Asian Option Pricing With Transaction Costs Under The Mixed Brownian-fractional Brownian Model

Classic option pricing theory is built on the basis of efficient market hypothesis, that market has no arbitrage, logarithm stock price after a discount of risk-free rate follows a martingale process; this suggests the stock price increments in most cases should be mutually independent. However, empirical analysis and behavioral finance research results display: the stock returns have long-range dependence and momentum effect-the stock price increments are not independent and has a positive autocorrelation. Therefore it is necessary for us to study the option pricing problem when the stock returns have long-range dependence and momentum effect.Empirical analysis shows Brownian-fractional Brownian model can fit the long-term dependence and the momentum effects of stock returns well. Based on behavioral finance point of view, this paper gets an Asian option pricing formula under the mixed Brownian-fractional Brownian model with transaction costs in the discrete time setting.As one of the most active exotic options, Asian-style option is widely used in the financial market. It is a path-dependent option whose payoff depends on the average of underlying asset price over some pre-set period of time, and it is more complicated to price Asian-style option than standard options because of the path-dependence.Based on the anchoring and adjustment heuristic in behavioral finance, this paper obtains an Asian option pricing formula under the mixed Brownian-fractional Brownian model with transaction costs in the discrete-time trade setting by a mean-self-financing Delta-hedging argument; results show that time scalesδt and long-range dependence of stock returns play an important role in option pricing with transaction costs. In particular, the numerical results display that there exists fundamental difference between continuous-time trade and discrete-time trade even ifδt is small enough.This paper contains the following contents:The first chapter, as the introduction of this paper, describes the study background and the progress, briefly introduces some of the concepts of the Asian option and fractional Brownian motion, and some of the behavioral financial theory involved in the paper.The second chapter describes the option pricing model with transaction costs in more detail. First, we review the deduction process of the classic Black-Scholes model, and then we make a more comprehensive review to the option pricing model with transaction costs.The third chapter is the core of this paper, in which we deduce formula of Asian option pricing models with transaction costs systematically. By a mean-self-financing Delta-hedging argument in a discrete time setting, a geometric average Asian call option pricing formula is obtained. We show that time scalesδt and Hurst exponent H play an important role in option pricing with transaction costs.Chapter IV is the empirical part, we obtain how scaling and long-range dependence affect option pricing by numerical analysis. Results display that there exists fundamental difference between continuous-time trade and discrete-time trade even ifδt is small enough.