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

Wilmott Journal, Vol. 2, No. 3, 2010

19 Pages Posted: 12 Nov 2008 Last revised: 14 Jun 2016

Shuichi Ohsaki

Merrill Lynch & Co.

Takaaki Ozeki

Mizuho-DL Financial Technology Co., Ltd.

Yuji Umezawa

Mizuho-DL Financial Technology Co., Ltd.

Akira Yamazaki

Hosei University - Graduate School of Business Administration

Date Written: November 10, 2008

Abstract

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.

Keywords: Implied Volatility, Default Probability, Arbitrage-Free Condition

JEL Classification: G13, G14, G33, C50

Suggested Citation

Ohsaki, Shuichi and Ozeki, Takaaki and Umezawa, Yuji and Yamazaki, Akira, A Note on the Black-Scholes Implied Volatility with Default Risk (November 10, 2008). Wilmott Journal, Vol. 2, No. 3, 2010. Available at SSRN: https://ssrn.com/abstract=1299322

Shuichi Ohsaki (Contact Author)

Merrill Lynch & Co.

Tokyo
Japan

Takaaki Ozeki

Mizuho-DL Financial Technology Co., Ltd. ( email )

Japan

Yuji Umezawa

Mizuho-DL Financial Technology Co., Ltd. ( email )

Japan

Akira Yamazaki

Hosei University - Graduate School of Business Administration ( email )

Japan

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