Closed-form Transformations from Risk-neutral to Real-world Distributions

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.

‘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.

Empirical risk aversion functions-estimates and assessment of their reliability

Pricing Kernels with Stochastic Skewness and Volatility Risk

We present a pricing kernel that summarizes well the main features of the dynamics of interest rates and risk in postwar U.S. data and use it to uncover how the pricing kernel has moved with the short rate in this data. Our findings imply that standard monetary models miss an essential link between the central bank instrument and the economic activity that monetary policy is intended to affect, and thus we call for a new approach to monetary policy analysis. We sketch a new approach using an economic model based on our pricing kernel. The model incorporates the key relationships between policy and risk movements in an unconventional way: the central bank's policy changes are viewed as primarily intended to compensate for exogenous business cycle fluctuations in risk that threaten to push inflation off target. This model, while an improvement on standard models, is considered just a starting point for their revision. It leads to critical questions that researchers need to answer as they continue to revise their approach to monetary policy analysis. Copyright © 2009 by the National Bureau of Economic Research.

We introduce a pricing kernel with time-varying volatility risk aversion to explain the observed time variations in the shape of the pricing kernel. When combined with the Heston-Nandi GARCH model, this framework yields a tractable option pricing model in which the variance risk ratio (VRR) emerges as a key variable. We show that the VRR is closely linked to economic fundamentals, as well as sentiment and uncertainty measures. A novel approximation method provides analytical option pricing formulas, and we demonstrate substantial reductions in pricing errors through an empirical application to the S&P 500 index, the CBOE VIX, and option prices.

The determinants and electoral consequences of asymmetric preferences

This paper studies rent-seeking contests where the contestants do not know the number of actively competing contestants. Two models are compared. In the first, all players are risk neutral; in the second, all have constant absolute risk aversion. If the expected fraction of active contestants is low, an increase in the number of potential contestants increases individual rent-seeking efforts. This effect is in contrast to the complete information case where individual rent-seeking efforts decrease in the number of contestants. The effect is more likely under risk neutrality, but also possible under risk aversion. Equilibrium rent seeking efforts are lower under risk aversion if and only if the expected fraction of active contestants is low.

In recent years a number of methods have been developed to infer implied state price densities (SPD) from cross sectional option prices, Chapter 7 and 8. Instead of comparing this density to a historical density extracted from the observed time series of the underlying asset prices, i.e. a risk neutral density to an actual density, Ait-Sahalia, Wang and Yared (2000) propose to compare two risk neutral densities, one obtained from cross sectional S&P 500 option data and the other from the S&P 500 index time series. Furthermore, they propose trading strategies designed to exploit differences in skewness and kurtosis of both densities. The goal of this article is to apply the procedure to the german DAX index. While the option implied SPD is estimated by means of the Barle and Cakici, Barle and Cakici (1998), implied binomial tree version, the time series density is inferred from the time series of the DAX index by applying a method used by Ait-Sahalia, Wang and Yared (2000). Based on the comparison of both SPDs the performance of skewness and kurtosis trades is investigated.

Smile from the past: A general option pricing framework with multiple volatility and leverage components

Affine multivariate GARCH models

The establishment of the fractional Black–Scholes option pricing model is under a major condition with the normal distribution for the state price density (SPD) function. However, the fractional Brownian motion is deemed to not be martingale with a long memory effect of the underlying asset, so that the estimation of the state price density (SPD) function is far from simple. This paper proposes a convenient approach to get the fractional option pricing model by changing variables. Further, the option price is transformed as the integral function of the cumulative density function (CDF), so it is not necessary to estimate the distribution function individually by complex approaches. Finally, it encourages to estimate the fractional option pricing model by the way of nonparametric regression and makes empirical analysis with the traded 50 ETF option data in Shanghai Stock Exchange (SSE).

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.

Implied volatility is an important element in risk management and option pricing. Black-Scholes model assumes a constant volatility, however, the evidence from financialmarket shows that the volatility is not constant but change with strike and time tomaturity. In this paper, the time to maturity is fixed and we will construct the implied volatility function of strike or moneyness. We can use regression method for estimation, but the data from financial market contains some noise and we need to apply smoothing techniques to estimate this implied volatility function. The standard non- and semiparametric regression methods don’t guarantee the resulting IV functions are arbitrage free, so we will insert our estimation result to Black and Scholes model and calculate the state price density (SPD). In a Black-Scholes model it is lognormal distribution with constant volatility parameter. In practice as volatility changes the distribution deviates from log-normality. We estimate volatilities and SPDs using EUREX option data on the DAX index by using different smoothing techniques. Our estimation will be carried out through the strike direction and moneyness direction. We will briefly introduce Local polynomials as one method. The most important smoothing techniques we will applied in this paper is B-splines, with the usage of roughness penalty, which allows a flexible choice on the degree of smoothness, and is promising for future research work on the arbitrage free constraint of implied volatility.

The literature discusses risk aversion as one of the behavioral determinants of technology adoption. However, little attention has been paid to measuring ambiguity aversion of poor people in developing countries or in finding the role of ambiguity aversion in technology adoption. Risk experiments in the previous studies have been designed in such a way that individuals face the risky and/or ambiguous situations alone. Individuals in the real world, especially farmers in developing countries, are likely to get information from peers before making any decision regarding a new innovation that has an ambiguous nature. This paper addresses two broad issues. The first issue is to measure the risk and ambiguity preferences of Bangladeshi rural farmers. The paper investigates whether the attitudes toward uncertainty (risk and ambiguity) differ when farmers face the uncertainty alone versus when they are allowed to communicate with peer groups of 3 or 6. It also investigates whether farmers’ demographic characteristics affect their attitudes toward uncertainty or not. A second issue is to find whether measures of attitudes toward uncertainty is same across different groups of subjects using experimental lotteries. To do so, this paper replicates the same experiments with groups of students in two universities in Bangladesh. Finally, the paper also investigates whether demographic variables affect the attitudes toward risk and ambiguity aversion or not. It finds that risk attitudes of farmers and students are same when deciding alone. However, farmers tend to show higher variation in risk aversion than students sample when deciding in a group of 3. In the latter case, farmers tend to show less risk aversion than students. While disaggregating the measured risk attitudes across gender, female students tend to show more risk aversion as well as higher variation in risk aversion than male students in the sample. The study also finds that The study also finds that students’ and farmers’ demographic characteristics affect both risk and ambiguity aversion.