The Gram-charlier Model Of Up And Out Call Barrier Option

As early as 1973,the Black-Scholes option pricing formula based on the non-dividend payout option was deduced from Fisher Black and Myron Scholes.In the same year,Robert Merton proposed a generalized model.In the option pricing model,assuming that the return rate of the underlying assets is in accordance with the geometric Brownian motion process and the volatility is constant,and only concerned with the mean and variance of the underlying asset price distribution.However,in the actual situation,the distribution of the underlying asset return process is like "spikes thick tail".Therefore,based on the traditional Black-Scholes-Merton model,this paper does not assume that the return on assets is geometric Brownian motion,but the expression rate of the underlying asset is expressed by high-order statistics,and then the Gram-Charlier series theorem is used to relate the statistic to the skewness and kurtosis of the underlying asset,so the probability density function of the underlying asset contains the skewness and the kurtosis coefficient,and then according to the Black-Scholes formula,and deduces the pricing formula of the European call option and up-out barrier option based on skewness and kurtosis adjustment.Finally,taking the actual closing price of ETF50 fund as an example,the paper uses MATLAB to calculate the price of up-out barrier option in Black-Scholes option pricing model and Gram-Charlier pricing model.