Can Standard Preferences Explain the Prices of Out-of-The-Money S&P 500 Put Options?

43 Pages Posted: 11 Aug 2005

See all articles by Luca Benzoni

Luca Benzoni

Federal Reserve Bank of Chicago - Research Department

Pierre Collin-Dufresne

Ecole Polytechnique Fédérale de Lausanne; Swiss Finance Institute; National Bureau of Economic Research (NBER)

Robert S. Goldstein

University of Minnesota - Twin Cities - Carlson School of Management; National Bureau of Economic Research (NBER)

Date Written: September 15, 2005

Abstract

Prior to the stock market crash of 1987, Black-Scholes implied volatilities of S&P 500 index options were relatively constant across moneyness. Since the crash, however, deep out-of-the-money S&P 500 put options have become 'expensive' relative to the Black-Scholes benchmark. Many researchers (e.g., Liu, Pan and Wang (2005)) have argued that such prices cannot be justified in a general equilibrium setting if the representative agent has 'standard preferences' and the endowment is an i.i.d. process. Below, however, we use the insight of Bansal and Yaron (2004) to demonstrate that the 'volatility smirk' can be rationalized if the agent is endowed with Epstein-Zin preferences and if the aggregate dividend and consumption processes are driven by a persistent stochastic growth variable that can jump. We identify a realistic calibration of the model that simultaneously matches the empirical properties of dividends, the equity premium, the prices of both at-the-money and deep out-of-the-money puts, and the level of the risk-free rate.

A more challenging question (that to our knowledge has not been previously investigated) is whether one can explain within a standard preference framework the stark regime change in the volatility smirk that has maintained since the 1987 market crash. To this end, we extend the model to a Bayesian setting in which the agent updates her beliefs about the average jump size in the event of a jump. Note that such beliefs only update at crash dates, and hence can explain why the volatility smirk has not diminished over the last eighteen years. We find that the model can capture the shape of the implied volatility curve both pre- and post-crash while maintaining reasonable estimates for expected returns, price-dividend ratios, and risk-free rates.

Keywords: Volatility smile, volatility smirk, implied volatility, option pricing, portfolio insurance, market risk

JEL Classification: G12, G13

Suggested Citation

Benzoni, Luca and Collin-Dufresne, Pierre and Goldstein, Robert S., Can Standard Preferences Explain the Prices of Out-of-The-Money S&P 500 Put Options? (September 15, 2005). Available at SSRN: https://ssrn.com/abstract=775146 or http://dx.doi.org/10.2139/ssrn.775146

Luca Benzoni (Contact Author)

Federal Reserve Bank of Chicago - Research Department ( email )

230 South LaSalle Street
Chicago, IL 60604
United States
312-322-8499 (Phone)

HOME PAGE: http://lbenzoni.frbchi.googlepages.com/

Pierre Collin-Dufresne

Ecole Polytechnique Fédérale de Lausanne ( email )

Quartier UNIL-Dorigny, Bâtiment Extranef, # 211
40, Bd du Pont-d'Arve
CH-1015 Lausanne, CH-6900
Switzerland

Swiss Finance Institute

c/o University of Geneva
40, Bd du Pont-d'Arve
CH-1211 Geneva 4
Switzerland

National Bureau of Economic Research (NBER)

1050 Massachusetts Avenue
Cambridge, MA 02138
United States

Robert S. Goldstein

University of Minnesota - Twin Cities - Carlson School of Management ( email )

19th Avenue South
Minneapolis, MN 55455
United States
612-624-8581 (Phone)

National Bureau of Economic Research (NBER)

1050 Massachusetts Avenue
Cambridge, MA 02138
United States

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