Volatility And The Pricing Kernel

Using both S&P 500 option and recently introduced VIX option prices, we study pricing kernels and their dependence on multiple volatility factors. We first propose nonparametric estimates of marginal pricing kernels, conditional on the VIX and the slope of the variance swap term structure. Our estimates highlight the state-dependence nature of the pricing kernels. In particular, conditioning on volatility factors, the pricing kernel of market returns exhibit a downward sloping shape up to the extreme end of the right tail. Moreover, the volatility pricing kernel features a striking U-shape, implying that investors have high marginal utility in both high and low volatility states. This finding on the volatility pricing kernel presents a new empirical challenge to both existing equilibrium and reduced-form asset pricing models of volatility risk. Finally, using a full-fledged parametric model, we recover the joint pricing kernel, which is not otherwise identifiable.

I investigate a pricing kernel in which coskewness and the market volatility risk factors are endogenously determined. I show that the price of coskewness and market volatility risk are restricted by investor risk aversion and skewness preference. The risk aversion is estimated to be between two and five and significant. The price of volatility risk ranges from -1.5% to -0.15% per year. Consistent with theory, I find that the pricing kernel is decreasing in the aggregate wealth and increasing in the market volatility. When I project my estimated pricing kernel on a polynomial function of the market return, doing so produces the puzzling behaviors observed in pricing kernel. Using pricing kernels, I examine the sources of the idiosyncratic volatility premium. I find that nonzero risk aversion and firms' non-systematic coskewness determine the premium on idiosyncratic volatility risk. When I control for the non-systematic coskewness factor, I find no significant relation between idiosyncratic volatility and stock expected returns. My results are robust across different sample periods, different measures of market volatility and firm characteristics.

I derive pricing kernels in which the market volatility is endogenously determined. Using the Taylor expansion series of the representative investor's marginal utility, I show that the price of market volatility risk is restricted by the investor's risk aversion and skewness preference. The risk aversion is estimated to be between two and five and is significant. The price of the market volatility is negative. Consistent with economic theory, I find that the pricing kernel decreases in the market index return and increases in market volatility. The projection of the estimated pricing kernel onto a polynomial function of the market return produces puzzling behaviors, which can be observed in the pricing kernel and absolute risk aversion functions. The inclusion of additional terms in the Taylor expansion series of the investor's marginal utility produces a pricing kernel function of market stochastic volatility, stochastic skewness, and stochastic kurtosis. The prices of risk of these moments are restricted by the investor's risk aversion, skewness preference, and kurtosis preference. The prices of risk of these moments should not be confused with the price of risk of powers of the market return, such as co-skewness and co-kurtosis.

A basic tenet of lognormal asset pricing models is that a risky currency is associated with a low pricing kernel volatility. Empirical evidence implies that a risky currency is associated with a relatively high interest rate. Taken together, these two statements associate high-interest-rate currencies with low pricing kernel volatility. We document evidence suggesting that the opposite is true. We approximate the volatility of the pricing kernel with the volatility of the short-term interest rate. We find that, across currencies, relatively high interest rate volatility is associated with relatively high interest rates. This contradicts the prediction of lognormal models. One possible reason is that our approximation of the volatility of the pricing kernel is inadequate. We argue that this is unlikely, in particular for questions involving currencies. We conclude that lognormal models of the pricing kernel are inadequate for explaining currency risk.

We take a similar form of pricing kernel which developed by Christoffersen et al (2013) to extend the multiple volatility components model. By that way, we can obtain a more elaborate model which also explains some puzzles in the market. Apart from that, a surprise result is we don't need to estimate full parameters in model. Instead of that, we estimate the scaling factor which plays an important role when changing of measure. Empirical tests demonstrate the well ability of generalized model when reconcile time series properties of stock returns with the option prices. Furthermore, we also use the in-sample and out-sample for testing the predictability of the generalized model. The result shows the pricing kernel more or less enhancing the predictability than before..

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.

We nest multiple volatility components, fat tails and a U-shaped pricing kernel in a single option model and compare their contribution to describing returns and option data. All three features lead to statistically significant model improvements. A U-shaped pricing kernel is economically most important and improves option fit by 17% on average and more so for two-factor models. A second volatility component improves the option fit by 9% on average. Fat tails improve option fit by just over 4% on average, but more so when a U-shaped pricing kernel is applied. Overall these three model features are complements rather than substitutes: the importance of one feature increases in conjunction with the others.