An Adaptive Evolutionary Approach to Option Pricing via Genetic Programming

We propose a methodology of Genetic Programming to approximate the relationship between the option price, its contract terms and the properties of the underlying stock price. An important advantage of the Genetic Programming approach is that we can incorporate currently known formulas, such as the Black-Scholes model, in the search for the best approximation to the true pricing formula. Using Monte Carlo simulations, we show that the Genetic Programming model approximates the true solution better than the Black-Scholes model when stock prices follow a jump-diffusion process. We also show that the Genetic Programming model outperforms various other models in many different settings. Other advantages of the Genetic Programming approach include its robustness to changing environment, its low demand for data, and its computational speed. Since genetic programs are flexible, self-learning and self-improving, they are an ideal tool for practitioners. The Black-Scholes model is a landmark in contingent claim pricing theory and has found wide acceptance in financial markets. The search for a better option pricing model continues, however, as the Black-Scholes model was derived under strict assumptions that do not hold in the real world and model prices exhibit systematic biases from observed option prices. While many extensions and alternative models have been suggested, none seems to be complete (Rubinstein, 1997). We propose a new methodology of Genetic Programming for better approximating the elusive relationship between the option price and its contract terms and properties of the underlying stock price. This method requires minimal assumptions and can easily adapt to changing and uncertain economic environments. Many researchers have attempted to explain the systematic biases of the Black-Scholes model as an artifact of its assumptions. The most often challenged assumption is the normality of stock returns. Merton (1976) and Ball and Torous (1985) propose a Poisson jump-diffusion returns processes. French, Schwert and Stambaugh (1987) and Ballie & DeGennaro (1990) 1 Black-Scholes pricing biases have been related to volatility, strike price, time to maturity, volume and leverage. Black and Scholes (1972) and Galai (1977) show that it is possible to make excess returns by buying "undervalued" (relative to the Black-Scholes price) and selling "overvalued" options. Mac Beth and Merville (1979) find that Black-Scholes overprices out-of-the-money and underprices in-the-money options. Rubinstein (1985) finds the same pricing biases as Mac Beth and Merville (1979) in the first half of his data set and the opposite biases in the second half. 2 Normality of stock returns has been repeatedly rejected. Indeed, Kim and Kon (1994) have ranked candidates for return distributions and found normality to be the least likely. Their rankings are: 1) Intertemporal dependence models (ARCH, GARCH), 2) Student t, 3) Generalized mixture of normal distributions, 4) Poisson jump, and 5) Stationary normal.