Pricing EIA With Cliquet-style Guarantees By Frame Duality Projection

This thesis presents an efficient and accurate method for pricing equity-indexed annuities(EIAs)with cliquet-style payoff structures.First,we introduce the conceptions and the properties of Lévy processes,Cox-Ingersoll-Ross(CIR)process,and timechanged Lévy processes.We assume that the log-return of underlying asset follows the general Lévy processes and time-changed Lévy processes,respectively.Then,we introduce how to use the known information to derive the density function of the random variable.In this work,the frame duality projection,also known as PROJ method,is applied to expand the density function in series based on Riesz Theory,and the Fast Fourier Transform(FFT)algorithm is used to get the coefficients.Moreover,two kinds of payoff structure are displayed.One has both local and global restrictions on the rate of return,the other is multiplicative.For the general Lévy processes,the details about the calculation method are present and the error analysis is included.We utilize five-point Gaussian quadrature rule to approximate some complex integrals.For the time-changed Lévy process,the dynamic of volatility is depicted by CIR process.After obtaining the characteristic function of time-changed Lévy process,PROJ method is used to the terminal payoff to derive the price.Besides,to accelerate the convergence rate,we use spectral filter on the terminal payoff structure since the Gibbs phenomenon is obvious when the function is not smooth.With the help of these techniques,we derive the formula of the price of equity-indexed annuities under time-changed Lévy models within some fixed time intervals.The last part is to compare the results with the Monte Carlo simulations to testify the accuracy and efficiency of our algorithm.The influence of different parameters on the price is discussed and the corresponding figures are present.In short,we find that PROJ method has a good performance in annuity pricing.When the distribution of the random variable is unknow,PROJ method can utilize the known information to approximate the density,so as to derive the value of the annuities within tolerant error.At the same time,we can reduce the error by increasing the number of bases.In addition,compared with the results of Monte Carlo simulation,PROJ method can greatly shorten the operation time while ensuring the accuracy.