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.

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.

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.

We show that Standard & Poor's (S&P) 500 futures are pulled toward the at-the-money strike price on days when serial options on the S&P 500 futures expire (pinning) and are pushed away from the cost-of-carry adjusted at-the-money strike price right before the expiration of options on the S&P 500 index (anti-cross-pinning). These effects are driven by the interplay of market makers' rebalancing of delta hedges due to the time decay of those hedges as well as in response to reselling (and early exercise) of in-the-money options by individual investors. The associated shift in notional futures value is at least $115 million per expiration day.

Numerous hedge funds stop reporting each year to commercial databases, wreaking havoc with analyzing investment strategies that incur the unobserved delisting return. We use estimated portfolio holdings for funds-of-funds to back out estimated hedge-fund delisting returns. For all exiting funds, the estimated mean delisting return is insignificantly different from the average monthly return for live hedge funds. However, funds with poor prior performance and no clearly stated delisting reason had a significantly negative estimated mean delisting return of -5.97%, suggesting that a shock to their returns “tips them over the edge” and leads to delisting.