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.A relationship exists between aggregate (i.e., marketwide) risk-neutral and subjective probability distributions and risk aversion functions across wealth. In each state of the world the following relationship holds:risk-neutral probability = subjective probability × risk aversion adjustment.The risk-neutral probability is the price, multiplied by the riskfree return, that an investor would pay for receiving $1 in that state. We need to multiply by the riskfree return in order to have a proper probability distribution that sums to one. The subjective probability is simply the assessment of an investor of how likely a state is to occur. These two probabilities would be identical if the investor were indifferent to risk. However, the investor might value a dollar more highly in certain states, namely ones where wealth is low. The risk aversion adjustment indicates these preferences of the investor. Once we know both risk-neutral and subjective probability distributions, we can derive those preferences empirically.While the theoretical relationship is well known, this article is the first to recover risk aversion empirically from the two distributions. This methodology is also used in Aït-Sahalia and Lo (2000). Other, less closely related I would like to thank Kerry Back

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.

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The volatility smile changed drastically around the crash of 1987 and new option pricing models have been proposed in order to accommodate that change. Deterministic volatility models allow for more°exible volatility surfaces but refrain from introducing additional risk-factors.Thus, options are still redundant securities. Alternatively, stochastic models introduce additional risk-factors and options are then needed for spanning of the pricing kernel. We develop a statistical test based on this di®erence in spanning. Using daily S&P500 index options data from 1986-1995, our tests suggest that both in-and out-of-the-money options are needed for spanning. The¯ndings are inconsistent with deterministic volatility models but are consistent with stochastic models which incorporate additional priced risk-factors such as stochastic volatility, interest rates, or jumps.ii

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.A relationship exists between aggregate (i.e., marketwide) risk-neutral and subjective probability distributions and risk aversion functions across wealth. In each state of the world the following relationship holds:risk-neutral probability = subjective probability × risk aversion adjustment.The risk-neutral probability is the price, multiplied by the riskfree return, that an investor would pay for receiving $1 in that state. We need to multiply by the riskfree return in order to have a proper probability distribution that sums to one. The subjective probability is simply the assessment of an investor of how likely a state is to occur. These two probabilities would be identical if the investor were indifferent to risk. However, the investor might value a dollar more highly in certain states, namely ones where wealth is low. The risk aversion adjustment indicates these preferences of the investor. Once we know both risk-neutral and subjective probability distributions, we can derive those preferences empirically.While the theoretical relationship is well known, this article is the first to recover risk aversion empirically from the two distributions. This methodology is also used in Aït-Sahalia and Lo (2000). Other, less closely related I would like to thank Kerry Back