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

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.