Empirical Pricing Kernels and Investor Preferences

This paper analyzes empirical market utility functions and pricing kernels derived from the DAX and DAX option data for three market regimes. A consistent parametric framework of stochastic volatility is used. All empirical market utility functions show a region of risk proclivity that is reproduced by adopting the hypothesis of heterogeneous individual investors whose utility functions have a switching point between bullish and bearish attitudes. The inverse problem of finding the distribution of individual switching points is formulated in the space of stock returns by discretization as a quadratic optimization problem. The resulting distributions vary over time and correspond to different market regimes.

ABSTRACTThe shape of the pricing kernel has important implications for expected option returns. We shed light on the pricing kernel puzzle (i.e., mixed results regarding the shape of the pricing kernel) by examining S&P 500 index option returns and empirical pricing kernels across a wide range of expiration dates (from 1 month to 1 year). We document that at short (long) maturities, out‐of‐the‐money call option returns are negative (positive) and decrease (increase) with the strike price. At short maturities, empirical pricing kernels predominantly exhibit a W‐shape, while this pattern becomes less pronounced, evolving toward a monotonically decreasing curve at longer maturities. Our study suggests that the shape of pricing kernels and call option returns varies with option maturities, reflecting investors' heterogeneous beliefs about index returns across different time horizons.

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.

Empirical Pricing Kernels and Investor Preferences

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.

‘Risk Aversion, background Risk, and the Pricing Kernel’ looks in more detail at utility functions and their effect on the shape of the pricing kernel. The authors discuss the meaning of risk aversion and, in particular, ‘relative risk aversion’ and show that if relative risk aversion is constant at different levels of wealth, then the pricing kernel exhibits constant elasticity. They then show that the introduction of ‘background risk’, that is, non-hedgeable risks, causes the pricing kernel to exhibit declining elasticity. This effect on the pricing kernel is particularly significant for the pricing of options.

Purpose – The purpose of this paper is to understand that option pricing is the response of option implied volatility (IV) to macroeconomic announcements. Design/methodology/approach – The authors use high-frequency data on ASX SPI 200 index options to examine the response of option IV, as well as higher moments of the underlying return distribution, to macroeconomic announcements. Additionally, the authors identify the response of the moments as a function of moneyness of the options. Findings – The findings suggest that in-the-money and out-of-the money options have difference characteristics in their responses, leading to the conclusion that heterogeneity in investor beliefs and preferences affect option IV through the state price density (SPD) function. Originality/value – The research contributes to the literature that examines whether IV captures the beliefs of market participants about the likelihood of future states together with the preferences of market participants towards these states. In particular, the authors relate changes in option IV to changes in macroeconomic announcements, through the impact of these announcements on the moments of the SPD function.

Management Insights

Empirical risk aversion functions-estimates and assessment of their reliability

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.

The state price density is modeled as an exponential function of the underlying state variables, and the Esscher transform is used to specify the forward-risk-adjusted measure. With the aid of state price densities, Esscher transforms, and characteristic functions, this paper provides a consistent framework for pricing options on stocks, interest rates, and foreign exchange rates. The framework discussed is quite general and is related to many popular models.

In recent years portfolio optimization models that consider more criteria than the expected return and variance objectives of the Markowitz model have become popular. These models are harder to solve than the quadratic mean-variance problem. Two approaches to find a suitable portfolio for an investor are possible. In the multiattribute utility theory (MAUT) approach a utility function is constructed based on the investor's preferences and an optimization problem is solved to find a portfolio that maximizes the utility function. In the multiobjective programming (MOP) approach a set of efficient portfolios is computed by optimizing a scalarized objective function. The investor then chooses a portfolio from the efficient set according to his/her preferences. We outline these two approaches using the UTADIS method to construct a utility function and present numerical results for an example.

Determining the optimal portfolio is important in the investment process because it includes the selection of appropriate fund allocation to manage financial risk effectively. Although risk cannot be entirely eliminated, it is managed through strategic allocation based on investor preferences. Therefore, this research aimed to use mathematical models, including the power utility function, the natural logarithm utility function, and a combination of both, to capture varying degrees of risk aversion. The optimal allocation was obtained by analytically maximizing the expected end-of-period wealth utility under each specification, where the investor level of risk aversion was derived by determining the constant. The utility function that failed to produce closed-form solutions was solved through the use of a numerical method to approximate the optimal portfolio weight. Furthermore, numerical simulations were performed using data from two stocks in the e-commerce sector to prove the impact of parameter changes on investment decisions. The result showed explicit analytical values for each utility function, providing investors with a structured framework for determining optimal portfolio weights consistent with their risk profile.

Pricing Kernel Modeling

The general problem of asset pricing when the discount rate differs from the rate at which an asset's cash flows accrue is considered. A pricing kernel framework is used to model an economy that is segmented into distinct markets, each identified by a yield curve having its own market, credit and liquidity risk characteristics. The proposed framework precludes arbitrage within each market, while the definition of a curve-conversion factor process links all markets in a consistent arbitrage-free manner. A pricing formula is then derived, referred to as the across-curve pricing formula, which enables consistent valuation and hedging of financial instruments across curves (and markets). As a natural application, a consistent multi-curve framework is formulated for emerging and developed inter-bank swap markets, which highlights an important dual feature of the curve-conversion factor process. Given this multi-curve framework, existing multi-curve approaches based on HJM and rational pricing kernel models are recovered, reviewed and generalised, and single-curve models extended. In another application, inflation-linked, currency-based, and fixed-income hybrid securities are shown to be consistently valued using the across-curve valuation method.