Asian Option Pricing With Transaction Costs Under Time-changed Brownian Motion

Asian option is a kind of path-dependent options. Its returns at expiration date depend on the average of the underlying asset price for the option in the period of validity, thus to minimize the effects of the price fluctuation and make the Asian option cheaper and more suited to the needs of customers than standard option.Although B-S model is still the most classic and the most widely used tool for option pricing, empirical researches show that it can’t reflect a lot of typical characteristics of the price fluctuation. So the improvement on classical B-S model is very meaningful,such as fractional B-S model, subdiffusive B-S model, etc. The B-S model under time-changed Brownian motion is a class of subdiffusive B-S model. Therefore, this paper studies the problem of Asian option pricing under time-changed Brownian motion with transaction costs. The main contents are as follows:(1) The constant value periods for the model of time-changed geometric Brownian motion are described. In the presence of fixed proportional transaction costs,by a mean self-financing Dealt-hedging strategy in a discrete time setting, we establish the pricing model of geometric average Asian call option under the time-changed geometric Brownian motion. Through Fourier transform and Laplace transform, we get the pricing formula. Then the call-put parity formula and the pricing formula of Asian put option are deduced. Finally using Matlab software to make numerical experiments, the influence of various parameters on the option value is discussed.(2) The pricing model of geometric average Asian call option with fixed proportional transaction costs under time-changed geometric fractional Brownian motion is established. We use variable substitution to convert it into heat conduction equation,and solve the equation to get analytic expression of Asian call option. Then we deduce the parity equation and analytic expression of put option. Finally through numerical experiments, the influence of various parameters on the option value is analyzed.(3) Dealt-hedging strategy is used to establish the pricing model of arithmetic average Asian call option with monotonous transaction costs under time-changed geometric fractional Brownian motion. Using the method of dimension reduction, the three-dimensional problem is transformed into a two-dimensional problem. Then we use the upwind difference numerical format to solve the problem. And the stability and error of the numerical format are analyzed. Finally, we simulate the pricingformula and verify the effectiveness of the numerical format by use of Matlab software. The influence of various parameters on the option value is discussed.