Contingent claims valuation when the security price is a combination of an Ito process and a random point process

Abstract

This paper develops several results in the modern theory of contingent claims valuation in a frictionless security market with continuous trading. The price model is a semi-martingale with a certain structure, making the return of the security a sum of an Ito-process and a random, marked point process. Dynamic equilibrium prices are known to be of this form in an ArrowDebreu economy, so there is no real limitation in our approach. This class of models is also advantageous from an applied point of view. Within this framework we investigate how the model behaves under the equivalent martingale measure in the P ∗- equilibrium economy, where discounted security prices are marginales. Here we present some new results showing how the marked point process affects prices of contingent claims in equilibrium. We derive a new class of option pricing formulas when the Ito process is a general Gaussian process, one formula for each positive L 2[O, T]-function. We show that our general model is complete, although the set of equivalent martingale measures is not a singleton. We also demonstrate how to price contingent claims when the underlying process has after-effects in all of its parameters.