| A model for stock prices by T. W. Epps, based on an extension of the Galton-Watson branching process, is studied, especially in the context of option pricing. A proof of the existence of an equivalent martingale measure is established, and closed formulas for call and put prices are given. A key feature of the model is that the level of the stock price affects its volatility. Moreover, there is a positive probability that the stock price will fall to zero and remain there permanently. The options priced by this model exhibit the so-called volatility smirk; a proof of this fact is given, involving sums of hypergeometric functions. Finally, the issue of put-call symmetry is addressed in the context of both branching processes and more general models. A useful characterization of the probability distributions and densities of all models that do satisfy put-call symmetry is revealed, along with applications. |
