Models for option prices

Cox, Ross, and Rubinstein [6] introduced a binomial option price model and derived the seminal Black–Scholes pricing formula. In this paper we characterize all possible stock price models that can be approximated by the binomial models and derive the corresponding approximations for the pricing formulas. We introduce two additional randomizations in the binomial price models seeking more general and more realistic limiting models. The first type of model is based on a randomization of the number of price changes, the second one on a randomization of the ups and downs in the price process.As a result we also obtain price models with fat tails, higher peaks in the center, nonsymmetric etc., which are observed in typical asset return data. Following similar ideas as in [6] we also derive approximating option pricing formulas and discuss several examples.