ترس سقوط پست-'87 در بازار گزینه آینده S & P 500

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.

This article has presented evidence that post-87 distributions implicit in S&P 500 futures options are strongly negatively skewed, and has examined two competing hypotheses: a stochastic volatility model with negative correlations between index and volatility shocks, and a stochastic volatility jump-diffusion model with time-varying jump risk. The fundamental premise underlying the stochastic volatility model is confirmed: index and implicit volatility shocks are in fact strongly negatively correlated. However, this negative correlation is not sufficient of itself to generate sufficiently negative implicit skewness. An extremely high volatility of volatility is also necessary — implausibly high when judged against the time series properties of option prices. The stochastic volatility model also has some difficulty in matching observed option prices even with implausible parameter values, with an tendency to underprice 0–2 month options with low strike prices and overprice 2–6 month options with high strike prices. This article has also presented strong evidence against the hypothesized square root diffusion processes driving instantaneous volatility and jump risk. Such processes possess many desirable features (nonnegativity, leverage effects, analytic tractability), but cannot account for the large and typically positive implicit volatility shocks observed in the S&P 500 futures options market. Implicit factor evolutions appear better described by an asymmetric jump-diffusion. Whether such behavior is also observed for conditional volatilities of S&P 500 futures returns is an open question. Standard GARCH models substantially rule out conditional volatility jumps, while regime-switching models typically assume all volatility changes are jumps. The stochastic volatility/jump-diffusion model is more compatible with observed option prices, and generates more plausible stochastic volatility parameter values. However, the model still relies on a substantial amount of distributional randomization to match longer-maturity option prices; more so than justified by the observed volatility of implicit factors. The difficulty is that the volatility smile is equally pronounced for short- and long-maturity options, after scaling by the appropriate standard deviation at different maturities. Finite-variance jump explanations cannot match this pattern, because of rapid convergence towards lognormal distributions at longer maturities. Alternate infinite-variance processes that do not have this property are consequently worth exploring, such as McCullochs (1995) stable Paretian model. The substantial negative skewness and leptokurtosis implicit in actively traded short-maturity option prices appear fundamentally inconsistent with an absence of large weekly movements in S&P 500 futures returns over 1988–93. This incompatibility is most apparent in implicit risk-neutral distributions inferred from models with jumps, which fit short-maturity option prices quite well on average. These models assign a 90% risk-neutral probability of observing at least 1 weekly move of 10% in magnitude over 1988–93; none was observed. And while the stochastic volatility models yield a lower probability of large moves over 1988–93, the systematic option pricing errors of these models indicates they are understating the implicit risk-neutral probabilities of large moves at short horizons. Alternate explanations for the divergence between risk-neutral distributions and observed returns include peso problems, risk premia, and option mispricing. A log utility representative agent calibration indicates that peso problems alone cannot explain the divergence; large moves should have been observed over 1988–93 according to distributions inferred from option prices. However, more extreme risk aversion would presumably decrease the true/risk-neutral jump frequency ratio to the point where a peso problem explanation becomes viable. Whether this can be achieved under plausible levels of risk aversion is an open question. A more promising explanation lies in the industrial organization of stock index option markets. These markets have been functioning since the crash substantially as an insurance market for crash risk. Relatively few option marketmakers apparently have been writing crash insurance for a broad array of money managers, which may pose institutional difficulties for the risk-sharing assumptions underlying representative agent models. On the demand side, it is conceivable that especially risk-averse money managers have been willing to buy crash insurance that never seems to pay off. The puzzle is on the supply side. Why have new entrants not undercut the marketmakers, given no formal barriers to entry in option markets? Or, alternatively, why have existing marketmakers not devoted more capital to the profitable business of writing options? Perhaps the largest profit opportunities did not last long enough for new entry. While implicit crash fears were present throughout 1988–93, the largest implicit abnormalities declined substantially with the end of the Gulf war.