Pricing S&P 500 Index Put Options: Smiles, Skews, and Leverage

Posted: 16 Mar 2007

See all articles by Robert L. Geske

Robert L. Geske

University of California, Los Angeles (UCLA) - Finance Area

Yi Zhou

San Francisco State University

Date Written: January 2007


The primary purpose of this paper is to examine whether leverage has a significant statistical and economic effect on the pricing of S&P 500 index put options. The secondary purpose is to present information regarding the shape and persistent smile rather than skew of the implied volatility function. This is the first paper to directly test for leverage effects in stock index put options. To analyze these effects we use the Geske (1979) compound option model. The Geske model is closed form, implies stochastic equity volatility, is consistent with Modigliani and Miller, incorporates debt refinancing, and includes possibly differential default and bankruptcy. Black-Scholes (1973) is a special case of the Geske model. In this paper we show that during the years 1996-2004 the aggregate market based debt to equity (D/E) ratio of the firms comprising the S&P 500 equity index varies from about 40-120 percent. We believe that we are the first to present a market D/E ratio derived from option theory. We also present evidence that on an average of about 200,000 options during this 8 year period the implied volatility most often exhibits a smile not a smirk or skew. Next and more importantly we are the first to report the details of the statistically significant economic effects that market leverage has on pricing S&P 500 index put options. We measure that the Geske model improves the net option valuation of listed in the money (or out of the money) S&P 500 index put options on average by about 37% (19%) compared to Black-Scholes values. We demonstrate that the improvement is directly (and monotonically) related to both the time to expiration of the option and the amount of leverage in this market index. For options with longer expirations and/or periods of higher market leverage the improvement is greater, ranging from about 50% to 85%. We also demonstrate economic significance in basis points by showing that dealers making a book in index options can expect benefits of at least several 100 basis points using Geske instead of Black-Scholes. Finally we show that the per cent pricing errors compare very favorably with Heston-Nandi (2000).

Keywords: Derivatives, Stochastic Stock Volatility, Leverage

JEL Classification: G12

Suggested Citation

Geske, Robert L. and Zhou, Yi, Pricing S&P 500 Index Put Options: Smiles, Skews, and Leverage (January 2007). Available at SSRN:

Robert L. Geske (Contact Author)

University of California, Los Angeles (UCLA) - Finance Area ( email )

University of California UCLA)-Financial Economics
Los Angeles, CA 90095-1481
United States

Yi Zhou

San Francisco State University ( email )

College of Business
1600 Holloway Avenue
San Francisco, CA 94132
United States
(415) 338-2661 (Phone)
(415) 338-0596 (Fax)


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