Empirical Pricing Kernel Research Articles

Probability weighting functions implied in options prices

The empirical pricing kernels estimated from index options are non-monotone (Rosenberg and Engle, 2002; Bakshi, Madan, and Panayotov, 2010) and the corresponding risk-aversion functions can be negative (Aït-Sahalia and Lo, 2000; Jackwerth, 2000). We show theoretically that these and several other properties of empirical pricing kernels are consistent with rank-dependent utility model with probability weighting function, which overweights tail events. We also estimate the pricing kernels nonparametrically from the Standard & Poor's 500 index options and construct empirical probability weighting functions. The estimated probability weights typically have the inverse-S shape, which overweights tail events and is widely supported by the experimental decision theory.

Three Solutions to the Pricing Kernel Puzzle*

The pricing kernel is an important link between economics and finance. In standard models of financial economics, it is proportional to the aggregate marginal utility in the economy. We first show that none of the three standard assumptions (completeness, risk aversion, and correct beliefs) is needed for the pricing kernel to be generally decreasing. If at least one of the three assumptions is violated, the pricing kernel can have increasing parts. We explain the economic principles that lead to an increasing part in the pricing kernel and compare the resulting pricing kernels with the empirical pricing kernel estimated in Jackwerth (2000, Rev. Financ. Stud., 13, 433–451).

Empirical Pricing Kernel Estimation Using a Functional Gradient Descent Algorithm Based on Splines

We propose a new methodology to estimate the empirical pricing kernel implied from option data. In contrast to most of the studies in the literature that use an indirect approach, i.e. first estimating the physical and risk-neutral densities and obtaining the pricing kernel in a second step, we follow a direct approach. Departing from an adequate parametric and economically motivated pricing kernel, we apply a functional gradient descent (FGD) algorithm based on B-splines. This approach allows us to locally modify the initial pricing kernel and hence to improve the final estimate. We empirically illustrate the estimation properties of the method and test its predictive power on S&P 500 option data, comparing it as well with other recent approaches introduced in the empirical pricing kernel literature.

Cautious Risk-Takers: Investor Preferences and Demand for Active Management

Actively managed mutual funds have distinct return distributions from their passive benchmarks and our theoretical analysis using tail-sensitive risk preferences suggests that active value and growth funds may serve to reduce downside risk and capture upside potential, respectively. Furthermore, tail-sensitivity measures estimated from the empirical pricing kernel have significant explanatory power for active fund flows, even after controlling for business cycles and market-wide sentiment. Finally, active funds, unlike their passive counterparts, have exposures to option strategies hedging downside risk or capturing upside potential.

Probability Weighting Functions Implied in Options Prices

The empirical pricing kernels estimated from index options are non-monotone (Rosenberg and Engle, 2002; Bakshi, Madan, and Panayotov, 2010) and the corresponding risk aversion functions can be negative (Ait-Sahalia and Lo, 2000; and Jackwerth, 2000). We show theoretically that these and several other properties of empirical pricing kernels are consistent with rank-dependent utility model with probability weighting function which over-weights tail events. We also estimate the pricing kernels non-parametrically from the S&P 500 index options and construct empirical probability weighting functions. The estimated probability weights typically have the inverse-S shape which overweights tail events and is widely supported by the experimental decision theory.

Canonical Option Pricing and Greeks with Implications for Market Timing

We introduce canonical representation of a call option premium which solves two open problems in option pricing theory. The first problem was posed by (Kassouf, 1969, pg. 694) who sought “theoretical substantiation” for his robust option pricing power law which eschewed assumptions about risk attitudes, rejected risk neutrality, and made no assumptions about stock price distribution. The second problem was posed by (Scott, 1987, pp. 423-424) who could not find a unique solution to the call option price in his option pricing model with stochastic volatility–without appealing to an equilibrium asset pricing model posed by Hull and White (1987) – and concluded: “[w]e cannot determine the price of a call option without knowing the price of another call on the same stock”. First, we show that under certain conditions derivative assets are superstructures of the underlying. Hence any option pricing or derivative pricing model in a given number field, based on an anticipating variable in an extended field, with coefficients in a subfield containing the underlying, is admissible for option pricing and or market timing. This explains the taxonomy of variegated option pricing formulae presented in Haug and Taleb (2008) critique of the Black and Scholes (1973); Merton (1973) formula. To wit, we claim that investor sentiment is an algebraic number. Accordingly, any polynomial which satisfies those criteria is admissible for price discovery and or market timing. Therefore, at least for empirical purposes, elaborate equations of mathematical physics or otherwise are unnecessary for pricing derivatives because much simpler adaptive polynomials in suitable algebraic numbers are functionally equivalent. Second, we prove, analytically, that Kassouf (1969) power law specification for option pricing is functionally equivalent to Black and Scholes (1973); Merton (1973) in an algebraic number field containing the underlying. In fact, our canonical representation of call option price is (i) convex in time to maturity, and (ii) depends on an algebraic number for the underlying – with coefficients based on observables in a subfield. Thus, paving the way for Wold decomposition of option prices, and providing theoretical contrast with Duan (1995) heuristic GARCH option pricing model. Third, our canonical representation theory has an inherent regenerative multifactor decomposition of call option price that (1) induces a duality theorem for call option premium, and (2) permits estimation of risk factor exposure for Greeks by standard [polynomial] regression procedures. Again, providing a theoretical (a) basis for option pricing of Greeks, and (b) solving Scott’s dual call option problem a fortiori by virtue of Riesz representation theory. Moreover, we show how our regenerative decomposition extends to (Wang and Caflish, 2010, pp. 4-5) Snell‘s envelope representation of a dual [American] call option premium. Fourth, when the Wold decomposition procedure is applied we are able to construct an empirical pricing kernel for call option based on residuals from a model of risk exposure to persistent and transient risk factors. This approach is distinguished from Chernov (2003) two stage procedure.

Three Solutions to the Pricing Kernel Puzzle

The pricing kernel is an important link between economics and finance. In standard models of financial economics it is proportional to the aggregate marginal utility in the economy. We first how that none of the three standard assumptions (completeness, risk aversion, and correct beliefs) is needed for the pricing kernel to be generally decreasing. If at least one of the three assumptions is violated, the pricing kernel can have increasing parts. We explain the economic principles that lead to an increasing part in the pricing kernel and compare the resulting pricing kernels with the empirical pricing kernel estimated in Jackwerth (2000).

Empirical pricing kernels obtained from the UK index options market

Empirical pricing kernels for the UK equity market are derived as the ratio between risk-neutral densities, inferred from FTSE 100 index options, and historical real-world densities, estimated from time series of the index. The kernels thus obtained are almost compatible with a risk averse representative agent, unlike similar estimates for the US market.

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.

Shape Invariant Modelling Pricing Kernels and Risk Aversion

Pricing kernels play a major role in quantifying risk aversion and investors' preferences. Several empirical studies reported that pricing kernels exhibit a common pattern across different markets. Mostly visual inspection and occasionally numerically summarise are used to make comparison. With increasing amount of information updated every day, the empirical pricing kernels can be viewed as an object evolving over time. We propose a systematic modelling approach to describing the evolution of the empirical pricing kernels. The approach is based on shape invariant models. It captures the common features contained in the shape of the functions and at the same time characterises the variability between the pricing kernels based on a few interpretable parameters. The method is demonstrated with the European options and returns values of DAX index.

The payoff and implied pricing kernel in REITs

This article explores the hybrid character (i.e. the resemblance of both stock and bond) of Real Estate Investment Trust (REIT) through the implied pricing kernel behind REITs prices. We use the Empirical Pricing kernel method (Rosenberg and Engle, 2002) to explore their Payoff probability density and extract the implied pricing kernel. To estimate payoff probability density, we use asymmetric GARCH model. Results indicate that implied pricing kernels flatten in all ranges of low rate of returns and decrease exponentially in ranges of high rate of returns. This means the REIT pricing kernel resembles a bond when rate of return is low, and a stock when it is high. The pattern is consistent between 1970 and 2000.

Testing Monotonicity of Pricing Kernels

The behaviour of market agents has always been extensively covered in the literature. Risk averse behaviour, described by von Neumann and Morgenstern (1944) via a concave utility function, is considered to be a cornerstone of classical economics. Agents prefer a fixed profit over uncertain choice with the same expected value, however lately there has been a lot of discussion about the reliability of this approach. Some authors have shown that there is a reference point where market utility functions are convex. In this paper we have constructed a test to verify uncertainty about the concavity of agents’ utility function by testing the monotonicity of empirical pricing kernels (EPKs). A monotone decreasing EPK corresponds to a concave utility function while non-monotone decreasing EPK means non-averse pattern on one or more intervals of the utility function. We investigated the EPK for German DAX data for years 2000, 2002 and 2004 and found the evidence of non-concave utility functions: H0 hypothesis of monotone decreasing pricing kernel was rejected at 5% and 10% significance level in 2002 and at 10% significance level in 2000.

State Dependence Can Explain the Risk Aversion Puzzle

Risk aversion functions extracted from observed stock and option prices can be negative, as shown by Ait-Sahalia and Lo (2000), Journal of Econometrics 94: 9-51; and Jackwerth (2000), The Review of Financial Studies 13(2), 433-51. We rationalize this puzzle by a lack of conditioning on latent state variables. Once properly conditioned, risk aversion functions and pricing kernels are consistent with economic theory. To differentiate between the various theoretical explanations in terms of heterogeneity of beliefs or preferences, market sentiment, state-dependent utility, or regimes in fundamentals, we calibrate several consumption-based asset pricing models to match the empirical pricing kernel and risk aversion functions at different dates and over several years.

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 volatil- ity is used. All empirical market utility functions show a region of risk proclivity that is reproduced by adopting the hypothesis of heteroge- neous 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.

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.

Empirical Pricing Kernels and Investor Preferences

Empirical Pricing Kernels and Investor Preferences

Nonparametric Pricing of Multivariate Contingent Claims

This paper develops and implements a methodology for pricing multivariate contingent claims (MVCC s) based on semiparametric estimation of the multivariate risk-neutral density function.This methodology generates MVCC prices which are consistent with current market prices of univariate contingent claims.This method allows for completely general marginal risk-neutral densities and is compatible with all univariate risk-neutral density estimation techniques. The univariate risk-neutral densities are related by their risk-neutral correlation, which is estimated using time-series data on asset returns and an empirical pricing kernel (Rosenberg and Engle, 1999). This permits the multivariate risk-neutral density to be identified without requiring observation of multivariate contingent claims prices. The semiparametric MVCC pricing technique is used for valuation of one-month options on the better of two equity index returns. Option contracts with payoffs dependent on are four equity indexpairs are considered: S&P500 - CAC40, S&P500 - NK225, S&P500 - FTSE100, and S&P500 - DAX30. Five marginal risk-neutral densities (S&P500, CAC40, NK225, FTSE100, and DAX30) are estimated semiparametrically using a cross-section of contemporaneously measured equity index option prices in each market. A bivariate risk-neutral Plackett (1965) density is constructed using the given marginals and risk-neutral correlation derived using an empirical pricing kernel and the historical joint density of the index returns. Price differences from a lognormal pricing formulausing historical and risk-neutral return moments are found to be significant.

Empirical pricing kernels

This paper investigates the empirical characteristics of investor risk aversion over equity return states by estimating a time-varying pricing kernel, which we call the empirical pricing kernel (EPK). We estimate the EPK on a monthly basis from 1991 to 1995, using SP G13; C50

Option Hedging Using Empirical Pricing Kernels

This paper develops a method for option hedging which is consistent with time-varying preferences and probabilities. The preferences are expressed in the form of an empirical pricing kernel (EPK), which measures the state price per unit probability, while probabilities are derived from an estimated stochastic volatility model of the form GARCH components with leverage. State prices are estimated using the flexible risk-neutral density method of Rosenberg (1995) and a daily cross-section of option premia. Time-varying preferences over states are linked to a dynamic model of the underlying price to obtain a one-day ahead forecast of derivative price distributions and minimum variance hedge ratios. Empirical results suggest that risk aversion over S&P500 return states is substantially higher than risk aversion implied by Black-Scholes state prices and probabilities using long run estimates of S&P500 return moments. It is also found that the daily level of risk aversion is strongly positively autocorrelated, negatively correlated with S&P500 price changes,and positively correlated with the spread between implied and objective volatilities. Hedging results reveal that typical hedging techniques for out-of-the-money S&P500 index options, such as Black-Scholes or historical minimum variance hedging, are inferior to the EPK hedging method. Thus, time-varying preferences and probabilities appear to be an important factor in the day-to-day pricing of S&P500 options.