Pricing Kernel Puzzle Research Articles

Volatility-dependent probability weighting and the dynamics of the pricing kernel puzzle

In order to estimate volatility-dependent probability weighting functions, we obtain risk neutral and physical densities from the Pan (J Financ Econ 63(1):3–50, 2002. https://doi.org/10.1016/S0304-405X(01)00088-5) stochastic volatility and jumps model. Across volatility levels, we find pronounced inverse S-shapes, i.e. small probabilities are overweighted, and probability weighting almost monotonically increases in volatility, indicating higher skewness preferences and crash aversion in volatile market environments. Moreover, by estimating probabilistic risk attitudes, equivalent to the share of risk aversion related to probability weighting, we shed further light on the pricing kernel puzzle. While pricing kernels estimated from the Pan (J Financ Econ 63(1):3–50, 2002. https://doi.org/10.1016/S0304-405X(01)00088-5) model display the typical U-shape as documented in the literature, pricing kernels—net of probability weighting—are strictly monotonically decreasing and thus in line with economic theory. Equivalently, we find risk aversion to be positive across wealth levels. Our results are robust to alternative maturities, wealth percentiles, alternative functional forms, a nonparametric empirical setting and variations of the Pan (J Financ Econ 63(1):3–50, 2002. https://doi.org/10.1016/S0304-405X(01)00088-5) coefficient estimates.

Ross Recovery and the Contemporaneous Pricing Kernel

This chapter investigates the contemporaneous pricing kernel obtained by applying the recovery theorem to options on the S&P500 index. Similarly to the literature on the pricing kernel puzzle, I find evidence that the recovered pricing kernels are U-shape, but only some of the time. The different shapes observed for the recovered kernels appear to be clustered in time, indicating that they are a genuine feature of risk preferences and not just an artifact created by noisy pricing kernel estimates. Although the recovered contemporaneous kernels cannot significantly predict future realized returns over the whole distribution, they are successful for the first 3 quartiles. A practitioner needing an instantaneous measure of downside risk could therefore obtain such information by applying the recovery theorem to options data.

Option market trading activity and the estimation of the pricing kernel: A Bayesian approach

We propose a nonparametric Bayesian approach for the estimation of the pricing kernel. Historical stock returns and option market data are combined through the Dirichlet Process (DP) to construct an option-adjusted physical measure. The precision parameter of the DP process is calibrated to the amount of trading activity in deep-out-of-the-money options. We use the option-adjusted physical measure to construct an option-adjusted pricing kernel. An empirical investigation on the S&P 500 Index from 2002 to 2015 shows that the option-adjusted pricing kernel is consistently monotonically decreasing, regardless of the level of volatility, thus providing an explanation to the well known U-shaped pricing kernel puzzle.

The pricing kernel puzzle: A behavioral explanation

We show that if sophisticated institutional managers and individual investors perceive tail-risks differently, then a new explanation for the pricing kernel puzzle emerges. We show, by example, that even a tiny difference in tail-risk perception by the two investor types can explain the pricing kernel puzzle.

Behavioral in the Short-run and Rational in the Long-run? Evidence from S&P 500 Options

We estimate the pricing kernel from options on the S&P 500 index for different horizons and over time. This allows us to compare short-term and long-term pricing kernels and analyze their time-series variation. We show that the well documented pricing kernel puzzle–the non-monotonicity of the pricing kernel–only exists for short horizons. For longer horizons the puzzle disappears and the level, shape and time-series variation of the pricing kernel are in line with standard rational asset-pricing models. In contrast, we show that the empirical features of the short-term kernel can be explained by a behavioral asset-pricing model.

Implied risk aversion and pricing kernel in the FTSE 100 index

This paper studies the estimation of the pricing kernel and explains the pricing kernel puzzle found in the FTSE 100 index. We use prices of options and futures on the FTSE 100 index to derive the risk neutral density (RND). The option-implied RND is inverted by using two nonparametric methods: the implied-volatility surface interpolation method and the positive convolution approximation (PCA) method. The actual density distribution is estimated from the historical data of the FTSE 100 index by using the threshold GARCH (TGARCH) model. The results show that the RNDs derived from the two methods above are relatively negatively skewed and fat-tailed, compared to the actual probability density, that is consistent with the phenomenon of “volatility smile.” The derived risk aversion is found to be locally increasing at the center, but decreasing at both tails asymmetrically. This is the so-called pricing kernel puzzle. The simulation results based on a representative agent model with two state variables show that the pricing kernel is locally increasing with the wealth at the level of 1 and is consistent with the empirical pricing kernel in shape and magnitude.

Risk-Neutral Densities: A Review

Trading in options with a wide range of exercise prices and a single maturity allows a researcher to extract the market's risk-neutral density (RND) over the underlying price at expiration. The RND contains investors’ beliefs about the true probabilities blended with their risk preferences, both of which are of great interest to academics and practitioners alike. With a particular focus on US equity options, I review the historical development of this powerful concept, practical details of fitting an RND to options market prices, and the many ways in which investigators have tried to distill true expectations and risk premia from observed RNDs. I briefly discuss areas of active current research including the pricing kernel puzzle and the volatility surface, and offer thoughts on what has been learned about RNDs so far and fruitful directions for future research.

Structural Breaks in the Variance Process and the Pricing Kernel Puzzle

Numerous studies find S-shaped pricing kernels, which is conflicting with standard theory. In contrast to that, based on a novel GARCH model with structural breaks, I show that the pricing kernel is consistently U-shaped. The results are robust to variations in the methodology and hold for several major international stock market indices. The new U-shaped pricing kernel estimates help to explain cross-sectional stock return anomalies. Furthermore, a pricing kernel mimicking trading strategy yields sizable Sharpe ratios. Finally, the empirical results can be explained well by a model with a variance-dependent pricing kernel, but only if structural breaks are included in the model.

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.

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.

Pricing Kernel Monotonicity and Conditional Information

A large literature finds evidence that pricing kernels estimated nonparametrically from option prices and historical returns are not monotonically decreasing in market index returns. We propose a new nonparametric estimator of the pricing kernel that reflects the information available to investors who set option prices. In simulations, the estimator outperforms current techniques. Our empirical estimates using S&P 500 index option data from 1996-2012 and FTSE 100 index option data from 2002-2013 suggest that the “pricing kernel puzzle” is a byproduct of econometric technique rather than a behavioral or economic phenomenon.

Behavioural Biases and the Pricing-Kernel Puzzle

The pricing-kernel, when estimated from index options, appears to show risk-loving behaviour for small gains and losses. Although this puzzle could be caused by option buyers using subjective probability-weighting, such weighting generates an unrealistically large (and time-varying) risk-premium. A better explanation is that limited arbitrage allows the market to be segmented into put buyers on the downside and call buyers on the upside. If the put buyers are pessimistic and underconfident, but the call buyers are optimistic and overconfident, the perceived physical distribution is highly skewed. Using option prices over a period of 15 years, we show that the segmented model can solve the puzzle. It also demonstrates some well-documented features, including: a small aggregate risk-premium, loss-aversion, and an impact of volatility on the kernel.

Differential Awareness of Rare Disasters and the Pricing Kernel Puzzle

Risk-averse expected utility maximization implies that the pricing kernel must be a non-increasing function of aggregate wealth. However, empirical research has found that the pricing kernel frequently displays a locally increasing portion in aggregate wealth. This is known as the pricing kernel puzzle. In this article, we show that if individual investors and sophisticated institutional managers perceive tail-risks differently, then there exists a range of feasible option prices that produce a locally increasing pricing kernel similar to the empirically observed pricing kernel. We show, by example, that even tiny differences in state spaces as perceived by the two investor types are sufficient to generate a non-monotonic pricing kernel.

Log, Stock and Two Simple Lotteries

This work studies the problem of decision-making under risk by agents whose information processing abilities may be limited. The constructed theoretical framework grounds on findings from economic laboratory experiments, incorporates existing neuroscience knowledge, and is implemented using information-theoretic formalism. Activation of the above information-processing constraints distorts the subjective perception of the objective stochastic environment the agent operates in, and the constrained-optimal decision-making requires appropriate adjustments. In the selected application, a general equilibrium macro-finance model, such biases of subjective perspective as overconfidence, pessimism and categorization thus emerge endogenously. The theoretical implications receive empirical support in a mutually consistent way: according to (cross-checked) calibrations, they allow us to reverse-engineer and rationalize the phenomena known as the equity premium/risk-free rate puzzles; as well as contribute to the understanding of such regularities as the portfolio underdiversification puzzle, style investing and the non-monotone pricing kernel puzzle. On the other hand, these results also help rationalize, by formulating certain optimizing foundations behind, the experimental evidence that underlies the Allais paradox and that is systematized in, e.g., the (cumulative) prospect theory.

A Bayesian Estimate of the Pricing Kernel

The focus of this article is the Pricing Kernel (PK), the building-block of asset pricing theory. In the classical framework the shape of the PK is monotonically decreasing in the stock price, nevertheless empirical evidence suggests that the PK is locally increasing in the interval around the 0% return and has an irregular behaviour in the tails of the distribution. We argue that these deviations, known as pricing kernel puzzle, derive from a dis-homogeneity between the physical and the risk neutral (RN) measure and can be corrected by embedding some forward looking information into the physical measure. Our proposed methodology combines the information from historical returns with that coming from the RN measure through a non-parametric Poisson-Dirichlet process. As a result the irregular behaviour of the PK in the tails of the distribution is reduced and the monotonicity is ensured in almost all cases.

Reference-Dependent Preferences and the Empirical Pricing Kernel Puzzle*

Supported by several recent investigations, the empirical pricing kernel (PK) puzzle might be considered as a stylized fact. Based on an economic model with reference-dependent preferences for the financial investors, we emphasize a microeconomic view that explains the puzzle via state-dependent aggregate preferences. We also investigate how the shape of the PK estimated from option and stock market index returns changes in relation to the volatility risk premium.

The Pricing Kernel Puzzle: Survey and Outlook

We update the survey of the pricing kernel puzzle of Jackwerth (2004). 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. Explanations of the pricing kernel puzzle are discussed and we suggest areas where more research is needed.

The Pricing Kernel Puzzle in Forward Looking Data

The Pricing Kernel Puzzle in Forward Looking Data

An Empirical Test of Pricing Kernel Monotonicity

SummaryA large class of asset pricing models predicts that securities which have high payoffs when market returns are low tend to be more valuable than those with high payoffs when market returns are high. More generally, we expect the projection of the stochastic discount factor on the market portfolio—that is, the discounted pricing kernel evaluated at the market portfolio—to be a monotonically decreasing function of the market portfolio. Numerous recent empirical studies appear to contradict this prediction. The non‐monotonicity of empirical pricing kernel estimates has become known as the pricing kernel puzzle. In this paper we propose and apply a formal statistical test of pricing kernel monotonicity. We apply the test using 17 years of data from the market for European put and call options written on the S&P 500 index. Statistically significant violations of pricing kernel monotonicity occur in a substantial proportion of months, suggesting that observed non‐monotonicities are unlikely to be the product of statistical noise. Copyright © 2014 John Wiley & Sons, Ltd.

Pricing Kernel Monotonicity and Conditional Information

A large literature finds evidence that kernels estimated nonparametrically from option prices and historical returns are not monotonically decreasing in market index returns. We propose a new nonparametric estimator of the kernel that reflects the information available to investors who set option prices. In simulations, the estimator outperforms current techniques. Our empirical estimates using S&P 500 index option data from 1996-2012 and FTSE 100 index option data from 2002-2013 suggest that the pricing kernel puzzle'' is a byproduct of econometric technique rather than a behavioral or economic phenomenon.