rate of return in equilibrium. The portfolio consists of holding 1 unit of the option and −n units of the stock. Then conditional on being on state 0 at time t the change in the value of the dynamic portfolio is µ dc − ndS =

¶ σ02 2 µ0 c0S S0 + c0SS S0 + θ[c1 (S1 ) − c0 (S0 )] − nµ0 S0 dt 2 + (σ0 c0S S0 − nσ0 S0 ) dz. (7)

Analogously, conditional on being in state 1 at time t, the change in the value of the dynamic portfolio is µ dc − ndS =

¶ σ12 2 µ1 c1S S1 + c1SS S1 + φ[c0 (S0 ) − c1 (S1 )] − nµ1 S1 dt 2 + (σ1 c1S S1 − nσ1 S1 ) dz. (8)

The proportions of stock to options are adjusted to make the portfolios riskless.

In state 0 we choose n0 = c0S , while in state 1 we choose n1 =

c1S .Evidently the risk free portfolios contain different proportions of stock and options according to the state of nature. As S changes, and therefore 9

since c0S may change also, the proportions of assets in the portfolio will in general be continuously adjusted so as to keep it riskless. An investor holding a long position in this portfolio will require a risk adjusted return equal to the capital gain plus the stream of dividends n0 δ0 S0 in state zero and n1 δ1 S1 in state one, where δ0 and δ1 are the dividends payable as a fraction of the stock price in each state. (We define δ0 = 1 , k0

δ1 =

1 . k1

Note that k0 and k1 were derived in the previous section.)

Taking this payment into account the total return on the portfolio over a period of time dt, given that the economy was in state 0 at time t, is dc − c0S dS − δ0 c0S S0 dt which satisfies the relation µ 2 ¶ σ0 2 dc − c0S dS − δ0 c0S S0 dt = c0SS S0 + θ(c1 (S1 ) − c0 (S0 )) − δ0 c0S S0 dt 2 Using the standard hedging argument, we note that in each state the risk free portfolio should earn the risk free return r, and therefore the the return on it satisfies the relation dc − c0S dS − δ0 c0S S0 dt = r [c0 (S0 ) − c0S S0 ] dt, Then, by equating the right hand sides of the two foregoing equations, we obtain the two following differential equations: σ02 c0SS S02 + (r − δ0 )c0S S0 − rc0 (S0 ) + θ[c1 (S1 ) − c0 (S0 )] = 0 2

(9a)

for the case when the economy is in state 0 at time t, and σ12 c1SS S12 + (r − δ0 )c1S S1 − rc1 (S1 ) + φ[c0 (S0 ) − c1 (S1 )] = 0 2 10

(9b)

for the other case, i.e., the economy is in state 1 at time t. The trial solutions take the form c0 (S0 ) = AS0λ

and c1 (S1 ) = BS1λ

(10)

Substituting these solutions in equation (10) into equation (9) we obtain the following relationships [(r + θ) − (r − δ0 )λ −

σ02 k0 λ(λ − 1)]A( )λ = θB 2 k1

(11a)

[(r + φ) − (r − δ1 )λ −

k1 σ12 λ(λ − 1)]B( )λ = φA 2 k0

(11b)

and

which gives four solutions for λ, with a pair of values for each A and B. Ai /Bi is fixed for each λi , but not the absolute values. The λs satisfy [(r + θ) − (r − δ0 )λ −

σ02 σ2 λ(λ − 1)][(r + φ) − (r − δ1 )λ − 1 λ(λ − 1)] = θφ. 2 2

This fourth order polynomial equation in λ has four distinct real roots, two positive and two negative1 . We assign the four roots as λ1 > λ2 > 0 > λ3 > 1

The reason is as follows. Each of the terms in square brackets on the left hand side of

the equation takes the value zero for one positive and one negative value of λ, providing (ρ + θ) > 0 and (ρ + φ) > 0. The LHS is increasing in λ for sufficiently large λ, and goes to infinity as λ goes to infinity. For sufficiently small λ the LHS goes to infinity as λ goes to minus infinity. At λ = 0, the LHS takes the value (ρ + θ)(ρ + φ) which exceeds θφ. Consequently there are two positive and two negative values of λ which satisfy the equation. It can be shown that the two positive roots exceed one.

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λ4 . The solutions for c0 and c1 take the general form c0 (S0 ) = Σ4i=1 Ai (S0 )λi and c1 (S1 ) = Σ4i=1 Bi (S1 )λi . In order to determine the values of the constants Ai and Bi , a number of terminal conditions have to be applied. The first pair of these conditions sets the value of the option to zero when the stock price S takes the value zero. Viz., c0 (0) = 0 and c1 (0) = 0. These conditions arise from the fact that when S is very small the prospect of it rising to the exercise price is quite remote and then the value to the option should be near zero. This implies that A3 , A4 , B3 and B4 should be zero. Then the value of the option then becomes c0 (S0 ) = Σ2i=1 Ai S0λi c1 (S1 ) = Σ2i=1 Bi S1λi It remains to impose the value matching and smooth pasting conditions. At the critical price S i at which the option is exercised, the value of the stock must equal the value of the option ci (S i ) plus the strike price K, thus:

ci (S i ) = S i − K At the price S i , smooth-pasting conditions hold which require that the slope option-value functions ci (S i ) must equal the slope of the stock value function (Si ) and thus have a slope equal to 1: ciS (S i ) = 1. 12

When these conditions are imposed, the solution is characterized by the following system of equations: λ1

λ2

λ1

λ2

A1 S 0 + A2 S 0 = S 0 − K B1 S 1 + B2 S 1 = S 1 − K λ1 −1

λ1 A1 S 0

λ1 −1

λ 1 B1 S 1

λ2 −1

+ λ2 A2 S 0

λ2 −1

+ λ 2 B2 S 1

=1 =1

This is a system of 4 equations in 4 unknowns (A1 , A2 , S 0 , S 1 ) which can be solved for given Bi . From equations (11) Bi can then be solved as follows: [(r + θ) − (r − δ0 )λi − Bi = θ

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σ02 λ (λ 2 i i

− 1)]

Ai (

k0 λi ) , i = 0, 1. k1

An illustrative example

In order to illustrate the implications of the Markov-switching model of stock prices for the value of a perpetual call option, we obtain estimates of the foregoing model using annual data for the NYSE composite Index from 1990 to 1997. The data are obtained from Shiller, 19892 with additional data 2

These data on US stock prices and dividends are listed in Shiller (1989), chapter 26,

where more details can be found. The stock prices are January values for the Standard and Poor Composite Stock Price Index. Each observation in the dividend series is an average for the year in question. Nominal stock prices and dividends are deflated by the

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obtained from Robert Shiller to extend the sample to 1997. Several authors, including Cecchetti, Lam, and Mark (1990), Bonomo and Garcia (1994) and Driffill and Sola (1998), have found that dividends are well modelled by a two-state Markov-switching model, and indeed that this can go a long way towards explaining the persistent deviation of US stock prices from those predicted by models which do not take account of regime-switching. For empirical analysis, we use a discrete-time version of the regimeswitching model developed in section 2. The first difference of the log of real dividends is denoted by ∆dt . The state of the economy at each point in time is denoted by xt which takes values {0, 1}. When xt = 0, ∆dt is assumed to be distributed N (α0 , (σ0d )2 ); and when xt = 1 it is distributed N (α1 , (σ1d )2 ). The states are assumed to follow a first order homogeneous Markov process with prob(xt = 0|xt−1 = 0) = p and prob(xt = 1|xt−1 = 1) = q.

The

evolution of the logarithm of real dividends can therefore be written as dt+1 = dt + α0 (1 − xt ) + α1 xt + (σ0d (1 − xt ) + σ1d xt )εt+1

(12)

where εt+1 is an i.i.d. standard normal variable. The stock price is assumed to be related to the dividend by St = Et e−ρ [St+1 + Dt ] producer price index (Shiller’s series 6 of prices for January each year) to get real stock prices and dividends.

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in which dt = log(Dt ). The fundamental solution for the stock price is St = k0 Dt when xt = 0, and St = k1 Dt when xt = 1. Note that E(Dt+1 |Dt , xt = 0) = Dt a0 E(Dt+1 |Dt , xt = 1) = Dt a1 1

1

2

2

where a0 = e(α0 + 2 σ0 ) and a1 = e(α1 + 2 σ1 ) . Then k0 and k1 satisfy k0 = e−ρ [1 + pk0 a0 + (1 − p)k1 a1 ] , −ρ

k1 = e

(13)

[1 + qk1 a1 + (1 − q)k0 a0 ] .

For the purpose of estimation, an error term σiP ut is introduced into the theoretically implied relationship between stock prices and dividends (Si /Di = ki ) in order to accommodate random departures from it which may be due to measurement errors etc. (Froot and Obstfeld, 1991).

The

estimated relationships are therefore St = ki + σiP ut Dt

in state i, f or i = 0, 1.

(14)

dt = αi + dt−1 + σi vt where ut and vt are i.i.d. standard normal variables while σiP and σi are positive constants.3 The estimation procedure assumes that agents in the financial markets know the actual state of the system, xt , at each point in time, whereas 3

Notice that we have augmented the model with random disturbances which may be

interpreted as measurement errors.

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the econometrician does not, and has to make inferences of it based on the observable history of the system, i.e., the information contained in the history of stock prices and dividends. We estimate the model subject to the theoretical restrictions on k0 , k1 implied by (13).4 The estimation procedure is described briefly in Appendix 1. The constant discount factor is chosen as the geometrical average ρ = 0.0675. The estimates are set out in Table 1. State 0 is a low growth/high volatility state, with the growth rate of dividends α b0 = −0.004 and the standard 4

To carry out specification tests for this model we compute the residuals as prt −

b where prt = (Pt /Dt ) and ψb are estimates of the parameter vector, E(prt |Ωt : ψ), ψ = (µ0 , µ1 , σ0 , σ1 , σ0P , σ1P , p, q). The conditional expectation of prt is constructed by multiplying the probabilities of the states obtained from the filter by the theoretically predicted values of the price-dividend ratio in each state. Based on the sample estimates b the predicted values of prt can be written as ψ, b = P (st = 0|It )(b E(prt |It : ψ) k0 ) + P (st = 1|It )(b k1 ). We then standardize the residuals by dividing them by the conditional standard deviation. Tests for AR (Godfrey-Breusch) and ARCH (Engle) errors are then performed. In Table 1 we show heteroskedasticity and autocorrelation-consistent standard errors. ML estimation was carried out using a variable-metric algorithm that approximates the Hessian according to the BFGS update. The pre-whitened quadratic spectral kernel with datadependent bandwidth discussed in Andrews (1991) and Andrews and Monahan (1992) was used for the covariance matrix estimator.

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error σ b0 = 0.15. The implied price/dividend ratio is b k0 = 19.1. State 1 is high growth/low variance. We have α b1 = 0.04 and σ b1 = 0.04. The implied price/dividend ratio is b k1 = 30.1. The probability of staying in each state per period is pb = 0.987 for state 0 and qb = 0.984 for state 1. For the no-switching model the corresponding estimates are α b = 0.018, σ b = 0.13, and b k = 22.8. The test statistics for the regime-switching model show that while there remains evidence of autocorrelation in the errors – on the basis of tests for 1st order and 4th order autocorrelation – the errors appear to be purged of ARCH effects, having allowed for regime-switching. The autocorrelation seems to result from rapid growth in stock prices in the last few years of the sample which produces a string of growing positive errors (as Figure 1a shows). By comparison with the results for the regime-switching model, the results reported in the right hand column of Table 1, for a model with no regime-switching, show the presence of strong ARCH effects (the consequence of not allowing for regime-switching) and also bigger test statistics for AR(1) and AR(4) errors. The plot of the data and estimates in Figure 1a shows how very much better the regime-switching model fits than does the model without regime-switching. The underestimate of stock prices in the 1990s is very clear. Figure 1b shows the filtered probabilities of having been in each state during the sample period. For most of the period, the data indicate that the

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economy was in state 0. The episodes in state 1 are between approximately 1958 and 1973 and from 1987 onwards. The first episode corresponds well with the postwar “Golden Age”, the second with the revival of growth in the late 1980s and the United States’ long sustained growth under Greenspan’s tenure at the Fed – the new Golden Age. The estimated parameters are used to price a notional option on the stock index. To produce parameters corresponding to the theoretical continuoustime model in section above from the discrete-time empirical estimates small modifications are made.

The drift of dividends in the continuous-time

process (b µi in state i) is set at µ bi = α bi + σ bi2 /2. The probabilities of leaving states 0 and 1 in a time interval dt, which are denoted by θ and φ respectively in the continuous time model are set equal to θ = (1− pb)/b p and φ = (1− qb)/b q. Otherwise the estimated parameters from the discrete-time model carry over to the continuous-time model. The implied values for the rate of (log real) dividend growth are µ b0 = 0.00756, µ b1 = 0.0410, and in the no-switching model µ b = 0.0256. The transition probabilities are φ = 0.0131, and θ = 0.0161. The roots (λj ) from the option-pricing model are λ1 = 2.468 and λ2 = 1.824. (In the no-switching model we have λ = 1.647.) The dividendprice ratios are δ0 = 0.0433 and δ1 = 0.0259. The filtered probabilities indicate that the economy was in state 1 in 1997 and the real stock price index stood at 5.90 at that point. Consider

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an option on the real stock price index with a strike price of 5.90 (i.e., the option is at-the-money at the 1997 data point). The price of the option, assuming the economy is in state 1, is 1.618. Were the economy in state 0 it would be 2.029. In the no-switching model it would be 1.647. Thus allowing for regime-switching may make a considerable difference to the price of the option. The option price in state 1 is a little lower than in the no-switching model largely because the volatility of stock prices is lower.

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Simulations.

So as to explore the implications of changes in key parameters of this model, we have computed solutions for a range of values. The baseline takes the parameter values estimated above and considers an option with a strike price that makes it “at the money” on the basis of the 1997 value for the index. In figures 2 – 7 we have varied: the variance of stock prices in each state, the drift of prices in each state, the discount factor, and finally the strike price. In each figure, the left hand panels give results for the regime switching model, while the right-hand panels give corresponding results for models without regime switching. The upper panels in each figure give the results for the exercise price; the lower panels for the option value when the stock price equals the 1997 data point in our sample.

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To price the options in

the simulations we used the risk-free rate of interest for 1997, denoted by r, and equal to 0.0617. The exercise price of the option is uniformly higher in state 0 – the low growth/high variance state – than it is in state 1 – the high growth/low variance state. On the basis of the no-switching model, the exercise price lies between them. In figure 2, an increase in the variance of stock prices in state 0 (from 0.0230 to 0.0250 in the figure) raises the exercise price in each state, and increases the option value, as would be expected. An increase in the variance of stock prices in state 1 (figure 3) raises the exercise price in state 1 but lowers it (very marginally) in state 0. This is attributed to the increase in state-1 variance making the two states more similar, causing the option values to converge. When the drift in state 0 increases (figure 4) the exercise prices in the two states converge. Once again the increase in state-0 drift makes the two states more similar. The option values nevertheless increase in both states. When the drift in stock prices in state 1 is increased (figure 5), then exercise prices and option values in both states rise. Increasing the discount factor, as is done in figure 6 naturally causes both exercise prices and option values in both states to fall. In figure 7 the strike price is increased from 5.5 to 10.0. The exercise price rises in both states, and the value of the option at the 1997 price falls in both states, again, broadly as would be expected. In order to establish whether the comparative statics depend on whether

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the option is in, at, or out of the money, we have carried out analogous exercises – varying the drifts, variances, and discount factor – for options with a range of different strike prices. The qualitative results do not change.

6

Conclusions

This paper has developed a valuation framework for Merton-style options such as perpetual American call options when the dividend dynamic of the underlying asset is governed by a regime switching process. Our valuation procedure involves an application of smooth-pasting techniques along the lines of Merton (1973) to solve a system of second-order stochastic differential equations. Our work also involves a development of a discrete-time dividend model in order to estimate the parameters needed in the option model. To illustrate the use of the option valuation under regime-switching models we have carried out a numerical analysis. The experiment reveals significant discrepancies between the option price based on the correctly-specified (regime-switching) model and that based on the conventional no-switching model or single regime model. More precisely, the no-switching model gave a price of 1.647 for the perpetual American call option at-the-money, whereas the regime-switching model for the same option gives a price of 2.029 in state 0 and 1.618 in state 1, implying a discrepancy of more than 23 percent in state

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0. This clearly underscores the importance of option pricing under regime switching models. It is important to note that when options are American style, single regime option pricing models lead to not only a mis-pricing outcome but also badly-timed exercise decisions. Finally, the sensitivity analysis carried out in this paper indicates that option prices are sensitive to certain parameters. There are several avenues to which the option valuation framework under regime-switching can be extended. Firstly, the analysis can be applied to the portfolio allocation problem. Secondly, more specific real options can be modelled with regime-switching processes. Finally, more importantly, there is a room for an extension of numerical approach to finite-life American options.

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Figure 2: Sensitivity with respect to Variances (State 0)

Figure 3: Sensitivity with respect to Variance (State 1)

Figure 4: Sensitivity with respect to Drift of Prices (State 0)

Figure 5: Sensitivity with respect to Drift of Prices (State 1)

Figure 6: Sensitivity with respect to Discount Factor

Figure 7: Sensitivity with respect to Strike Price

References [1] Andrews, D.W.K., 1991. “Heteroskedasticity and AutocorrelationConsistent Covariance Matrix Estimation”, Econometrica 59, 817-858. [2] Andrews, D.W.K and J.C.Monahan, 1992. “An improved Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator”, Econometrica 60, 953-966. [3] Bakshi, G.S., C. Cao and Z. Chen, 1997. “Empirical Performance of Alternative Option Pricing Models,” Journal of Finance 52, No. 5, 1997, 2003-2049. [4] Black, F. and M. Scholes, 1973, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy 3, 637-654. [5] Bollen, N.P.B. 1998, “Valuing Options in Regime-Switching Models”, Journal of Derivatives 6, 38-49. [6] Bollen, N.P.B., S.F. Gray and R.E. Whalley, 2000, “Regime-Switching in Foreign Exchange Rates: Evidence from Currency Option Prices,” Journal of Econometrics 94, 239-276. [7] Bonomo, M., and R. Garcia, 1994, “Can a Well-Fitted Equilibrium Asset-Pricing Model Produce Mean Reversion?,” Journal of Applied Econometrics 9, 19-29. 30

[8] Cecchetti, Stephen G., Pok-Sam Lang and Nelson C. Mark, 1990, “Mean Reversion in Equilibrium Asset Prices,” American Economic Review 80, 398-418. [9] Chourdakis, K. M. and E. Tzavalis, 2000, “Option Pricing Under Discrete Shifts in Stock Returns”, Working Paper, QM College, University of London. [10] Driffill, J. and Martin Sola, 1998, “Intrinsic Bubbles and Regime Switching,” Journal of Monetary Economics 42, (1998) 357-373. [11] Duan, J., 1995, “The GARCH Option Pricing Model”, Mathematical Finance 5, 13-32. [12] Duan, J., I. Popova and P. Ritchken, 1999, Option Pricing Under Regime Switching, mimeo, Department of Finance, Hong Kong University of Science & Technology, Hong Kong. [13] Elliott, RJ, L. Aggoun, and JB Moore, 1997. Hidden Markov Models: Estimation and Control. New York: Springer. [14] Engel, C. and J. D. Hamilton, 1990. “Long Swings in the Dollar: Are They in the Data and Do Markets Know It?”, The American Economic Review 80(4), 689-713.

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[15] Hamilton, James D., 1989, A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle, Econometrica 57, 357-384. [16] Hamilton, J.D., 1996. “Specification Testing in Markov-Switching Time Series Models”, Journal of Econometrics 70, 127-157. [17] Heston, Steven 1993, “A Closed-form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options”, Review of Financial Studies 6, 327-343. [18] Merton, Robert C., 1973, “Theory of Rational Option Pricing”, Bell Journal of Economics 4, 141-183. [19] Merton, Robert C., 1976, “Option Pricing When the Underlying Stock Returns are Discontinuous”, Journal of Financial Economics 4, 125-144. [20] Naik, V. 1993, “Option Valuation and Hedging Strategies with Jumps in the Volatility of Assets Returns,” Journal of Finance 48, 1969-1984. [21] Shiller, Robert, 1989, Market Volatility, MIT Press, Cambridge, Massachusetts. [22] White, Halbert, 1987, “Specification Testing in Dynamic Models”, in Truman F. Bewley, ed., Advances in Econometrics, Fifth World Congress, Vol. I, Cambridge: Cambridge University Press. 32

Table 1: Estimated models for log real dividends Regime-Switching α0 α1 σ0 σ1 σ0P σ1P k0 k1 p q Log. lik. AIC SIC HQ AR(1) AR(4) ARCH(1) ARCH(4)

No Regime Switching -0.00394 (0.0116) 0.0402 ( 0.00374) 0.152 (0.0239) 0.0414 (0.0108) 3.375 (0.424) 5.496 (2.144 ) 19.135 (3.172)* 30.120 (5.325)* 0.987 (0.0378) 0.984 (0.0488) -18.674 53.346 74.026 61.711 23.4 26.3 3.82 5.32

α

0.0175 (0.0120)

σ

0.127 (0.0374)

σP

6.761 (1.357)

k

22.759 (6.457)*

-83.293 172.586 180.341 175.723 64.8 66.8 26.6 29.7

Notes: The figures in parentheses are autocorrelation and heteroskedasticity consistent standard errors. The Akaike, Schwarz, and Hannan-Quinn model selection criteria are calculated as AIC = −2`m + 2g, SIC = −2`m + 2g ln(T ) and HQ = −2`m + 2g ln(ln(T )) respectively where `m is the maximum value of the Gaussian log-likelihood function and g is the number of freely estimated parameters. The standard errors with an asterisk ∗ are asymptotic standard errors of restricted parameters calculated in the following way; given x b = x(b µ0 , µ b1 , σ b0 , σ b1 , θb0 , θb1 , pb, qb), asymptotic variance of x b is V ar(b x) = Jb ∂x ∂x 0 b b b V ar(b µ0 , µ b1 , σ b0 , σ b1 , θ0 , θ1 , pb, qb)J , where J = [ ∂µ1 , ..., ∂q ], and x ∈ {k0 , k1 , λ}. To calculate the standard error of c1 use cb1 = c1 (b c0 , µ b0 , µ b1 , σ b0 , σ b1 , θb0 , θb1 , pb, qb, and J = ∂c1 , ∂c1 , ..., ∂c1 . ∂c0

33

∂µ0

∂q