Fractional Fourier Transform For Pricing American Double-Barrier Option

Barrier options have developed rapidly since the 1990s.Barrier options are one of the path-dependent option with the most trading volume in the over-the-counter market.With the rapid development of barrier options,it is urgent to conduct theoretical research on barrier options.The pricing research of barrier options has important value both in theory and in the real market.In the current research of option pricing,there are many literatures on studying European barrier options,and there are few studies on American barrier options.The early implementation characteristics of American barrier options can avoid some risks.However,it is a free and nonlinear boundary problem in mathematical models so it can not be solved in the analytical form.The uncertainty of free boundaries makes it difficult to solve differential equations,so that it can only be solved by numerical methods.For many more complex pricing systems,such as the Levy process,the American double barrier option pricing problem is more complicated.The numerical method is applied to the American barrier option pricing problem.On the one hand,it is based on the risk neutral method.On the other hand,the partial differential equation is written on basis of the B-S equation,and then the numerical method is used to solve it.Fourier transform theory is used in this thesis to price options and study the pricing problem of American double barrier options under Levy process.This dissertation is divided into three parts:The first part introduces the Euler method under the fast Fourier transform,and uses the Fourier transform to transform the partial differential equation satisfying the price of the American double barrier option into the initial value problem of the ordinary differential equation.The second part introduces the CONV method under the fractional Fourier transform,which converts the transition probability of the value of the option from t_k+1 to t_k.As a result that Fourier transform of the function convolution is the product of the function Fourier transform,the Fourier transform can be used to discretize the integral function,which in turn iterates out the value of the option at time t₀.The third part introduces the C-N difference method.The differential equations of the option value are discretized into difference equations by using the finite difference method.By constructing the difference format,the problem of the partial differential equations is solved into algebraic equations,and the equations are solved with the discrete solution consisting of an approximation solution.Through comparing the three methods,the calculation accuracy and time of the three methods are analyzed using the binary tree method with 1600 steps as the standard,and the following results are obtained:In terms of calculation accuracy,compared with Euler method and C-N difference method,CONV method has higher calculation accuracy,followed by Euler method,and the lowest accuracy is C-N difference method.Considering that the research object is American double barrier Options,due to the existence of barrier,various numerical methods have errors in discretizing stock prices.In the CONV method,when the fractional Fourier transform is used to solve the problem,the accuracy of the CONV method also changes with the change of the value of N.The specific expression is:the larger N,the higher the accuracy.In the comparison of operation time,among the four methods,the Euler method under the fast Fourier transform uses the shortest time,followed by the finite difference method,and the CONV method under the fractional Fourier transform takes a long time.The binary tree method with 1600 steps takes a lot of time steps,so it takes far more than the other three methods.