The Statistical Properties Of Q-Gaussian Process And Its Application In Option Pricing

Option pricing is one of the hot topics in modern financial mathematics.The classical Black-Scholes price model laid the foundation for the theory of that stock prices obey geometric Brownian motion.However,the sharp peaks and thick tails as well as the long-term memory of the asset price distribution in the actual financial market contradict the classical model assumptions,resulting in the deviation of option prices.As a modification of Black-Scholes model,the price model based on Tsaliis non-extensive physics not only enriches the theory of asset price model,but also effectively reduces the deviation caused by Black-Scholes model.Firstly,this paper discusses the non-extensive Tsallis theory and introduces the concept of q-Gaussian process.The probability density function is used to calculate the mean and variance functions of the process.Since only one-step transfer density of the process can be obtained,the non-Markovian property of the q-Gaussian process is discussed by numerical calculation.The results show that the process has martingale property and self-similarity.In view of the fact that this process has the same long dependence as fractional Brownian motion,this paper compare the two processes and explore the relationship between the parameters of the two processes.Secondly,since the abnormal diffusion characteristics of the q-Gaussian process can well characterize the distribution of financial asset prices,this paper studies the pricing of European option driven by the q-Gaussian process.Using equivalent martingale measure transformation and (?) formula,the stock price formula based on q-Gaussian process is obtained.Because of the unsolvable integral problem in the subsequent pricing process of the European option,this paper proposes to use the least square method to solve the problem,thus obtaining the closed solution of the European option price.Numerical simulation results show that the least square method is superior to the other two approximation methods proposed by Borland in 2002 and2004.Finally,this paper reuses the q-Gaussian driven pricing model to price geometric average Asian options.Different from the pricing process of European option,the problem of double integral approximation is more difficult to solve when pricing Asian option.The least square method is also used to approximate the integral problem and the closed solution of geometric average Asian option is obtained.Compared with the pricing results of Zhao and Wang respectively,the numerical simulation shows that the least squares approximation method is also applicable to the pricing of Asian options,and more accurate results are obtained.