An Empirical Study Of European Option Pricing And Implied Volatility Under The Heston Model

Black and Scholes proposed the famous BS model for pricing European option in 1973, which set off a revolution. But the outcome coming from this model was very different from the real thing as there were a lot of restrictions in it. Especially one of assumptions of this model:the volatility of stock price movement is a constant, which obeys a geometric Brownian motion. In a way, this assumption is against with real thing. Actually, the implied volatility is related with the maturity date and the striking price. It often presents like smile and tilted.In view of this drawback of BS model, some scholars used the stochastic volatility models to replace the constant volatility. Heston model is one of those models. It describes volatility smile very well. Its core idea is to add a stochastic process of volatility on the base of BS model so that the volatility of model becomes stochastic. There is no analytic solution of Heston model in most cases except that the underlying asset is a single one and the option is a kind of European option.Firstly, we introduce the basic theory and the analytic solution of BS model, then point out drawbacks of it and explain the necessity of other model. After that, we replace BS model with stochastic volatility model, taking Heston model as an example, present the theory and property of it in detail. At last we show the analytic solution of Heston model.Since Heston model's analytic solution is too complicated to calculate, taking European call option as an example, we take numerical methods of Gaussian Quadrature and Fast Fourier Transform to compute the price of option. Then we show the implied volatility after introducing the numerical methods to solve this model. According to the definition of implied volatility, we use the bisection method to calculate the implied volatility.After that, we imply the meaning of parameters estimation. Then we show some classical methods on parameters calibration of Heston model, discuss then-advantages and disadvantages, in view of this, we introduce the method of differential evolution which we use to calibrate the parameters of Heston model.According to FTSE 100 index option, we do some empirical investigating using numerical methods of Gaussian Quadrature and Fast Fourier Transform to compute the price of European option under the Heston model and calibrate the parameters of this model. Then we research the implied volatility of European option under the sample data, proving the existence of volatility skew.At last, the outcome shows the robustness of the program, presenting that this model and method can be taken into pricing the European option, which is helpful to the risk management.