Empirical Pricing Kernel Research Articles

Empirical Pricing Kernel and Option-Implied Risk Aversion in China 50 ETF

ABSTRACT Based on an analysis of the China 50 ETF options and their underlying assets, we measure the empirical pricing kernel and implied risk aversion. By employing a Markov-switching GARCH model, the estimated results show a monotonically decreasing pricing kernel under a high-volatility regime and a U-shaped pricing kernel under a low-volatility regime. The implied risk aversion is inversely S-shaped under both high- and low-volatility regimes. However, the implied risk aversion under the low-volatility regime has a wide range. Investors’ risk aversion perspective helps explain patterns of pricing kernels and risk aversion estimates. Finally, we find that implied risk aversion is predictive of short-term (excess) market returns based on in-sample and out-of-sample tests.

Cautious Risk Takers: Investor Preferences and Demand for Active Management

ABSTRACTDespite their mediocre mean performance, actively managed mutual funds are distinct from passive funds in their return distributions. Active value funds better hedge downside risk, while active growth funds better capture upside potential. Since such performance features may appeal to investors with tail‐overweighting preferences, we show that preferences for downside protection and upside potential estimated from the empirical pricing kernel can help explain active fund flows in the value and growth categories, respectively. This effect of investor risk preferences varies significantly with funds' downside‐hedging and upside‐capturing ability, with levels of active management, and across retirement and retail funds.

Probability weighting functions obtained from Hong Kong index option market

In this paper we estimate the pricing kernel from the Hong Kong index option market and obtain the empirical probability weighting functions based on the rank-dependent expected utility. The empirical pricing kernel is estimated semi-parametrically as the ratio of the risk-neutral and objective densities. We employ a two-step estimation procedure to estimate the objective and risk-neutral densities under a consistent parametric framework of the non-affine generalised autoregressive conditional heteroskedasticity (G.A.R.C.H.) diffusion model. In the first step, we develop a continuous particle filters-based maximum likelihood estimation method to estimate the objective parameters of the G.A.R.C.H. diffusion model using the Hang Seng Index (H.S.I.) returns. In the second step of our estimation, we depart from the usual pure calibration approach and use the H.S.I. option prices to estimate the risk-neutral parameters of the G.A.R.C.H. diffusion model by constraining certain parameters to be consistent with the time-series behaviour of H.S.I. returns. Based on the estimated objective and risk-neutral parameters, the objective and risk-neutral densities are obtained by inverting the corresponding characteristic functions. Empirical results indicate that the empirical pricing kernel estimated from the Hong Kong index option market is non-monotonic and the estimated probability weighting functions are S-shaped, which implies that investors underweight small probability events and overweight large ones.

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.

The Pricing Kernel Puzzle: A Real Phenomenon or a Statistical Artifact?

AbstractA large literature finds evidence that the pricing kernels estimated from option prices and historical returns are not monotonically decreasing in market returns. We show that the inevitable coalescence of contingencies, especially in the left‐tail, associated with estimating a distribution function may give rise to a nonmonotonic empirical pricing kernel even if the actual pricing kernel is monotonic. Hence, the observed nonmonotonicity of pricing kernels may be a statistical artifact rather than a real phenomenon. We argue that empirical work should explicitly correct for this effect by widening the option pricing bounds associated with monotonic pricing kernels.

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.

Uniform Confidence Bands for Pricing Kernels

Pricing kernels implicit in option prices play a key role in assessing the risk aversion over equity returns. We deal with non-parametric estimation of the pricing kernel (Empirical Pricing Kernel) given by the ratio of the risk-neutral density estimator and the subjective density estimator. The former density can be represented as the second derivative w.r.t. the European call option price function, which we estimate by non-parametric regression. The subjective density is estimated non-parametrically too. In this framework, we develop the asymptotic distribution theory of the EPK in the L1 sense. Particularly, to evaluate the overall variation of the pricing kernel, we develop a uniform confidence band of the EPK. Furthermore, as an alternative to the asymptotic approach, we propose a bootstrap confidence band. The developed theory is helpful for testing parametric specifications of pricing kernels and has a direct extension to estimating risk aversion patterns. The established results are assessed and compared in a Monte-Carlo study. As a real application, we test risk aversion over time induced by the EPK.

A Microeconomic Explanation of the EPK Paradox

Supported by several recent investigations the empirical pricing kernel paradox might be considered as a stylized fact. In Chabi-Yo et al. (2008) simulation studies have been presented which suggest that this paradox might be caused by regime switching of stock prices in financial markets. Alternatively, we want to emphasize a microeconomic view. Based on an economic model with state dependent utilities for the financial investors we succeed in explaining the paradox by changes of the risk attitudes. Theoretically, the change behaviour is compressed by the pricing kernels. As a starting point for empirical insights we shall develop and investigate inverse problems in terms of data fits for estimated basic values of the pricing kernel.

Empirical Pricing Kernels: A Tale of Two Tails and Volatility?

By considering options on the S&P 500 it is shown how (i) missing option data, (ii) misestimated subjective probabilities, and (iii) a time varying variance (risk-premium) lead to different empirical pricing kernels. Accounting for all these aspects, the empirical return pricing kernel is w-shaped in calm periods and u-shaped in turbulent times. Finding w-shaped return pricing kernels, this paper is the first to reveal the interconnectedness between two asset pricing puzzles: non-monotonically decreasing return pricing kernels and u-shaped volatility pricing kernels. Based on the non-parametric estimates, a parametric option pricing model that matches the stylized facts in the return and volatility dimension is proposed. Moreover, it is shown how a simple ambiguity aversion economy generates w- and u-shaped return pricing kernels.

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.

Sentiment Lost: The Effect of Projecting the Empirical Pricing Kernel Into a Smaller Filtration Set

Supported by empirical examples, this paper provides a theoretical analysis to show what is the impact of an improper calibration of the physical measure on the estimation of the empirical pricing kernel. While extracting the risk-neutral measure from option data provides a naturally forward looking measure, extracting the real world probability from a stream of historical returns is only partially informative, thus suboptimal with respect to investors’ future beliefs. In virtue of this disalignment, most of papers present in literature are then affected by a non-homogeneity bias.From a probabilistic viewpoints, the missing beliefs are totally unaccessible stopping times on the coarser filtration set. As a consequence, an absolutely continuous local or strict local martingale, once projected on it, becomes continuous with jumps.As a result of a non fully informative physical measure, the proposed empirical pricing kernel is no longer a true martingale, as required by the classical theory, but a strict local martingale with consequences on the probabilistic nature of the relative risk neutral measure.Finally we show how the implied options’ moments help in reducing the degree of inaccessibility and shorten the distance between what is theoretically required and empirically accessible.

Pricing Kernel Modeling

We propose a new method to estimate the empirical pricing kernel based on option data. We estimate the pricing kernel nonparametrically by using the ratio of the risk-neutral density estimator and the subjective density estimator. The risk-neutral density is approximated by a weighted kernel density estimator with varying unknown weights for different observations, and the subjective density is approximated by a kernel density estimator with equal weights. We represent the European call option price function by the second order integration of the risk-neutral density, so that the unknown weights are obtained through one-step penalized least squares estimation with the Kullback-Leibler divergence as the penalty function. Asymptotic results of the resulting estimators are established. The performance of the proposed method is illustrated empirically by simulation and real data application studies.

Pricing kernel modeling

We propose a new method to estimate the empirical pricing kernel based on option data. We estimate the pricing kernel nonparametrically by using the ratio of the risk-neutral density estimator and the subjective density estimator. The risk-neutral density is approximated by a weighted kernel density estimator with varying unknown weights for different observations, and the subjective density is approximated by a kernel density estimator with equal weights. We represent the European call option price function by the second order integration of the risk-neutral density, so that the unknown weights are obtained through one-step penalized least squares estimation with the Kullback-Leibler divergence as the penalty function. Asymptotic results of the resulting estimators are established. The performance of the proposed method is illustrated empirically by simulation and real data application studies.

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.

Options on Leveraged ETFs: A Window on Investor Heterogeneity

A risk-neutral probability distribution (RND) for future S&P 500 returns extracted from index options contains investors' true expectations and also their risk preferences. But the empirical pricing kernel that emerges in a representative agent framework, which suppresses investor differences, is inconsistent with investor rationality. The anomalous shape can be generated easily in a model with heterogeneous investors facing limits to arbitrage, however. We explore investor heterogeneity directly via RNDs extracted from options on three exchange traded funds with leveraged short and long exposures to the S&P 500 index, and find large differences that are not arbitraged away. For example, holders of ETFs with short exposure to the market value payoffs in negative return states significantly more than those with long exposure do. Separating true expectations from risk premia requires further assumptions, so we consider polar cases in which either all investors have the same expectations but different risk preferences, or the reverse. The results are largely consistent with the expectations differences we anticipate from investors choosing short or leveraged long market exposure. We then look at changes in RNDs to see how realized returns affect investors with different expectations. A large negative realized return raises the median future return expected by bulls but lowers it for bears. Uncertainty over future returns widens for both types of investors when they are wrong and narrows when they are right.

Testing monotonicity of pricing kernels

The behaviour of market agents has been extensively covered in the literature. Risk averse behaviour, described by Von Neumann and Morgenstern (Theory of games and economic behavior. Princeton University Press, Princeton, 1944) via a concave utility function, is considered to be a cornerstone of classical economics. Agents prefer a fixed profit over an uncertain choice with the same expected value, however, lately there has been a lot of discussion about the empirical evidence of such risk averse behaviour. Some authors have shown that there are regions where market utility functions are locally convex. In this paper we construct a test to verify uncertainty about the concavity of agents’ utility function by testing the monotonicity of empirical pricing kernels (EPKs). A monotonically decreasing EPK corresponds to a concave utility function while a not monotonically decreasing EPK means non-averse pattern on one or more intervals of the utility function. We investigate the EPKs for German DAX data for the years 2000, 2002 and 2004 and find evidence of non-concave utility functions: the null hypothesis of a monotonically decreasing pricing kernel is rejected for the data under consideration. The test is based on approximations of spacings through exponential random variables. In a simulation we investigate its performance and calculate the critical values (surface).

Uniform Confidence Bands for Pricing Kernels

Pricing kernels implicit in option prices play a key role in assessing the risk aversion over equity returns. We deal with nonparametric estimation of the pricing kernel (Empirical Pricing Kernel) given by the ratio of the risk-neutral density estimator and the subjective density estimator. The former density can be represented as the second derivative w.r.t. the European call option price function, which we estimate by nonparametric regression. The subjective density is estimated nonparametrically too. In this framework, we develop the asymptotic distribution theory of the EPK in the L1 sense. Particularly, to evaluate the overall variation of the pricing kernel, we develop a uniform confidence band of the EPK. Furthermore, as an alternative to the asymptotic approach, we propose a bootstrap confidence band. The developed theory is helpful for testing parametric specifications of pricing kernels and has a direct extension to estimating risk aversion patterns. The established results are assessed and compared in a Monte-Carlo study. As a real application, we test risk aversion over time induced by the EPK.

The economics of options-implied inflation probability density functions

Recently a market in options based on consumer price index inflation (inflation caps and floors) has emerged in the US. This paper uses quotes on these derivatives to construct probability densities for inflation. We study how these probability density functions respond to news announcements and find that the implied odds of deflation are sensitive to certain macroeconomic news releases. We also estimate empirical pricing kernels using these option prices along with time series models fitted to inflation. The options-implied densities assign considerably more mass to extreme inflation outcomes (either deflation or high inflation) than do their time series counterparts. This yields a U-shaped empirical pricing kernel, with investors having high marginal utility in states of the world characterized by either deflation or high inflation.

Reference Dependent Preferences and the EPK Puzzle

Supported by several recent investigations, the empirical pricing kernel (EPK) puzzle might be considered a stylized fact. Based on an economic model with state dependent preferences for the financial investors, we want to emphasize a microeconomic view that succeeds in explaining the puzzle. We retain the expected utility framework in a one period model and illustrate the case when the state is defined with respect to a reference point. We further investigate how the model relates the shape of the EPK to the economic conditions.

Reference Dependent Preferences and the EPK Puzzle

Supported by several recent investigations, the empirical pricing kernel (EPK) puzzle might be considered a stylized fact. Based on an economic model with state dependent preferences for the financial investors, we want to emphasize a microeconomic view that succeeds in explaining the puzzle. We retain the expected utility framework in a one period model and illustrate the case when the state is defined with respect to a reference point. We further investigate how the model relates the shape of the EPK to the economic conditions.