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 ${Z}_{T}error\; "Character\; not\; currently\; supported:\; (FullDesc)"{\mathbb{I}}_{[\nu,+\infty )}$. 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.

Keywords

Brownian random bridge / Semimartingale / Local time / Compensator process / Information-based asset pricing / Credit risk / Default time / Totally inaccessible stopping time

Acknowledgements

I would like to express my deep gratitude to the referee for carefully reading the manuscript and providing invaluable comments.

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