Lévy Random Bridges and the Modelling of Financial Information

Abstract

The information-based asset-pricing framework of Brody, Hughston and Mac- rina (BHM) is extended to include a wider class of models for market information. In the BHM framework, each asset is associated with a collection of random cash flows. The price of the asset is the sum of the discounted conditional expecta- tions of the cash flows. The conditional expectations are taken with respect to a ¯ltration generated by a set of 'information processes'. The information pro- cesses carry imperfect information about the cash flows. To model the flow of information, we introduce in this paper a class of processes which we term Levy random bridges (LRBs). This class generalises the Brownian bridge and gamma bridge information processes considered by BHM. An LRB is defined over a finite time horizon. Conditioned on its terminal value, an LRB is identical in law to a Levy bridge. We consider in detail the case where the asset generates a single cash flow XT occurring at a fixed date T. The flow of market information about XT is modelled by an LRB terminating at the date T with the property that the (random) terminal value of the LRB is equal to XT . An explicit expression for the price process of such an asset is found by working out the discounted conditional expectation of XT with respect to the natural filtration of the LRB. The prices of European options on such an asset are calculated.