publicité

Option Pricing Formula for Generalized Stock Models Xin Gao, Xiaowei Chen Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China [email protected], [email protected] Abstract The stock model and option pricing problem are central contents in modern finance. In this paper, generalized stock model for financial market is proposed and the European option pricing formula for the generalized stock model is computed. Keywords: finance, fuzzy process, option pricing, Liu process 1 Introduction Brownian motion was introduced to finance by Bachelier [1]. Samuelson [17] [18] proposed the argument that geometric Brownian motion is a good model for stock prices. In the early 1970s, Black and Scholes [3] and, independently, Metron [13] used the geometric Brownian motion to determine the prices of stock options. Stochastic financial mathematics was founded based on the assumption that stock price follows geometric Brownian motion. The Black-Scholes formula has become an indispensable tool in today’s daily financial market practice. Different from randomness, fuzziness is another type of uncertainty in real world. In order to deal with the evolution of fuzzy phenomena with time, Liu [11] proposed a fuzzy process, a differential formula and a fuzzy integral. Later, the community renamed them Liu process, Liu formula and Liu integral due to their importance and usefulness, just like Brownian motion, Ito formula and Ito integral. Some researches surrounding the subject have been made. You [19] studied differential and integral of multi-dimensional Liu process. Qin [15] considered some properties of analytic functions of complex Liu process. Dai [4] gave a reflection principle related to Liu process. As a different doctrine, Liu [11] presented an alternative assumption that stock price follows geometric Liu process. Moreover, a basic stock model for fuzzy financial market was also proposed by Liu [11]. We call it Liu’s stock model in order to differentiate it from Black-Scholes stock model. Qin and Li [16] presented the European options pricing formula for Liu’s stock model. However, the time-varying interest rate, time-varying stock drift and time-varying stock diffusion factors were not considered in Liu’s stock model. In real market, time-varying factors are necessary to be considered. We build a generalized stock model in the presence of a timevarying interest rate, time-varying stock drift and time-varying stock diffusion according to the reality. Considering the option pricing problem is a fundamental problem in financial market, we investigate the European option pricing formulas for the generalized stock model. The rest of this paper is organized as follows. Some basic concepts about Liu process are recalled and Liu’s stock model is introduced in Section 2. The generalized stock model based on the generalized geometric Liu process is proposed in Section 3. European call and put option price formulas are derived in Sections 4 and 5. Finally, some conclusions are listed. 2 Liu’s Stock Model Definition 1 (Liu [11]) A fuzzy process Ct is said to be a Liu process if (i) C0 = 0, (ii) Ct has stationary and independent increments, 1 (iii) every increment Ct+s − Cs is a normally distributed fuzzy variable with expected value et and variance σ 2 t2 whose membership function is µ µ µ(x) = 2 1 + exp π|x − et| √ 6σt ¶¶−1 , −∞ < x < ∞. Liu process is said to be standard if e = 0 and σ = 1. If Ct is a Liu process, then the fuzzy process Xt = exp(Ct ) is called a geometric Liu process. It was assumed that stock price follows geometric Brownian motion, and Black-Scholes stock model was then founded based on this assumption. Liu [11] presented an alternative assumption that stock price follows geometric Liu process. Liu [11] presented a basic stock model for fuzzy financial market in which the bond price Xt and the stock price Yt follow ( dXt = rXt dt (1) dYt = eYt dt + σYt dCt where r is the riskless interest rate, e is the stock drift, σ is the stock diffusion, and Ct is a standard Liu process. It is just a fuzzy counterpart of Black-Scholes stock model [3]. 3 Generalized Stock Model Let Ct be a standard Liu process, and let e(t) and σ(t) be functions. Then the fuzzy differential equation dYt = e(t)Yt dt + σ(t)Yt dCt has the solution µZ Yt = Y0 exp Z t e(s)ds + 0 t ¶ σ(s)dCs , 0 which is called the generalized geometric Liu process. The stock price given by Yt has instantaneous stock drift e(t) and stock diffusion σ(t). Both the instantaneous stock drift and stock diffusion are allowed to be time-varying. This process includes all possible models of a stock price process that is always positive, has no jumps, and is driven by a single standard Liu process. If e and σ are constant, it degenerates into Liu’s stock model. Qin and Li [16] presented the European options pricing formula for Liu’s stock model. When the interest rate is time-varying denoted by r(t), and the stock price is a generalized geometric Liu process, we obtain a new stock model for fuzzy financial market in which the bond price Xt and the stock price Yt follow: ( dXt = r(t)Xt dt (2) dYt = e(t)Yt dt + σ(t)Yt dCt which is called as the generalized stock model. 4 European Call Option Pricing Formula A European call option gives the holder the right, but not the obligation, to buy a stock at a specified time for a specified price. In this section, we consider European call option pricing problem for the generalized stock model. Considering the generalized stock model, we assume that a European call option has strike price K and expiration time T . Then the payoff from buying a European call ³option is (YT´−K)+ . RT Considering the time value of money, the present value of this payoff is exp − 0 r(t)dt (YT − K)+ . 2 Definition 1 European call option price f c for the generalized stock model is defined as Ã Z ! Ã ÃZ ! !+ Z T T T f c = exp − r(t)dt E Y0 exp e(t)dt + σ(t)dCt − K 0 0 (3) 0 where K is the strike price at time T . In order to calculate the option price, we solve Equation (3) and give an integral form as follows: Theorem 1 European call option price f c for the generalized stock model is defined as ÃZ !Z ( ÃZ ! ) T +∞ T f c = Y0 exp (e(t) − r(t))dt Cr exp σ(t)dCt ≥ u du. ¡ RT ¢ 0 K exp − 0 e(t)dt /Y0 (4) 0 Proof: By the definition of expected value of fuzzy variable, we have Ã Z ! Ã ÃZ ! !+ Z T T T f c = exp − r(t)dt E Y0 exp e(t)dt + σ(t)dCt − K 0 = Ã Z exp − 0 T 0 Ã Z = exp − T 0 0 ÃZ T = Y0 exp !Z (e(t) − r(t))dt 0 0 ( +∞ ¡ RT 0 5 0 Ã !Z ÃZ ! !+ Z T +∞ T r(t)dt Cr Y0 exp e(t)dt + σ(t)dCt − K ≥ r dr 0 0 0 !Z ( ÃZ ! ) Z T +∞ T r(t)dt Cr Y0 exp e(t)dt + σ(t)dCt − K ≥ r dr K exp − 0 ¢ ÃZ Cr exp ! T σ(t)dCt e(t)dt /Y0 ) ≥ u du. 0 European Put Option Pricing Formula A European put option gives the holder the right, but not the obligation, to sell a stock at a specified time for a specified price. In this section, we consider European put option pricing problem for the generalized stock model. Definition 2 European put option price f p for the generalized stock model is defined as Ã Z ! Ã ÃZ !!+ Z T T T f p = exp − r(t)dt E K − Y0 exp e(t)dt + σ(t)dCt 0 0 (5) 0 where K is the strike price at time T . Theorem 2 European put option price f p for the generalized stock model is defined as ¡ RT ¢ ÃZ !Z ( ÃZ ! ) T K exp − e(t)dt /Y0 T 0 (e(t) − r(t))dt Cr exp σ(t)dCt ≤ u du. (6) f p = Y0 exp 0 −∞ 0 Proof: By the definition of expected value of fuzzy variable, we have Ã Z ! Ã ÃZ !!+ Z T T T f p = exp − e(t)dt + σ(t)dCt r(t)dt E K − Y0 exp 0 0 Ã Z = exp − 0 T 0 Ã !Z ÃZ !!+ Z T +∞ T Cr e(t)dt + σ(t)dCt ≥ r dr r(t)dt K − Y0 exp 0 0 0 3 Ã Z = exp − !Z T r(t)dt 0 ÃZ = Y0 exp ÃZ T !Z ¡ RT K exp − (e(t) − r(t))dt 0 Z T Cr K − Y0 exp 0 0 6 ( +∞ e(t)dt + 0 ¢ ( e(t)dt /Y0 ! T σ(t)dCt 0 ÃZ Cr exp −∞ ) ≥r dr ! T σ(t)dCt ) ≤ u du. 0 Conclusions In this paper, we proposed a generalized stock model for financial market and investigated the option pricing problems based on it. European call and put option price formulas for the generalized stock model were defined and computed. Acknowledgments This work was supported by National Natural Science Foundation of China Grant No.60425309. References [1] Bachelier L, Théorie de la spéculation, Ann. Sci. École Norm. Sup., vol.17, pp.21-86, 1900. [2] Baxter M, and Rennie A, Financial Calculus: An Introduction to Derivatives Pricing, Cambridge University Press, 1996. [3] Black F and Scholes M, The pricing of option and corporate liabilities, Journal of Political Economy, vol.81, pp.637-654, 1973. [4] Dai W, Reflection principle of Liu process, http://orsc.edu.cn/process/071110.pdf, 2007. [5] Etheridge A, A Course in Financial Calculus, Cambridge University Press, 2002. [6] Hull J, Options, Futures and Other Derivative Securities, 5th ed., Prentice-Hall, 2006. [7] Li X, Expected value and variance of geometric Liu process, http://orsc.edu.cn/process /071123.pdf, 2007. [8] Li X, and Liu B, A sufficient and necessary condition for credibility measures, International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems, vol.14, no.5, pp.527-535, 2006. [9] Liu B, Uncertainty Theory, Springer-Verlag, Berlin, 2004. [10] Liu B, Uncertainty Theory, 2nd ed., Springer-Verlag, Berlin, 2007. [11] Liu B, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, vol.2, no.1, pp.3-16, 2008. [12] Liu B, and Liu YK, Expected value of fuzzy variable and fuzzy expected value models, IEEE Transactions on Fuzzy Systems, vol.10, no.4, pp.445-450, 2002. [13] Merton R, Theory of rational option pricing, Bell Journal Economics & Management Science, vol.4, no.1 pp.141-183, 1973. [14] Peng J, Credibilistic stopping problems for fuzzy stock market, http://orsc.edu.cn/process /071125.pdf, 2007. [15] Qin Z, On analytic functions of complex Liu process, http://orsc.edu.cn/process/071026. pdf, 2007. 4 [16] Qin Z, and Li X, Option Pricing Formula for Fuzzy Financial Market, Journal of Uncertain Systems, Vol.2, No.1, pp.17-21, 2008. [17] Samuelson P A, Proof that properly anticipated prices fluctuate randomly, Ind. Manage. Rev., Vol.6, pp.41-50, 1965. [18] Samuelson P A, Mathematics of speculative prices, SIAM Rev., Vol.15, pp.1-42, 1973. [19] You C, Multi-dimensional Liu process, differential and integral, http://orsc.edu.cn/process /071015.pdf, 2007. [20] Zadeh LA, Fuzzy sets, Information and Control, vol.8, pp.338-353, 1965. 5