Finite Difference Method for Pricing of Indonesian Option under a Mixed Fractional Brownian Motion

This paper deals with an Indonesian option pricing using mixed fractional Brownian motion to model the underlying stock price. There have been researched on the Indonesian option pricing by using Brownian motion. Another research states that logarithmic returns of the Jakarta composite index have long-range dependence. Motivated by the fact that there is long-range dependence on logarithmic returns of Indonesian stock prices, we use mixed fractional Brownian motion to model on logarithmic returns of stock prices. The Indonesian option is different from other options in terms of its exercise time. The option can be exercised at maturity or at any time before maturity with profit less than ten percent of the strike price. Also, the option will be exercised automatically if the stock price hits a barrier price. Therefore, the mathematical model is unique, and we apply the method of the partial differential equation to study it. An implicit finite difference scheme has been developed to solve the partial differential equation that is used to obtain Indonesian option prices. We study the stability and convergence of the implicit finite difference scheme. We also present several examples of numerical solutions. Based on theoretical analysis and the numerical solutions, the scheme proposed in this paper is efficient and reliable.