European Option Pricing Methods Based On Fourier Cosine Method

Compared with American options and Bermuda options that have uncertain exercise time,European options,which can only exercise at maturity date,are widely used in international financial markets because of their easy operation and direct pricing.They are also chosen by Shanghai 50 ETF,which is the only floor traded stock option in China.In terms of pricing methods,since most of the extended models describing the underlying asset price process do not have analytic solutions,it is very important to study the numerical pricing method of European optionsFirstly,this paper briefly introduces several main numerical methods at present,and analyses their advantages and disadvantages from two aspects of error and time complexity,then draws out the main research object of this paper-Fourier Cosine method.Secondly,under the assumption of Black-Scholes model,European option pricing based on Fourier COS method is carried out,and the theoretical feasibility of this method is deduced and proved in detail.Then,by applying the risk-neutral pricing principle,the improved pricing formula of COS method is finally obtained,which is faster than the original method.After obtaining the numerical results,this paper continues to make a comparative analysis of the factors affecting the pricing effect of COS method.Firstly,we fix the original parameters unchanged.Through numerical verification,we find that the truncation coefficient L has a great impact on the overall method,and also revise the fixed value of L to the adaptive range.Secondly,we turn to analyze the influence of the parameters,showing that the option expiration date T and the number of Fourier expansion terms N have a significant impact on L,and the suggestion of increasing the number of terms N from 64 to 128 is put forward,which enlarges the adaptive range of L while increasing the stability of the model.Finally,this paper attempts to extend the COS method on BS model to Heston model and VG model,and solve some theoretical derivation and numerical analysis problems.