Fractional Liu Process and Applications to Finance

Transcription

1 Fractional Liu Process and Applications to Finance Zhongfeng Qin, Xin Gao Department of Mathematical Sciences, Tsinghua University, Beijing 84, China Abstract In this paper, we consider fractional Liu process. First, the membership functions, expected values and variances of arithmetic and geometric fractional Liu process are given. Then we suppose that stock price follows geometric fractional Liu process and formulate fractional Liu s stock model. Based on this model, European option pricing formulas are obtained. Keywords: Liu process, stock price. Introduction Fractional Brownian motion deals with long-range dependence while still assuming a Gaussian process. Nevertheless, it offers the promise of giving simple, tractable solutions to pricing financial options and presents a natural way of modeling long-range dependence. Similarly, fractional Liu process is considered. First, its membership function is provided. Then fractional Liu process is introduced to model stock price. Based on this assumption, European option pricing formulas are obtained. The rest of this paper is organized as follows. Section 2 recalls some definitions and results of fuzzy calculus. Section 3 introduces the relative concepts of fractional Liu process. Fractional Liu s stock model is formulated and European option pricing formulas for this model are given in Section 4. Finally, some conclusions are listed. 2 Fuzzy Calculus Definition Liu [3]) A fuzzy process C t is said to be a Liu process if i) C, ii) C t has stationary and independent increments, iii) every increment C t+s C s is a normally distributed fuzzy variable with expected value et and variance 2 t 2 whose membership function is Liu process is said to be standard if e and. )) π x et µx) 2 + exp, < x <.

2 3 Fractional Liu Process Definition 2 Let C t be a standard Liu process. Then a fractional Liu process with index α is defined by where < α <. F t t t s) α dc s ) Theorem Assume that F t is a fractional Liu process with index α. At each t, F t is a normally distributed fuzzy variable with expected value and variance t 2 2α / α) 2. Definition 3 Let F t be a fractional Liu process with index α. Then the fuzzy process is called an arithmetic fractional Liu process. A t et + F t 2) Theorem 2 For each t >, A t has a normal membership function )) π α) z et µz) 2 + exp, z >. 3) α Proof: If z >, then it follows that The proof is complete. µz) 2Cr{A t z}) 2Cr{et + F t z}) { 2Cr F t z et }) ) z et µ Ft )) π α) z et 2 + exp. α Remark : For each t >, it follows from Theorem 2) that E[A t ] et, and V [A t ] 2 t 2 2α α) 2. Definition 4 Let F t be a fractional Liu process with index α. Then the fuzzy process is called a geometric fractional Liu process. G t exp et + F t ) 4) Theorem 3 For each t >, G t has a lognormal membership function )) π α) ln z et µz) 2 + exp, z >. 5) α Proof: If z >, then it follows that The proof is complete. µz) 2Cr{G t z}) 2Cr{et + F t ln z}) { }) ) ln z et ln z et 2Cr F t µ Ft )) π α) ln z et 2 + exp. α 2

3 Theorem 4 For each t >, the expected value of G t is where α / α). E[G t ] csc) exp Proof: According to the definition of expected value of fuzzy variable, we have E[G t ] Cr{G t x} { Cr F t + exp Let α / α). If < π, then we obtain E[G t ] csc) exp ) π2 et, < π 6) } ln x et π α)ln x et) α )). )) πln x et) + exp expπet/) expπet/) + x π2 et Otherwise, we have E[G t ] +. The proof is complete. π/ ). 4 A New Stock Model In this section, the bond price X t and the stock price Y t are assumed to be governed by dx t rx t dt 7) dy t ey t dt + Y t df t where r is the riskless interest rate, e is the stock drift, is the stock diffusion, and F t is a fractional Liu process with index α. In this model, the market is comprised of a riskless cash bond and a risky tradable stock. The stock price follows a geometric fractional Liu process Y t expet + F t ). European Options Considering this new stock model, we assume that a European call option has strike price K and expiration time T. Its pricing formula is given as follows. Theorem 5 European call option price formula for this stock model is f Y exp rt ) π α) K/Y + exp 6T α ln x et ) ). 8) 3

4 Proof: It follows from the definition of expected value that f exp rt )E[Y T K) + ] exp rt ) exp rt ) Y exp rt ) Y exp rt ) Cr{Y T K) + x} Cr{Y T K x} K/Y Cr{expeT + F T ) u}du π α) K/Y + exp 6T α ln x et ) ). Similarly, we consider the pricing formula of a European call option with strike price K and expiration time T for fractional Liu s stock model. Its pricing formula is given as follows. Theorem 6 European put option price formula for this stock model is f Y exp rt ) π α) K/Y + exp Proof: It follows from the definition of expected value that Acknowledgements f exp rt )E[K Y T ) + ] exp rt ) exp rt ) Y exp rt ) Y exp rt ) K/Y Cr{K Y T ) + x} Cr{K Y T x} K/Y 6T α et ln x) ). 9) Cr{expeT + F T ) u}du ). + exp π α) et ln x) 6T α This work was supported by National Natural Science Foundation of China Grant No References [] Hu Y, and ksendal, Fractional white noise calculus and applications to finance, Infinite Dimensional Analysis, Quantum Probability and Related Topics, Vol.6, No., -32, 23. [2] Li X, and Liu B, A sufficient and necessary condition for credibility measures, International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems, Vol.4, No.5, , 26. [3] Liu B, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, Vol.2, No., 3-6, 28. 4

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