ON DETERMINANTS AND SENSITIVITIES OF OPTION PRICES IN DELAYED BLACK-SCHOLES MODEL A. B. M. Shahadat Hossain, Sharif Mozumder ABSTRACT This paper investigates determinant-wise effect of option prices when the stock market is replicated by the classical Black-Scholes (B-S) and the delayed Black-Scholes (DB-S) models. We discuss the effects of different choices of delay characterizing functions in DB-S model compare to the B-S model. In so doing, we propose some rational functions which characterize the delay in DB-S model and satisfy the structural hypothesis of DB-S dynamics. A strong observation that stands out from this analysis is that figuring out a proper delay function is anything but obvious. stochastic delay model has been proposed in 1. Introduction recent literature [1] [4].This model derived an In 1973, Fischer Black and Myron Scholes proposed first analytic option pricing model known explicit formula for pricing European call option when the underlying stock price follows nonlinear as the B-S Model. This model has been stochastic functional differential equation with a subsequently treated as the benchmark model in the literature. It is still widely used to calculate the fair price of a European option. Though this model is the origin of most of the option pricing models, it sometimes gives inaccurate prices when we test it against real market data. This is because B-S model is premised on number of very rigid assumptions which are barely justified in real market. e.g. assumption of constant volatility is a great contrast in the market and is associated with fixed delay. It is assumed in the delayed model [1] that the parameters of the model will allow sufficient flexibility for a better fit than that obtained by the B-S model. We investigate the relative pricing, using a cross-section of real market data. Hence the objective of this paper is to investigate the variation of the option prices along various price-determinants of the model. A number of functions used in delay characterization of DB-S model have been investigated in this other qualitative misspecification of price regard. Some practical issues regarding the use of patterns. To address volatility related mispricing, DB-S model have been reported. These remarks go 37
against the spirit of unconstrained application of DB-S model which is assumed from theoretical point of views. the volatility for the delayed option pricing formula [1]. We have generated Mathematica codes to produce the numerical illustration. The B-S model assumes price process follows a geometric Brownian motion and derivatives traded on the stock have fair prices under riskneutral valuation. It is the basis of modern option pricing theory and it derives the theoretical option prices using five key determinants: stock price, strike price, volatility, time to expiration, and short-term (risk free) interest rate. DB-S model considers the fact that the evoluation of volatility takes strongly into account the knowledge of the past variations [10]. These past variations are feeded into current volatility; in particular volatility of prices delayed by a factor gets feeded into current volatility. The delay effect is described by a function which plays significant role in evolution of prices. In this paper, we investigate some simple functions which describe the delay effect in price evolution and notice that some choices of though not contrasting with hypothesis of the model are not practically appealing. We then see the option price variation, depending on various determinants, for our illustrative choices of investigating functions. Each choice of such function acts as an additional determinant in DB-S model. In each illustration, a particular form of rational function is considered which yields This paper is structured as follows. In section 2, we describe the dynamics and pricing formulas under B-S and DB-S models. For a general delay characterization function g the pricing in DBS model is revisited. The hypothesis under which the choices are made is revisited in section 3. In section we also consider a delayed B-S model and calculate the fair price of a European call option [1] but use the same assumptions like B-S. We compare this model with the B-S one by using the computer algebra system (CAS) Mathematica [8]. 2 The B-S and DB-S Option Pricing Formulas The dynamics of a risky asset, such as stock, can be described by the following Stochastic Differential equation (SDE): [ ] (2.1) where ( ) is a Brownian motion under the given probability measure, called the probability measure of the risk-neutral world; the parameter is the risk-free interest rate and denotes the volatility of the risky asset. 38
The value of a European call option at time with exercise price, current stock price, maturity, constant interest-rate and volatility : (2.2) where. of all continuous functions [ ], is a Banach space with the norm [ ] ; is continuous; ([ ] ) is - measurable with respect to the Borel -algebra of ([ ] ); is a standard Brownian motion process (one-dimensional) adapted to the filtration [ ]; the process ([ ] ) stands for the segment [ ]. Here is the standard normal cumulative distribution function, given by dx, is the probability that a variable with a standard normal distribution with mean 0 and variance 1, i.e. will be less than. On the other hand, the dynamics of the risky asset in delayed B-S model is derived from the following stochastic functional differential equation (SFDE) [5]: ( ( )) [ ]; [ ] (2.3) where are positive constants with [ ] ([ ] ) is a continuous functional where ([ ] ), denotes the space In particular, we now define the functional, where are positive constants with a { } Hence the equation (2.3) SFDE reduces to the following stochastic delay differential equation (SDDE) [5]: ( ( )) [ ]; [ ] (2.4) Now if we consider a market consisting of a riskless asset (e.g. a bond or bank account) with rate of return (i.e., and a single stock whose price at time satisfies the SDDE (2.4) where a.s.. Furthermore, in the SDDE we assume that the delays are positive. Then a B-S type formula derived in [1] (see pp.482-486) for the fair price of European call option,, 39
written on the stock with exercise price and maturity time prior to maturity is given by: ( ) [ ] ( in } (2.5) where ( ( ( ) ) ( ) If and, then ( ( ( ) ( ( )) ( ( )) ) ) where is given by ( ) ( ) and 1/σ [ og 1/σ [ og Note that if for all then Equation reduces to the classical B-S formula. In contrast with B-S formula, the fair price in the delayed model depends not only on the stock price at the present time, but also on the whole segment { [ ]}.(Of course [ ] [ ] since and ). For and, we can develop a recursive procedure to calculate (2.5) by backward. steps of length from the expiration time of the option.coupled with numerical approximations, this recursive procedure can be used to compute the option price at any time [ ].We will only consider the conditional distribution of the solution process for [ ]. 3 Hypothesis of The function in DB-S model plays the key role in model performance. This, however, has to satisfy some properties, namely: (i) (ii) (iii) is monotonically increasing (i.e. (iv) approaches a constant value as However, in the original work the authors [1] did not produce any numerical result. Moreover, no published article is available in the literature where empirical investigation is conducted using DB-S model. We observed that not every choice of satisfying above properties really works. In fact, figuring out working s is itself a good research. Trial and error methods can be applied to see what kind of produces meaningful prices which compare well with respect to market prices and/or other trivial model prices such as those obtained from B-S model. Another important point that comes to the scenario is that calibrating DB-S 40
model with several s using market prices might be a practical approach, it might allow the market itself to talk for any particular form of at a particular context. In this article, we figure out some rational expression for which worked well with standard parameter. It must be mentioned that the parameters, and coming with different choices of can also be calibrated from a cross-section of market data. However, in this article we figure out consistent values of and simply by trial and error method. We consider the following rational forms of : where (i), (ii) and (iii). are positive constants. Illustrative values of parameters and are ( a=1, b=1), for which our choices of and are plotted in Figure 3.1. Figure 3.1: Illustrative example of functions characterizing delays in DBS model. We will show in the next section that and play a very important role in pricing and sensitivities in DBS model. 4 Determinants of Option Prices The formulas (2.2) and (2.5), obtained for European call option by B-S and DB-S respectively, has five determinants in common which affect the prices; these are current stock price, the strike price, the maturity time, the volatility of the stock price, and the risk-free interest rate. However, as discussed in previous section, the function itself is a determinant for the DB-S model. For our choices and of this additional determinant we produce the relative sensitivities of call prices with respect to other common determinants for both the B-S and DB-S 41
models. Figure 4.1 presents, the sensitivities for whereas Figures 4.2 and 4.3 present the similar sensitivities for and respectively. Here, we observed changes in option prices by considering changes in one of these determinants and keeping other determinants fixed. Figure 4.1 shows relative sensitivities of call prices corresponding to individual changes in various determinants and our choice of for. Similar sensitivity features are presented in Figure 4.2 and Figure 4.3 for the choices of and. The set of individual values of the determinants which we used in the B-S (solid line) and DB-S (dashed line) models are:. The high volatility, as high as, should not be surprising especially considering the recent tumultuous markets. (a) (b) (c) (d) Figure 1. Effect of changes factors for B-S (solid line) and that for DB-S (dashed line) With with fixed and. Standard features of sensitivities are well-observed for both B-S and DB-S models. However, what is more important to take note of is the sensitivity with respect to the determinant g. 42
(a) (b) (c) (d) Figure 2. Effect of changes factors for B-S (solid line) and that for DB-S (dashed line) with with fixed and. (a) (b) 43
(c) (d) Figure 3. Effect of changes factors for B-S (solid line) and that for DB-S (dashed line) with with fixed and. the best in any particular context. Moreover it is 5. Conclusion In this paper sensitivities of option prices, with respect to various determinants, have been studied when stock market is assumed to follow BS and DBS dynamics. In so doing we offer three delay characterizing functions as illustrative examples of general delay characterization in DB-S model, a variant of celebrated B-S model. In the possible to determine the most suitable values of parameters, appearing in a particular form of, from market data. Overall this paper observed that price sensitivities with respect to various usual determinants are well comparable under both B-S and DB-S model, except that sensitivities with respect to interest rate heavily counts on the choice of. original paper [1], after exploring the theoretical framework of DB-S model, the authors didn't care much about the practical features of their characterization. This paper reveals that not all the functions satisfying the hypothesized conditions in theoretical framework do work in practice. It further notices that in fact figuring out a working (characterizing the delay in DB-S model) is anything but obvious. Trying number of alternatives in trial and error method is crucial and this paper proposes three such alternatives to try References [1] Arriojas M., Hu Y., Mohammed S.-E. A. and Pap G., A Delayed Black and Scholes Formula, Stochastic Analysis and Applications, 25(2007), 471-492. [2] Kemajou, Elisabeth and Mohammed, Salah-Eldin and Tambue, Antoine, A Stochastic Delay Model For Pricing Debt and Loan Guarantees: Theoretical Results, preprint, 2012. for. It is observed that itself is a determinant for [3] D.T. Breeden and R.H. Litenberger, Prices DB-S model and it is likely that market will talk for of State-Contingent Claims Implicit in 44
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