The pricing kernel puzzle in forward looking data

The pricing kernel puzzle concerns the locally increasing empirical pricing kernel, which is inconsistent with a risk-averse representative investor in a single period, single state variable setting. Some recent papers worry that the puzzle is caused simply by the mismatch of backward looking subjective and forward looking risk-neutral distributions of index returns. By using a novel test and forward looking information only, we generally confirm the existence of a u-shaped pricing kernel puzzle in the S&P 500 options data. The evidence is weaker for tests against an alternative with a risk-neutral investor and for longer horizons.     

Pinning in the S&P 500 futures

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.     

Recovering risk aversion from option prices and realized returns

A relationship exists between aggregate risk-neutral and subjectiveprobability distributions and risk aversion functions. We empiricallyderive risk aversion functions implied by options prices andrealized returns on the S&P500 index simultaneously. Theserisk aversion functions dramatically change shapes around the1987 crash: Precrash, they are positive and decreasing in wealthand largely consistent with standard assumptions made in economictheory. Postcrash, they are partially negative and partiallyincreasing and irreconcilable with those assumptions. Mispricingin the option market is the most likely cause. Simulated tradingstrategies exploiting this mispricing show excess returns, evenafter accounting for the possibility of further crashes, transactioncosts, and hedges against the downside risk.     

The price of a smile: hedging and spanning in option markets

The volatility smile changed drastically around the crash of1987, and new option pricing models have been proposed to accommodatethat change. Deterministic volatility models allow for moreflexible volatility surfaces but refrain from introducing additionalrisk 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 difference in spanning.Using daily S&P 500 index options data from 1986-1995, ourtests suggest that both in- and out-of-the-money options areneeded for spanning. The findings are inconsistent with deterministicvolatility models but are consistent with stochastic modelsthat incorporate additional priced risk factors, such as stochasticvolatility, interest rates, or jumps.     

The pricing kernel puzzle: survey and outlook

It has been a while since the literature on the pricing kernel puzzle was summarized in Jackwerth (Option-implied risk-neutral distributions and risk-aversion, The Research Foundation of AIMR, Charlotteville, 2004). That older survey also covered the topic of risk-neutral distributions, which was itself already surveyed in Jackwerth (J Deriv 2:66–82, 1999). Much has happened in those years and estimation of risk-neutral distributions has moved from new and exciting in the last half of the 1990s to becoming a well-understood technology. Thus, the present survey will focus on the pricing kernel puzzle, which was first discussed around 2000. We document the pricing kernel puzzle in several markets and present the latest evidence concerning its (non-)existence. Econometric studies are detailed which test for the pricing kernel puzzle. The present work adds much breadth in terms of economic explanations of the puzzle. New challenges for the field are described in the process.