Pricing with Variance Gamma Information

Abstract

In the information-based pricing framework of Brody, Hughston & Macrina, the market filtration {Ft}t≥0 is generated by an information process {ξt}t≥0 defined in such a way that at some fixed time T an FT-measurable random variable XT is “revealed”. A cash flow HT is taken to depend on the market factor XT, and one considers the valuation of a financial asset that delivers HT at time T. The value of the asset St at any time t∈[0,T) is the discounted conditional expectation of HT with respect to Ft, where the expectation is under the risk neutral measure and the interest rate is constant. Then ST−=HT, and St=0 for t≥T. In the general situation one has a countable number of cash flows, and each cash flow can depend on a vector of market factors, each associated with an information process. In the present work we introduce a new process, which we call the normalized variance-gamma bridge. We show that the normalized variance-gamma bridge and the associated gamma bridge are jointly Markovian. From these processes, together with the specification of a market factor XT, we construct a so-called variance-gamma information process. The filtration is then taken to be generated by the information process together with the gamma bridge. We show that the resulting extended information process has the Markov property and hence can be used to develop pricing models for a variety of different financial assets, several examples of which are discussed in detail.

Keywords

• information-based asset pricing
• Lévy processes
• gamma processes
• variance gamma processes
• Brownian bridges
• gamma bridges