Recovering Probabilities and Risk Aversion from Option Prices and Realized Returns

Jackwerth, Jens Carsten and Rubinstein, Mark (2003): Recovering Probabilities and Risk Aversion from Option Prices and Realized Returns. Published in: The Legacy of Fisher Black

Abstract

This paper summarizes a program of research we have conducted over the past four years. So far, it has produced two published articles, one forthcoming paper, one working paper currently under review at a journal, and three working papers in progress. The research concerns the recovery of market-wide risk-neutral probabilities and risk aversion from option prices. The work is built on the idea that risk-neutral probabilities (or their related state-contingent prices) are an amalgam of consensus subjective probabilities and consensus risk aversion. The most significant departure of this work is that it reverses the normal direction of previous research, which typically moves from assumptions governing subjective probabilities and risk aversion, to conclusions about risk-neutral probabilities. Our work is partially motivated by the conspicuous failure of the Black-Scholes formula to explain the prices of index options in the post-1987 crash market. First, we independently estimate risk-neutral probabilities, taking advantage of the now highly liquid index option market. We show that, if the options market is free of general arbitrage opportunities and we can at least initially ignore trading costs, there are quite robust methods for recovering these probabilities. Second, with these probabilities in hand, we use the method of implied binomial trees to recover a consistent stochastic process followed by the underlying asset price. Third, we provide an empirical test of implied trees against alternative option pricing models (including “naïve trader” approaches) by using them to forecast future option smiles. Fourth, we argue that realized historical distributions will be a reliable proxy for certain aspects of the true subjective distributions. We then use these observed aspects together with the option-implied risk-neutral probabilities to estimate market-wide risk aversion.

• Item Type: MPRA Paper
• Language: English
• Keywords: Risk Aversion; Option; Realized Returns
• Subjects: D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D81 - Criteria for Decision-Making under Risk and Uncertainty

References

Black, F. and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy 81, No. 3 (May-June 1973), pp. 637-659.

Breeden, D.T. and R.H. Litzenberger, “Prices of State-Contingent Claims Implicit in Option Prices,” Journal of Business 51, No. 4 (October 1978), pp. 621-651.

Cox, J.C., “The constant elasticity of variance option pricing model” Journal of Portfolio Management (Special Issue: A Tribute to Fischer Black, December 1996), pp. 15-17.

Cox, J.C, S.A. Ross and M. Rubinstein, “Option Pricing: A Simplified Approach,” Journal of Financial Economics 7, No. 3 (September 1979), pp. 229-263.

Dumas, B., J. Fleming and R. Whaley, “Implied Volatility Functions: Empirical Tests,” Journal of Finance 53, No. 6 (1998), pp. 2059-2106.

Gemmill, G. and N. Kamiyama, “International Transmission of Option Volatility and Skewness: When You’re Smiling, does the Whole World Smile?” City University Business School, London, working paper (February 1997)

Heston, S.L. “A Closed-Form Solution for Options with Stochastic Volatility and Applications to Bond and Currency Options,” Review of Financial Studies 6, No. 2 (Summer 1993), pp. 327-343.

Jackwerth, J., “Generalized Binomial Trees,” Journal of Derivatives 5, No. 2 (1997), pp. 7-17.

Jackwerth, J., “Recovering Risk Aversion from Option Prices and Realized Returns,” Review of Financial Studies 13, No. 2 (2000), 433-451.

Jackwerth, J.C., “Do We Live in a Lognormal World?” University of Wisconsin at Madison, working paper in progress (February 1997).

Jackwerth, J.C., “Geometric and Probabilistic Bounds on Option Prices,” University of Wisconsin at Madison, working paper in progress (March 1997).

Jackwerth, J.C. and M. Rubinstein, “Recovering Probabilities from Option Prices,” Journal of Finance 51, No. 5 (December 1996), pp. 1611-1631.

Jackwerth, J.C. and M. Rubinstein, “Recovering Stochastic Processes from Option Prices,” University of Wisconsin at Madison and University of California at Berkeley, working paper in progress (2001).

Leland, H., “Beyond Mean-Variance: Risk and Performance Measurement in a Nonsymmetrical World,” Financial Analyst Journal 55, (1999), pp. 27-36.

Merton, R.C., “Option Pricing When Underlying Stock Returns are Discontinuous,” Journal of Financial Economics 3, No. 1 (January-March 1976), pp. 125-144.

Rubinstein, M., “The Valuation of Uncertain Income Streams and the Pricing of Options,” Bell Journal of Economics and Management Science 7, No. 2 (Autumn 1976), pp. 407-425.

Rubinstein, M., “Nonparametric Tests of Alternative Option Pricing Models Using All Reported Trades and Quotes on the 30 Most Active CBOE Option Classes from August 23, 1976 through August 31, 1978,” Journal of Finance 40, No. 2 (June 1985), pp. 455-480.

Rubinstein, M., “Displaced Diffusion Option Pricing,” Journal of Finance 38, No. 1 (March 1983), pp. 213-217. Rubinstein, M., “Implied Binomial Trees,” Journal of Finance 49, No. 3 (Presidential Address to the American Finance Association, July 1994), pp. 771-818.

Toft, K.B. and B. Prucyk, “Options on Levered Equity: Theory of Empirical Tests,” Journal of Finance 52, No. 3 (July 1997), pp. 1151-1180.