Post-’87 crash fears in the S&P 500 futures option market. (English) Zbl 0942.62118

Summary: Post-crash distributions inferred from S&P 500 future option prices have been strongly negatively skewed. This article examines two alternate explanations: stochastic volatility and jumps. The two option pricing models are nested, and are fitted to S&P 500 futures options data over 1988-1993. The stochastic volatility model requires extreme parameters (e.g., high volatility of volatility) that are implausible given the time series properties of option prices. The stochastic volatility/jump-diffusion model fits option prices better, and generates more plausible volatility process parameters. However, its implicit distributions are inconsistent with the absence of large stock index moves over 1988-93.


62P05 Applications of statistics to actuarial sciences and financial mathematics
91B28 Finance etc. (MSC2000)
62P20 Applications of statistics to economics


Full Text: DOI


[1] Aı̈t-Sahalia, Y.; Lo, A.W., Nonparametric estimation of state-price densities implicit in financial asset prices, Journal of finance, 53, 499-547, (1998)
[2] Bakshi, G.; Cao, C.; Chen, Z., Empirical performance of alternative option pricing models, Journal of finance, 52, 2003-2049, (1997)
[3] Barone-Adesi, G.; Whaley, R.E., Efficient analytic approximation of American option values, Journal of finance, 42, 301-320, (1987)
[4] Bates, D.S., The crash of ’87was it expected? the evidence from options markets, Journal of finance, 46, 1009-1044, (1991)
[5] Bates, D.S., Dollar jump fears, 1984-1992distributional abnormalities implicit in currency futures options, Journal of international money and finance, 15, 65-93, (1996)
[6] Bates, D.S., Jumps and stochastic volatilityexchange rate processes implicit in PHLX deutsche mark options, Review of financial studies, 9, 69-107, (1996)
[7] Bates, D.S., Testing option pricing models, (), 567-611
[8] Bates, D.S., The skewness premiumoption pricing under asymmetric processes, Advances in futures and options research, 9, 51-82, (1997)
[9] Black, F., 1976. Studies of stock price volatility changes. Proceedings of the 1976 Meetings of the American Statistical Association, pp. 177-181.
[10] Black, F.; Scholes, M., The pricing of options and corporate liabilities, Journal of political economy, 81, 637-659, (1973) · Zbl 1092.91524
[11] Campbell, J.Y.; Hentschel, L., No news is good newsan asymmetric model of changing volatility in stock returns, Journal of financial economics, 31, 281-318, (1992)
[12] Canina, L.; Figlewski, S., The informational content of implied volatility, Review of financial studies, 6, 659-682, (1993)
[13] Chaudhury, M.M.; Wei, J., Upper bounds for American futures optionsa note, Journal of futures markets, 14, 111-116, (1994)
[14] Cochrane, J.H., Saá-Requejo, J., 1996. Beyond arbitrage: good-deal asset price bounds in incomplete markets. National Bureau of Economic Research working paper 5489.
[15] Cox, J.C.; Ingersoll, J.E.; Ross, S.A., A theory of the term structure of interest rates, Econometrica, 53, 385-407, (1985) · Zbl 1274.91447
[16] Das, S.R., Sundaram, R.K., 1997. Taming the skew: higher-order moments in modeling asset price processes in finance. National Bureau of Economic Research working paper 5976.
[17] Day, T.E.; Lewis, C.M., Stock market volatility and the information content of stock index options, Journal of econometrics, 52, 267-287, (1992)
[18] Dempster, A.P.; Laird, N.M.; Rubin, D.B., Maximum likelihood from incomplete data via the EM algorithm, Journal of the royal statistical society B, 39, 1-38, (1977) · Zbl 0364.62022
[19] Derman, E.; Kani, I., Riding on a smile, Risk, 7, 32-39, (1994)
[20] Diz, F., Finucane, T.J., 1993. Do the options markets really overreact? Journal of Futures Markets 13, 298-312.
[21] Dumas, B.; Fleming, J.; Whaley, R.E., Implied volatility functions: empirical tests, Journal of finance, 53, 2059-2106, (1998)
[22] Dupire, B., Pricing with a smile, Risk, 7, 18-20, (1994)
[23] Fackler, P.L.; King, R.P., Calibration of option-based probability assessments in agricultural commodity markets, American journal of agricultural economics, 72, 73-83, (1990)
[24] Fleming, J., The quality of market volatility forecasts implied by S&P 100 index option prices, Journal of empirical finance, 5, 317-345, (1998)
[25] Franks, J.R.; Schwartz, E.S., The stochastic behaviour of market variance implied in the prices of index options, The economic journal, 101, 1460-1475, (1991)
[26] Galai, D., The components of the return from hedging options against stocks, Journal of business, 56, 45-54, (1983)
[27] Gemmill, G., Did option traders anticipate the crash? evidence from volatility smiles in the U.K. with U.S. comparisons, Journal of futures markets, 16, 881-898, (1996)
[28] Gennotte, G.; Leland, H.E., Market liquidity, hedging, and crashes, American economic review, 80, 999-1021, (1990)
[29] George, T.J.; Longstaff, F.A., Bid – ask spreads and trading activity in the S&P 100 index options market, Journal of financial and quantitative analysis, 28, 381-398, (1993)
[30] Geske, R., The valuation of compound options, Journal of financial economics, 7, 63-81, (1979)
[31] Grossman, S.J., An analysis of the implications for stock and futures price volatility of program trading and dynamic hedging strategies, Journal of business, 61, 275-298, (1988)
[32] Grossman, S.J.; Zhou, Z., Equilibrium analysis of portfolio insurance, Journal of finance, 51, 1379-1403, (1996)
[33] Hamilton, J.D., Time series analysis, (1994), Princeton University Press Princeton, NJ · Zbl 0831.62061
[34] Heston, S.L., A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of financial studies, 6, 327-344, (1993) · Zbl 1384.35131
[35] Hull, J.; White, A., The pricing of options on assets with stochastic volatility, Journal of finance, 42, 281-300, (1987)
[36] Ingersoll, J.E., Theory of financial decision making, (1987), Rowman and Littlefield Savage, MD
[37] Jacklin, C.; Kleidon, A.W.; Pfleiderer, P., Underestimation of portfolio insurance and the crash of October 1987, Review of financial studies, 5, 35-63, (1992)
[38] Kendall, M.G.; Ord, J.K.; Stuart, A., Kendall’s advanced theory of statistics, (1987), Oxford University Press New York
[39] McCulloch, J.H., Financial applications of stable distributions, (), 393-425
[40] Merton, R.C., Option pricing when underlying stock returns are discontinuous, Journal of financial economics, 3, 125-144, (1976) · Zbl 1131.91344
[41] Merville, L.J.; Pieptea, D.R., Stock-price volatility, Mean-reverting diffusion, and noise, Journal of financial economics, 242, 193-214, (1989)
[42] Nandi, S., 1996. Pricing and hedging index options under stochastic volatility: an empirical examination. Federal Reserve Bank of Atlanta working paper.
[43] Nelson, D.B., Conditional heteroskedasticity in asset returnsa new approach, Econometrica, 59, 347-370, (1991) · Zbl 0722.62069
[44] Palm, F.C.; Vlaar, P.J.G., Simple diagnostic procedures for modeling financial time series, Allgemeines statistisches archiv, 81, 85-101, (1997)
[45] Pearson, K., On a method of determining whether a sample of size n supposed to have been drawn from a parent population having a known probability integral has probably been drawn at random, Biometrika, 25, 379-410, (1933) · JFM 59.1174.02
[46] Platen, E.; Schweitzer, M., On feedback effects from hedging derivatives, Mathematical finance, 8, 67-84, (1998) · Zbl 0908.90016
[47] Poterba, J.; Summers, L., The persistence of volatility and stock market fluctuations, American economic review, 76, 1142-1151, (1986)
[48] Rubinstein, M., Displaced diffusion option pricing, Journal of finance, 38, 213-217, (1983)
[49] Rubinstein, M., Implied binomial trees, Journal of finance, 49, 771-818, (1994)
[50] Ruud, P.A., Extensions of estimation methods using the EM algorithm, Journal of econometrics, 49, 305-341, (1991) · Zbl 0742.62106
[51] Shimko, D., Bounds of probability, Risk, 6, 33-37, (1993)
[52] Schmalensee, R.; Trippi, R.R., Common stock volatility expectations implied by option premia, Journal of finance, 33, 129-147, (1978)
[53] Shumway, R.H.; Stoffer, D.S., An approach to time series smoothing and forecasting using the EM algorithm, Journal of time series analysis, 3, 253-264, (1982) · Zbl 0502.62085
[54] Silva, E.M.; Kahl, K.H., Reliability of Soybean and corn option-based probability assessments, Journal of futures markets, 13, 765-779, (1993)
[55] Smith, J.Q., Diagnostic checks of non-standard time series models, Journal of forecasting, 4, 283-291, (1985)
[56] Stein, J.C., Overreactions in the options market, Journal of finance, 44, 1011-1023, (1989)
[57] Taylor, S.J.; Xu, X., The term structure of volatility implied by foreign exchange options, Journal of financial and quantitative analysis, 29, 57-74, (1994)
[58] Trautmann, S., Beinert, M., 1995. Stock price jumps and their impact on option valuation. University of Mainz (Germany) working paper.
[59] Watson, M.W.; Engle, R.F., Alternative algorithms for the estimation of dynamic factor, MIMIC, and varying coefficient regression nodels, Journal of econometrics, 23, 385-400, (1983) · Zbl 0534.62083
[60] Whaley, R.E., Valuation of American call options on dividend-paying stocks, Journal of financial economics, 10, 29-58, (1982)
[61] Whaley, R.E., Valuation of American futures optionstheory and empirical tests, Journal of finance, 41, 1, 127-150, (1986)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.