Pricing Options on Defaultable Stocks

In this note, we develop stock option price approximations for a model which takes both the risk o default and the stochastic volatility into account. We also let the intensity of defaults be influenced by the volatility. We show that it might be possible to infer the risk neutral default intensity from the stock option prices. Our option price approximation has a rich implied volatility surface structure and fits the data implied volatility well. Our calibration exercise shows that an effective hazard rate from bonds issued by a company can be used to explain the implied volatility skew of the implied volatility of the option prices issued by the same company.

💡 Research Summary

The paper develops a tractable approximation for pricing European options on stocks that are subject to both default risk and stochastic volatility, and it shows how the risk‑neutral default intensity can be inferred directly from option prices. The authors begin by reviewing the literature on structural and intensity‑based credit‑risk models as well as stochastic‑volatility models, pointing out that most existing frameworks treat default intensity as a constant or exogenous process, thereby ignoring the empirically observed link between a firm’s credit risk and its equity volatility.

To address this gap, the authors specify a joint dynamics for the stock price (S_t), its variance (V_t), and the default intensity (\lambda_t). Under the risk‑neutral measure the stock follows
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