| Summary: | Although the Black and Scholes (1973) model achieved great success in option pricing theory, the two obvious phenomena have received much attention in past decades (Kou, 2002). One is the asymmetric leptokurtic features; the other is the volatility “smiles”. To modify the Black and Scholes (1973) model, we introduce the Kou (2002) double exponential jump-diffusion model. It is more consistent with the price process than the Black and Scholes (1973) model. The Kou (2002) model not only contains the "normal" continuous process but also include "abnormal" jumps caused by outside news.
Furthermore, to describe the stochastic volatility, we introduce GARCH (1,1) for its good performance on estimation of volatility. We update the Black and Scholes (1973) and
the Kou (2002)model by substituting the constant volatility for the stochastic one.
To compare the model, we choose Monte Carlo simulation to estimate the theoretical option price of S&P 500 in the empirical study. The performance of the Kou (2002) model
is much better than the Black and Scholes (1973) model. The performance Black and Scholes (1973) & GARCH (1,1) model is the worst, and the Kou (2002) & GARCH (1,1) model is not better than the Kou (2002) model.
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