A Note on the Black-Scholes Implied Volatility with Default Risk

This paper focuses on a theoretical aspect of relations between the Black-Scholes implied volatility and the default probability in a general framework that the stock price is fixed at zero after default occurs. It is shown that the default probability of the company under a risk-neutral measure significantly links to the implied volatility skew at extremely small strike. Moreover, it is proved that the divergence speed of the implied volatility must be determined uniquely in any defaultable economy under arbitrage-free condition. Finally, through a numerical test, we show whether our modelfree formula is applicable or not in practice