A relationship exists between aggregate risk-neutral and subjective probability distributions and risk aversion functions. We empirically derive risk aversion functions implied by option prices and realized returns on the S&P500 index simultaneously. These risk aversion functions dramatically change shapes around the 1987 crash: Precrash, they are positive and decreasing in wealth and largely consistent with standard assumptions made in economic theory. Postcrash, they are partially negative and partially increasing and irreconcilable with those assumptions. Mispricing in the option market is the most likely cause. Simulated trading strategies exploiting this mispricing show excess returns, even after accounting for the possibility of further crashes, transaction costs, and hedges against the downside risk.

This article derives underlying asset risk-neutral probability distributions of European options on the S&P 500 index. Nonparametric methods are used to choose probabilities that minimize an objective function subject to requiring that the probabilities are consistent with observed option and underlying asset prices. Alternative optimization specifications produce approximately the same implied distributions. A new and fast optimization technique for estimating probability distributions based on maximizing the smoothness of the resulting distribution is proposed. Since the crash, the risk-neutral probability of a three (four) standard deviation decline in the index (about -36 percent (-46 percent) over a year) is about 10 (100) times more likely than under the assumption of lognormality.

A relationship exists between aggregate risk-neutral and subjective probability distributions and risk aversion functions. We empirically derive risk aversion functions implied by options prices and realized returns on the S&P500 index simultaneously. These risk aversion functions dramatically change shapes around the 1987 crash: Precrash, they are positive and decreasing in wealth and largely consistent with standard assumptions made in economic theory. Postcrash, they are partially negative and partially increasing and irreconcilable with those assumptions. Mispricing in the option market is the most likely cause. Simulated trading strategies exploiting this mispricing show excess returns, even after accounting for the possibility of further crashes, transaction costs, and hedges against the downside risk.

[Widespread violations of stochastic dominance by 1-month S&P 500 index call options over 1986-2006 imply that a trader can improve expected utility by engaging in a zero-net-cost trade net of transaction costs and bid-ask spread. Although precrash option prices conform to the Black-Scholes-Merton model reasonably well, they are incorrectly priced if the distribution of the index return is estimated from time-series data. Substantial violations by postcrash OTM calls contradict the notion that the problem lies primarily with the left-hand tail of the index return distribution and that the smile is too steep. The decrease in violations over the postcrash period of 1988-1995 is followed by a substantial increase over 1997-2006, which may be due to the lower quality of the data but, in any case, does not provide evidence that the options market is becoming more rational over time.]

ABSTRACT This article derives underlying asset risk‐neutral probability distributions of European options on the S&P 500 index. Nonparametric methods are used to choose probabilities that minimize an objective function subject to requiring that the probabilities are consistent with observed option and underlying asset prices. Alternative optimization specifications produce approximately the same implied distributions. A new and fast optimization technique for estimating probability distributions based on maximizing the smoothness of the resulting distribution is proposed. Since the crash, the risk‐neutral probability of a three (four) standard deviation decline in the index (about −36 percent (−46 percent) over a year) is about 10 (100) times more likely than under the assumption of lognormality.

This article derives underlying asset risk-neutral probability distributions of European options on the S&P 500 index. Nonparametric methods are used to choose probabilities that minimize an objective function subject to requiring that the probabilities are consistent with observed option and underlying asset prices. Alternative optimization specifications produce approximately the same implied distributions. A new and fast optimization technique for estimating probability distributions based on maximizing the smoothness of the resulting distribution is proposed. Since the crash, the risk-neutral probability of a three (four) standard deviation decline in the index (about −36 percent (−46 percent) over a year) is about 10 (100) times more likely than under the assumption of lognormality.

We model a firm’s value process controlled by a manager maximizing expected utility from restricted shares and employee stock options. The manager also controls allocation of his outside wealth, which allows partially hedging of his exposure to firm risk. Managerial control increases the expected time to exercise for his employee stock options. It also reduces the gap between his certainty equivalent and the firm’s Fair Value for his compensation, but that gap remains substantial. Managerial control also causes traded options to exhibit an implied volatility smile. With costly control the same basic patterns remain, but the manager’s risk-taking is dampened.

Abstract We investigate incentive effects of a typical hedge fund contract for a manager with power utility. With a one-year horizon, the manager displays risk taking that varies dramatically with fund value. We extend the model to multiple yearly evaluation periods and find that the manager's risk taking is rapidly moderated if the fund performs reasonably well. The most realistic approach to modeling fund closure uses an endogenous shutdown barrier where the manager optimally chooses to shut down. The manager increases risk taking as fund value approaches that barrier, and this boundary behavior persists strongly with multiyear horizons.

We construct a panel of S&P 500 Index call and put option portfolios, daily adjusted to maintain targeted maturity, moneyness, and unit market beta, and test multi-factor pricing models. The standard linear factor methodology is applicable because the monthly portfolio returns have low skewness and are close to normal. We hypothesize that any one of crisis-related factors incorporating price jumps, volatility jumps, and liquidity (along with the market) explains the cross-sectional variation in returns. Our hypothesis is not rejected, even when the factor premia are constrained to equal the corresponding premia in the cross-section of equities. The alphas of short-maturity out-of-the-money puts become economically and statistically insignificant. (JEL G11, G13, G14)

Starting with the fundamental relation that state prices are the product of physical probabilities and the stochastic discount factor, Ross (2015) shows that, given strong assumptions, knowing state prices suffices to back out physical probabilities and the stochastic discount factor at the same time. We find that such recovered physical distributions based on the S&P 500 index are incompatible with future returns and fail to predict future returns and realized variances. These negative results are even stronger when we add economically reasonable constraints. Simple benchmark methods based on a power utility agent or the historical return distribution cannot be rejected.