On an Extension of the Brownian Bridge with Applications in Finance

TL;DR

This paper extends the Brownian bridge model to include more general information processes, incorporating default risk and different noise types, with applications in credit risk modeling and finance.

Contribution

It introduces a generalized information-based asset-pricing framework that models default risk and complex noise processes, expanding the original Brownian bridge approach.

Findings

Modeling default risk with a combined Brownian bridge and Lévy process.

Conditions for maintaining the Markov property in extended models.

Explicit modeling of cash flow and bankruptcy time in credit assets.

Abstract

The main purpose of this paper is to extend the information-based asset-pricing framework of Brody-Hughston-Macrina to a more general set-up. We include a wider class of models for market information and in contrast to the original paper, we consider a model in which a credit risky asset is modelled in the presence of a default time. Instead of using only a Brownian bridge as a noise, we consider another important type of noise. We model the flow of information about a default bond with given random repayments at a predetermined maturity date by the so called market information process, this process is the sum of two terms, namely the cash flow induced by the repayment at maturity and a noise, a stochastic process set up by adding a Brownian bridge with length equal to the maturity date and a drift, linear in time, multiplied by a time changed L\'evy process. In this model the information concerning the random cash-flow is modelled explicitly but the default time of the company is not since the payment is contractually set to take place at maturity only. We suggest a model, in which the cash flow and the time of bankruptcy are both modelled, which covers contracts, e.g. defaultable bonds, to be paid at hit. From a theoretical point of view, this paper deals with conditions, which allow to keep the Markov property, when we replace the pinning point in the Brownian bridge by a process. For this purpose, we first study the basic mathematical properties of a bridge between two Brownian motions.