Hedging of options when the price process has jumps whose arrival rate depends on the price history

A traditional model for financial asset prices is that of a solution of a stochastic differential equation, driven by Brownian motion and Lebesgue measure; that is, a standard diffusion. The classic Black-Scholes model is a special case of this rubric. In some situations, however, such a model is inappropriate. In particular, empirical work has led researchers to conclude that appropriate models often contain price processes with jumps. This is reflected both in simple observations of price processes, and in statistical analysis of tail distributions (that is, the existence and persistence of ‘heavy tails’, that diffusion models do not have). Further, when modeling implied volatility surfaces, models that allow for jumps fit the data better than do models that do not, especially when times are close to maturity. Building on the pioneering work of R. Frey, we consider models where the price process of a risky asset can have jumps following a specific structure, as well as a diffusion component. Such models create interesting problems, since one can no longer use the theory of complete markets, but instead must rely on alternatives, such as the construction of minimal martingale measures. We show how this can be done and how options can be priced in this framework. We go further, however, and we consider the case where the jumps of the price process can depend on the history of the process; there are sound economic reasons for considering such models, and while they lead to further complications in the analysis, they are nevertheless still tractable, as we show. The possibility that the jumps depend on the past history of the process is new for this type of model