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Pricing of european options when the underlying stock price follows a linear birthdeath process a

b

Ralf Korn , Markus Kreer & Mark Lenssen

c

a

Fachbereich Mathematik, Johannes Gutenberg – Universitat Mainz, Mainz, D-55099, Germany b

Fixed Income & Derivatives Research, UBS, Zurich, CH-8021, Switzerland

c

Global Product Development, Treasury, NatWest Markets, 135, London, EC2M 3UR, United Kingdom Available online: 27 Jul 2011

To cite this article: Ralf Korn, Markus Kreer & Mark Lenssen (1998): Pricing of european options when the underlying stock price follows a linear birth-death process, Communications in Statistics. Stochastic Models, 14:3, 647-662 To link to this article: http://dx.doi.org/10.1080/15326349808807493

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COMMUN. STATIST.-STOCHASTIC MODELS, 14(3), 647-662 (1998)

Pricing of European Options when the Underlying Stock Price Follows a Linear Birth-Death Process

Ralf KORN Fachbereich Mathematik Johannes Gutenberg - Universitat Mainz, D-55099 Mainz, Germany Markus KREER Fixed Income & Derivatives Research, UBS, CH-8021 Zurich, Switzerland Mark LENSSEN Global Product Development, Treasury, NatWest Markets, 135 Bishopsgate, London EC2M 3UR, United Kingdom

Abstract This investigation considers a possible approach to price options if the underlying stock jumps up or down or remains unchanged, extending ideas of Cox and Ross (1976) to a more general jump model with state-dependent jump intensities. Provided that in addition to the stock itself one option on this stock is traded in the market, we can show by valuation of arbitrage arguments that the price of European options is then determined uniquely. Keywords valuation by arbitrage, martingale representation for point processes, equivalent martingale measures, birth-death processes 1. INTRODUCTION The problem of option pricing if stock prices do not follow the commonly assumed geometric Brownian motion (as discussed e.g. in the seminal paper by Black and Scholes [4]) was first brought to general attention by Cox and Ross [8]. Especially, they deduced explicit pricing formulae for European vanilla

Copyright Q 1998 by Marcel Dekker, Inc.

648

KORN,KREER, AND LENSSEN

options on stocks following simple jump processes. The jump processes in Cox and Ross [8] have the common feature that jumps to more than one value were excluded. This simplification made it possible to find a preference-free equilibrium price for European options by pure arbitrage considerations. All subsequent publications on this topic we are aware of deal with this latter class of jump processes (Poisson-type birth processes, e.g. Page and Sanders [16], Elliot and Kopp [9]). There are related papers where the relative price changes (and not the price changes in absolute values !) are modelled via jump -diffusion models (see e.g. Aase [I] or Bardhan and Chao [2]) which differ from our modelling approach . Cox and Ross [8] already mentioned the possibility to solve the option valuation problem for random jumps to more than one value by the introduction of additional stocks "to complete the market". This idea was e.g. taken up by Jones [12] who considered a jump-diffusion model with Poisson-type jumps (i.e. constant jump intensities). In order to investigate the effect of jumps on the pricing of options, he set up a risk-free hedge portfolio consisting of the risk-fiee bond, the stock itself and in addition several options on the stock (apparently motivated by the praxis of option-traders).

In the following paper we investigate the possibility to price European options if the change of the underlying's price is described as a sum of two independent jump processes with state-dependent jump intensities. One may thmk of the underlying for example as a traded stock whose price S can go up, go down or stay the same. To complete the market we require in addition the existence of M e r traded (derivative) securities, whose only randomness is affected by the underlying stock. In our case we complete the market by introduction of a so called European style low exercise price option, henceforth called LEPO. The idea of these financial securities originates fkom the LEPOs which are traded e.g. at the Swiss Options and Financial Futures Exchange (see also SOFFEX [18]). Of course, we follow the more state of the art-approach of contingent claim valuation using the concept of equivalent martingale measures introduced in Harrison and Pliska [lo]. In discrete time our problem corresponds to a trinomial tree (see e.g. Boyle [5], Omberg [15]. Perrakis [16] derives preference-free upper and lower bounds on European options for processes of trinomial type (also in the limit At + 0). However, these preference-free bounds are not very tight. In our paper we present a simple model for which explicit option pricing formulae can be given. Our main tools are standard methods from the theory of birth-death processes, results about existence of equivalent martingale measures, and martingale representation results for point processes.

PRICING OF EUROPEAN OPTIONS

The plan of the paper is as follows: in Section 2 we briefly specify the stochastic process which we use for modelling the stock price movements. In Section 3 we present the pricing formula for European style options. As a byproduct we show that the fair price of a European option must satisfy an infinite system of ordinary differential equations which is the analogue to the Black and Scholes equation in our model. The existence of a unique replication strategy (consisting of trading in the bond, the underlying stock and the LEPO put in a suitable way) for a European option will be the subject of Section 4. Section 5 concludes the paper and outlines open problems and possible future research. 2. THE STOCK PRICE MODEL We model the movements of the stock price S = St by a jump process in continuous time t with zero dnft, using the symbolic notation due to Cox and Ross [8], as follows

where h and q are some positive constants which shall be named transition rates. In other words, for tick size equal to 1 this model describes the stock price dynamics tick by tick. More formally, given some T > 0 we describe the movements of the stock price S t in continuous time t E[O, TI as follows where Nt(l) and Nt(2) are jump processes on the time interval [O, TI with jump size 1 and intensities ASt and qSt , respectively. As usual, the jump processes are defined on the probability space (Q, j , P) where the internal history is given by the filtration (ft )tEIO,T] generated by Nt(l) and Nt(2). For convenience we also assume f = fT. Obviously, the processes t

t

Nt(l) - j hS,ds 0

and

Nt(2) - /qS,ds 0

are martingales with respect to the filtration (jt )tEIO,T] Taking the initial value So to be a non negative integer we may restrict the stock price process S = St to non-negative integer values only. For this purpose

650

KORN, KREER, AND LENSSEN

note that L O = 0, 7.0 = 0 guarantee that the stock price will never become negative (principle of limited liability), that is, S E {0,1,2,...). In case that at some time t*e[O, TI the stock price becomes zero, the choice of the transition rates guarantees that St will be zero for all later times t ~ [ t *TI , . In other words, our model includes the phenomenon of bankruptcy. Note that the linear dependence of the jump intensities on the stock price implies that the stock price makes a lot of movements if the price is high while it behaves relatively flat when the price is low. This can be compared to the characteristic of the usual geometric Brownian motion having increments which are proportional to the stock price. Also, our model could be regarded as a discretized (of the state space) version of the Cox and Ross square root model (where W is a Brownian motion) in the sense that we obtain it as a diffusion limit of our linear blrth-death process if we let the jump size AS go to zero while we simultaneously let the transition ratesh, q both approach infinity (and keeping the quotient hlq constant) such that the corresponding limits

-

p = lim [(A q) AS] , o

= lim

[(h+q) (AS)Z]

exists. We refer the interested reader to Cox and Ross [8] for the relevant arguments. We can thus also identify the relation between the drift and diffusion parameters in the Cox and Ross square root model and the transition rates h and q in our model. The difference between the transition rates determines the tendency of the stock price, while their sum is responsible for the volatility. To illustrate the typical sample path behaviour of a linear birth-death process, we display two simulated sample paths in Figures 1 and 2. In both these examples we have chosen T=l, h=0.6 and q=0.5. In the first example we have an initial stock price of So = 100. Note that the expectation of ST is given by E(ST) = So. eO.l(see e.g. Bailey [3]). In the second example we have an initial stock price of So = 30. Comparing both examples we see that in the first example the stock price has only short periods of constant prices. It has 133 price movements per year whle the stock price in the second example changes only 39 times per year. The periods during which the price remains constant are much longer. It is also possible to consider another tick size than one ; just multiply the right side of equation (1) by the desired size. Whereas Black and Scholes [4] started from a lognormal probability distribution with drift for the stock prices, in our case we have to derive the probability distribution of our stock price model. We denote the desired probability distribution of stock prices by pnj(t,t0) = P(St=jSto=n). This notation indicates


Figure 1 : Sample path of a birth-death process

Figure 2 : Sample path of a birth-death process

652


that the probability of the stock price being equal to some number j E {0,1,2,...) at time t depends on the initial value Sto= n at a previous time to. These probabilities p, j satisfy the well-known forward birth-death equations (keeping both to and Sto= n fixed, and writing s = t-to ). See also Bailey [3].

One can show that the total probability remains conserved, i.e. I: pnj(s) = 1 (e.g. Cox and Miller [7], Kreer [13] ). The probabilities pnj also satisfy the system of backward birth-death equations (keeping now t and St=j fixed, and writing s =

Note that our particular choice of transition rates generalises the linear birth process of Cox and Ross [8] by allowing for jumps down as well as up. The desired distribution of stock prices p,j will be obtained by solving the infinite set of coupled ordinary differential equations (2). M e r the specification of the stochastic process for the stock price movements we state finally our basic assumptions concerning the financial market in consideration. Our assumptions are - with the exception of the next-to-last one - standard and correspond e.g, to those presented in Hull [ l 11. Trading is continuous in [0, TI and takes place in a liquid frictionless market, that is to say, there are no transaction costs and taxes in the market. Short-selling with full use of proceeds is allowed. There are no arbitrage opportunities. In addition to the stock S the market trades a European style Low Exercise Price Option, F*, with strike 1 and maturity T. We may assume this LEPO F* to be a put option with strike 1. The rate r , at which individuals can borrow and lend freely, is constant. 3. DERIVATION O F AN EXPLICIT OPTION PRICING FORMULA

The no-arbitrage paradigm for pricing contingent claims B payable at time T states that the price ft of this contingent claim at time t < T must be given as

PRICING OF EUROPEAN OPTIONS f, =

B( e-r(T-')B I ft 1

(*I

where k is the expectation with respect to some equivalent martingale measure 6 (i.e. 6 is equivalent to our original probability measure P (i.e. the measures have the same null sets) and the discounted security prices St (of securities underlying the contingent claim) are martingales with respect to 6). See e.g. Harrison and Pliska [lo] for the presentation of the relevant arguments. By noting that (non-trivial) linear birth-death processes are supported on the non-negative integers we realise that their associated probability measures are necessarily equivalent (uniqueness of the associated measures corresponding to a pair of transition rates (A, q) is a consequence of e.g. Theorems 18.415 in Liptser and Shiryayev [14]). Proposition 3.1 will characterise all martingale measures 6 for our security price (that are associated with a linear birth-death process).

Proposition 3.1 A probability measure 6 associated to a linear birth-death process with transition rates i,is an equivalent martingale measure for the security price iff we have

4

Proof: Let first 6 be an equivalent martingale measure. Then we must have er('")

=

i( s(~)/s(s)1 f, )

for ~i s 5t iT

(5)

On the other hand, one finds in particular for the standard initial condhons fin,n(O,O)

= ~n,n(o,O)=

1

fin,j (0,O) = pn,j (0,O) = 0 for all other j

#

n,

that explicit calculation leads to (e.g. Bailey 1964)

for 0 1 s i t 5 T. Comparison of (6) with equation (5) yields assertion (4). To prove the other direction of our claims note that by using relation (6) it is easily seen that relation (4) implies that the associated measure i? is an equivalent martingale measure (equivalence of P and 6 is ensured by the remarks preceding the proposition).

0 Because we always assume a non-negative interest rate r for the bond price, Proposition 3.1 indicates that in a risk-neutral world (i.e. the discounted stock

654


price is a martingale) the transition rates for up-jumps are higher than those for down-jumps. We will M e r call transition rates satisfying relation (4) riskadjusted rates. As a consequence of Theorem 19.7 of Liptser and Shuyayev [14] we can even give the explicit form of @ via its Radon-Nikodym derivative with respect to P

It is hence possible to transform the birth-death process with transition rates 3L and q into another birth-death process with arbitrary positive transition rates and fi by the above change of measure. Proposition 3.1 relates the risk-adjusted transition rates iand 6. Combination of these facts shows that an equivalent martingale measure @ is uniquely determined if e.g. fi can be uniquely inferred from the market. So far we have shown that there exists an uncountable number of equivalent martingale measures, and what we need is a rule to figure out the "correct" one for our option pricing problem. But this is the point where the assumption of a traded LEPO put enters the game. Because this LEPO put is actually traded, it must have a price. Assuming that there are no arbitrage opportunities in the market the price must be of the form (*) for some equivalent martingale measure @ . If we can show that the corresponding parameter fi is uniquely determined by the market price of the LEPO put, then - to prevent arbitrage opportunities (We will refer to h s point later on in more detail)- the price of a European contingent claim with terminal payoff function B must be given as (conditional) expectation with respect to the same @.This unique dependence of fi on the market price of the LEPO put will be shown in Proposition 3.3 . To prove this proposition we need the following lemma which is a standard result in the theory of blrth-death processes and will enable us to give explicit representations for the price of European contingent claims. Lemma 3.2 The solution of the forward birth-death equations (2), with the riskadjusted rates iand fi satisfying (4), subject to the initial conditions BRn(t,t) = 1 ~ fixed) is given by and fin (t,t) = 0 for all other j # n, ( t [O,T]


where we have used the abbreviations

Proof: This standard result can be obtained by the method of generating functions and is gwen e.g. in Bailey [3].

0 Proposition 3.3 Let the market price F*t at time t < T of the LEPO put with strike 1 and maturity T satisfy the regularity condition er(T-')~*t 51 . Then the positive parameter fi is uniquely determined at time t by the market price of the LEPO put as solution of . for n 2 1 the where a is defined in equation (9) and a ~ ( 0 , l ) Furthermore, market price F*t = F*(T, t, St=n) is convex in a. Proof: Since F\ must be of the form (*) with 0= ifelse ~ r =O ST Lemma 3.2 (with s=T) implies that the price of the LEPO put must satisfy equation (10) for a suitable parameter fi. Using the regularity condition and the definition of a in (9) we see that (10) has a unique positive solution fi. The convexity of F*(T, t, St=n) is an immediate consequence of equation (10).

0 Note that the regularity condition required in Proposition 3.3 must be satisfied if the market contains no arbitrage because the terminal payoff of the LEPO put is at most 1. Combination of Lemma 3.2 and Proposition 3.3 now yields our main theorem which gives us explicit expressions for the price of European contingent claims. Theorem 3.4 a) The fair price f at time t of any European contingent claim with terminal payoff B (where B is f T -measurable and 6-integrable) is given by

656


where the corresponding parameter 6 is uniquely given by the price of the LEPO put. b) The fair price ft,, of a European contingent claim with payoff function gj = g(ST=j) payable at T (such that the moments of the terminal payoff are finite) at time t when the stock price St is equal to n is given by ft

=f

(T, t, St=n) = e-

r(T-t) rn

Z gj 6 n , j (T,t)

j=O

where Gn j(T,t) are given by equations (8) and (9). In particular, the price at time t of a European call option with final payoff gj = max( j-X, 0), where the positive integer X denotes the strike, has the form

while in the case of a European put option we have put (T,t,St= n) =

where a and p are given by equations (9) in Lemma 3.2. Equations (13) and (14) are related by put-call parity. c) If the fair price f,,, at time t of a European contingent claim with a terminal payoff g(ST) (and finite moments of this payoff ) is continuously differentiable with respect to t then satisfies the infinite system (15) of dfferential equations with initidterminal conditions (16) :

Proof: a) and b) are direct consequences of the foregoing Proposition 3.3 and Lemma 3.2). To prove c), note that for St 2 1 we have

PRICING O F EUROPEAN OPTIONS

Because e-*f(~,t,s, ) is a @-martingale,the coefficient of dt in the first line after the last equality sign must be zero. Combining this with relationship (4) proves equation (15). The boundary conditions (16) are obvious.

If we compare the equality (15) with the Black-Scholes equation

(where C(t,p) is the Black-Scholes price of a European option at time t when the current stock price is p and o is the volatility parameter of the stock price), we can directly see the similarity between our system of pricing equations and equation (17). The first two terms coincide. If we interpret the expressions in square brackets in the remaining two terms as finite difference approximations for the derivatives in the Black-Scholes equation we see that the parameter fi plays a similar role in our model as the volatility o in the Black-Scholes model. While option trading using Black-Scholes actually means tradmg of the volatility o, in our context of a birth-death process model option trading means trading the parameter fi. Also, the assumption of constant market coefficients in the BlackScholes approach is similar to our assumption of constant transition rates.

4. REPLICATION OF EUROPEAN OPTIONS In this section we will show that we have actually completed our market model by the introduction of the LEPO-put and its market price at time 0. Note that we need this market price in time 0 to determine the equivalent martingale measure. However, if the market has chosen this parameter via the price of the LEPO-put then the market model consisting of the bond, the stock and the LEPO-put is complete which will be shown in the proof of Theorem 4.1 Let the positive parameter fi be given. Then for every contingent claim with terminal payment B (where B is f T -measurable and @-integrable) there exists a replication strategy (gt, wt , cpt) describing the number of bonds, shares of the stock and number of LEPO-puts held at time t, i.e. the processes (6, y,q) are predictable and satisfy


i.e. the market model is complete. Proof: Let f t be the price of the contingent claim (according to Theorem 3.4) with fT = B a.s. For notational convenience, we will prove our theorem only for the case r = 0, i.e. the bond price is assumed to be constant. The general case can be proved with exactly the same method. The integral representation theorem for point process martingales (see Bremaud [6], p. 64-67) gives us the existence of two predictable processes Ht(i), i=1,2, satisfying T

j H t (i)lct (i)ds < m

as.

0

such that we have the following representation of ft

where bt (1) = it (2) are the @-intensitiesof N t(i) (remember r = 0). To find a replication strategy (~,v,(P)we assume (note that r = 0 implies dB = 0)

Comparing equations (19) and (20) we see that ( y , ~ must ) satisfjr the set of equations


- Ws + 'Ps [F*(TASS -1) - F*(TASS 11 = Hs (2) As long as we have S > 0 the first equation in (21) can be solved for cps (if we have S=O then the unique pair (~,cp)is (0,O)) giving 'Ps

=

(HS(~)+HS(~))L(U a *, -F

(22)

as

To show that (w,(P)also satisfy the third equation in (21) note that we have -Ws

+ 9 s [F*(T,s,Ss -1) - F*(T,s,Ss )I .

where the second but last equality sign is obtained by using the differential equation (15) in its form with r = 0. The bond component 6 is now defined in such a way that we have a self-financing strategy, i.e.

This theorem also shows that the LEPO-put and any European contingent claim on the underlying stock must be valued using the same equivalent martingale measure. If the contingent claim would be valued higher (lower) by using a different equivalent martingale measure as that used for valuing the LEPO-put, we could sell (buy) the contingent claim and buy (sell) the replicating portfolio of Theorem 4.1. But, this would provide us with an arbitrage opportunity which is excluded fiom our market model. 5. CONCLUSION

We have modelled stock prices by a jump model in which they can go up, go down or stay the same. This model was already suggested e.g. by Perrakis [17] to describe thinly traded stocks. We have completed the market by introducing a derivative security on the underlying stock, the LEPO put, whose price is determined in the market by the laws of supply and demand. Note that due to putcall parity we may also complete the market using a LEPO call (In fact we could use any contingent claim of the form g(ST) that has the feature that the parameter

660


fi

can be deduced from its market price). Thus the market determines the martingale measure which is to be used for arbitrage-free pricing of European style contingent claims. The parameter fi is critical in our framework and its role can be compared (in a wider sense) with the role of the volatility o in the BlackScholes world: both of these parameters enable traders to express their market views on the underlying stock in consideration. To see the economic relevance of the parameter fi, recall that the probability (in the risk-neutral world) of bankruptcy (that is the probability of extinction of the birth-death process) given the current price n is given by equations (8) and (9) as

(see also e.g. Bailey [3]). Clearly the higher the present stock price Sto' the lower the chance of bankruptcy. On the other hand we notice that the lower the risk-free rate r the more likely is the occurrence of bankruptcy . Further research concerned with jump processes as a mean of modelling financial markets will mainly have to consider two aspects: first one needs to dedicate some time to extend the above arguments for market completion using exchange-traded options with strike bigger than 1, i.e. to complete the market not using LEPOs. On the other side it will be both interesting and.important to analyse market data on the basis of jump processes. In particular exchange traded options on exchange traded stocks will be very promising candidates to investigate whether the pricing differences from the Black-Scholes price due to 'fat tails' can be explained in a more satisfying way by a jump methodology as described here. Acknowledgements Part of thls work was done during the first author's visit at the Dept. of Electrical and Electronic Engineering, Imperial College, London, financed by a research scholarship of the Deutsche Forschungsgemeinschafi which is gratefully acknowledged. RK and MK wish to thank Professors P.Embrechts (ETH Zurich) and W.Runggaldier (Padova) for stimulating discussions during a workshop in Mainz. Part of this work was carried out while the second author was at Research & Product Development, Trading Risk, NatWest Markets, 135 Bishopsgate, London EC2M 3UR, United Kingdom. Some helpful advises by Professor A.N.Shnyayev on Girsanov-type theorems on general point processes during his stay at the ETH Zurich are gratefully acknowledged.


661

MK and ML wish to thank Dr.Suni1 Gandhi (NatWest Capital Markets, London) for stimulating and helpful discussions. Last but not least we would like to thank two anonymous referees for many helpful suggestions and comments. References Aase, K. K., Contingent claim valuation when the security price is a combination of an It6 process and a random point process, Stoch. Proc. Appl. 28 (1988), 185-220. Bardhan, I. and Chao, X., Pricing options on securities with discontinuous returns, Stoch. Proc. Appl. 48 (1993), 123-137. Bailey, N.T., The Elements of Stochastic Processes, Wiley Classics Library Edition, New York 1990. Black, F. and Scholes, M., The pricing of options and corporate liabilities, J. Political Econ. 81 (1973), 637-659. Boyle, P.P., A lattice framework for option pricing with two state variables, J. Financial & Quantitative Analysis 23 (1 988), 1- 12. Bremaud, P. Point Processes and Queues, Springer Series in Statistics, Berlin 1981. Cox, D.R. and Mdler, H.D., The Theory of Stochastic Processes, Methuen & Co. Ltd, London 1965. Cox, J.C. and Ross, S.A., The valuation of options for alternative stochastic processes, J. Financial Econ. 3 (1976), 145-166. Elliot, R.J. and Kopp, P.E., Option pricing and hedge portfolios for Poisson processes, Stoch. Anal. Appl. 8 (1990), 157-167. [lo] Harrison, M.J. and Pliska, S.R., Martingales and stochastic integrals in the theory of continuous trading, Stoch. Proc. Appl. 11 (1981), 215-260. [ l 11 Hull, J.C., Options, Futures, and other Derivative Securities, Prentice-Hall International, 2nd Edition, Englewood Cliffs, New Jersey 1993. [12] Jones, E.P., Option arbitrage and strategy with large price changes, J. Financial Econ. 13 (1984), 91-113. [13] Kreer, M., Analytic birth-death processes: A Hilbert space approach, Stoch. Proc. Appl. 49 (1994), 65-74. [14] Liptser, R, and Shuyayev, A.N., Statistics of Random Processes (vol. II), Springer, Berlin 1978. [15] Omberg, E., Efficient discrete-timejump process models in option pricing, J. Financial & Quantitative Analysis 23 (1988), 161-174.

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[16] Page, F.H. and Sanders, A.B., General derivation of the jump process option pricing formula, J. Financial & Quantitative Analysis 21 (1986), 437-446. [17] Perrakis, S., Preference-fiee option prices when the stock returns can go up, go down, or stay the same, Advances in Futures and Options Res. 3 (1988), 209-235. [18] SOFFEX : Kontrakt-Spezifikationen,May 1994 Received: Revised: Accepted:

412211996 1/23/1997 411 111997