Information-Based Approach: Pricing of a Credit Risky Asset in the Presence of Default Time

TL;DR

This paper extends an information-based asset pricing model to include a stochastic bankruptcy time, deriving asset prices and option values with explicit formulas, emphasizing the default time's unpredictability in financial markets.

Contribution

It introduces a novel framework incorporating stochastic bankruptcy time into the information-based asset pricing model, with explicit computations of price dynamics and default compensators.

Findings

Derived explicit asset price process and option prices.

Showed the default time is totally inaccessible, modeling unpredictable default.

Provided explicit formulas for compensators of key stochastic processes.

Abstract

We extend the information-based asset-pricing framework by Brody, Hughston \& Macrina to incorporate a stochastic bankruptcy time for the writer of the asset. Our model introduces a non-defaultable cash flow $Z_T$ to be made at time $T$, alongside the time $\tau$ of a possible bankruptcy of the writer of the asset are in line with the filtration generated by a Brownian random bridge with length $\nu=\tau \wedge T$ and pinning point $\sigma Z_T$, where $\sigma$ is a constant. Quantities $Z_T$ and $\tau$ are not necessarily independent. The model does not depend crucially on the interpretation of $\tau$ as a bankruptcy time. We derived the price process of the asset and compute the prices of associated options. The dynamics of the price process satisfy a diffusion equation. Employing the approach of P.-A.~ Meyer, we provide the explicit computation of the compensator of $\nu$. Leveraging special properties of the bridge process, we also provide the explicit expression of the compensator of $Z_T\,\mathbb{I}_{[\nu,+\infty)}$. The resulting conclusion highlights the totally inaccessible property of the stopping time $\nu$. This characteristic is particularly suitable for financial markets where the time of default of a writer cannot be predictable from any other signal in the system until default happens.