A Jump Model with Binomial Volatility

We introduce a pricing model for equity options in which sample paths follow a variance-gamma (VG) jump model whose parameters evolve according to a two state Markov chain process. The model is capable of justifying the observed implied volatility skews for options at all maturities. Furthermore, the term structure of implied variance-rate appears to be an increasing function of the time to maturity, in agreement with empirical evidence. As in GARCH type models, jump sizes are positively correlated to volatility. Explicit extensions of the VG pricing formulas for European options are developed, in which complex combinatorial expressions arise whose valuation is hardly feasible. The main result of the article is a resummation algorithm based on the method of lines, which greatly reduces the algorithmic complexity of the pricing formulas. This algorithm is also the basis of approximate numerical schemes for American and Bermudan options, for which a state dependent exercise boundary can be computed. This work has been done jointly with S. Jaimungal and D. Rubisov and appeared in Quantitative Finance, April 2003.