IJPAM: Volume 58, No. 1 (2010)

THE SOLUTIONS TO A BI-FRACTIONAL
BLACK-SCHOLES-MERTON DIFFERENTIAL EQUATION

Jin-Rong Liang$^1$, Jun Wang$^2$, Wen-Jun Zhang$^3$
Wei-Yuan Qiu$^4$, Fu-Yao Ren$^5$
$^1$Department of Mathematics
East China Normal University
Shanghai, 200241, P.R. CHINA
$^{2,4,5}$Department of Mathematics
Fudan University
Shanghai, 200433, P.R. CHINA
$^2$e-mail: majwang@fudan.edu.cn
$^3$Department of Mathematics
Shenzhen University
Shenzhen, 518060, P.R. CHINA


Abstract.A model for option pricing of a two parameter $(\gamma,\alpha)$-fractional Black-Scholes-Merton differential equation is established based on the stock price modeled by $(dS_t)^{\alpha}=\mu (S_t)^{\alpha}(dt)^{\alpha}+\sigma
(S_t)^{\alpha}$ $dW_{\alpha}(t)$, where $\alpha >0,$ $\mu,\sigma$ are constants and $dW_{\alpha}(t)=\varepsilon (dt)^{\alpha/2},$ fractional Wiener process, $\varepsilon $ obeys standard normal distribution. We solve the bi-fractional Black-Scholes-Merton differential equation obtained under the key boundary condition $C(S, t) = \max (S - K, 0)$ for call option and $P(S, t) = \max (K - S, 0)$ for put option at time $T$, the maturity date of the option, and obtain the explicit option pricing formulas for European call option and put option for $\gamma>0,
1\leq \alpha \leq 2.$

Received: December 22, 2009

AMS Subject Classification: 35K57, 35K99

Key Words and Phrases: option pricing, Black-Scholes-Merton differential equation, fractional derivatives, Taylor series of fractional order

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 58
Issue: 1