Option Pricing Under Stochastic Crashes

The crashes in the financial markets tend to lead to some departure of the option value from the existing option pricing models, such as on October 19, 1987, the U.S. stock market experienced a crash, so a direct result of the stock index options is a departure from the Black-Scholes model. This paper gives some attempts of option pricing in the financial markets that have the crash and future crash.Firstly, according to empirical studies that give the dynamic structure after a crash in the financial markets, and assuming that after the crash there is a Poisson jump process, and that its magnitude of jump subjects to double exponential distribution, the paper derives the option pricing partial differential equations in the financial markets after a crash; and then regarding random jump each time that the market suffered as a crash, a simple pricing model is presented, and using the Monte Carlo method to compute the European call option price; Finally, with the GARCH model to estimate volatility, improves the estimating methods of the second part in the idea. In this paper, the return of asset generating from the stochastic difference equations, compared to the normal distribution, have thicker left tail and higher peak. Also, found that compared the results presented in this paper with the classic Black-Scholes formula, with the horizontal axis as the strike price the line is gentler, the classic Black-Scholes model gives higher price with the low strike price than the model presented in this paper and because of low strike prices Black-Scholes model overestimates the option values.