Mikhail Chernov

A study towards a unified approach to the joint estimation of objective and risk neutral measures for the purpose of options valuation

The purpose of this paper is to bridge two strands of the literature, one pertaining to the objective or physical measure used to model an underlying asset and the other pertaining to the risk-neutral measure used to price derivatives. We propose a generic procedure using simultaneously the fundamental price and a set of option contracts. We use Hestons (1993, Review of Financial Studies 6, 327--343) model as an example, and appraise univariate and multivariate estimation of the model in terms of pricing and hedging performance. Our results, based on the SP Efficient method of moments; State price densities; Stochastic volatility models; Filtering

Model Specification and Risk Premia: Evidence from Futures Options

This paper examines model specification issues and estimates diffusive and jump risk premia using S&P futures option prices from 1987 to 2003. We first develop a time series test to detect the presence of jumps in volatility, and find strong evidence in support of their presence. Next, using the cross section of option prices, we find strong evidence for jumps in prices and modest evidence for jumps in volatility based on model fit. The evidence points toward economically and statistically significant jump risk premia, which are important for understanding option returns.

Sources of entropy in representative agent models

We propose two metrics for asset pricing models and apply them to representative agent models with recursive preferences, habits, and jumps. The metrics describe the pricing kernels dispersion (the entropy of the title) and dynamics (time dependence, a measure of how entropy varies over different time horizons). We show how each model generates entropy and time dependence and compare their magnitudes to estimates derived from asset returns. This exercise -- and transparent loglinear approximations -- clarifies the mechanisms underlying these models. It also reveals, in some cases, tension between entropy, which should be large enough to account for observed excess returns, and time dependence, which should be small enough to account for mean yield spreads.

Empirical reverse engineering of the pricing kernel

Abstract This paper proposes an econometric procedure that allows the estimation of the pricing kernel without either any assumptions about the investors preferences or the use of the consumption data. We propose a model of equity price dynamics that allows for (i) simultaneous consideration of multiple stock prices, (ii) analytical formulas for derivatives such as futures, options and bonds, and (iii) a realistic description of all of these assets. The analytical specification of the model allows us to infer the dynamics of the pricing kernel. The model, calibrated to a comprehensive dataset including the S&P 500 index, individual equities, T-bills and gold futures, yields the conditional filter of the unobservable pricing kernel. As a result we obtain the estimate of the kernel that is positive almost surely (i.e. precludes arbitrage), consistent with the equity risk premium, the risk-free discounting, and with the observed asset prices by construction. The pricing kernel estimate involves a highly nonlinear function of the contemporaneous and lagged returns on the S&P 500 index. This contradicts typical implementations of CAPM that use a linear function of the market proxy return as the pricing kernel. Hence, the S&P 500 index does not have to coincide with the market portfolio if it is used in conjunction with nonlinear asset pricing models. We also find that our best estimate of the pricing kernel is not consistent with the standard time-separable utilities, but potentially could be cast into the stochastic habit formation framework of Campbell and Cochrane (J. Political Economy 107 (1999) 205).

Term Structure and Volatility: Lessons from the Eurodollar Markets

We evaluate the ability of several affine models to explain the term structure of the interest rates and option prices. Since the key distinguishing characteristic of the affine models is the specification of conditional volatility of the factors, we explore models which have critical differences in this respect: Gaussian (constant volatility), stochastic volatility, and unspanned stochastic volatility models. We estimate the models based on the Eurodollar futures and options data. We find that both Gaussian and stochastic volatility models, despite the differences in the specifications, do a great job matching the conditional mean and volatility of the term structure. When these models are estimated using options data, their properties change, and they are more successful in pricing options and matching higher moments of the term structure distribution. The unspanned stochastic volatility (USV) model fails to resolve the tension between the futures and options fits. Unresolved tension in the fits points to additional factors or, even more likely, jumps, as ways to improve the performance of the models. Our results indicate that Gaussian and stochastic volatility models cannot be distinguished based on the yield curve dynamics alone. Options data are helpful in identifying the differences. In particular, Gaussian models cannot explain the relationship between implied volatilities and the term structure observed in the data.

Crash Risk in Currency Returns

We develop an empirical model of bilateral exchange rates. It includes normal shocks with stochastic variance and jumps in an exchange rate and in its variance. The probability of a jump in an exchange rate corresponding to depreciation (appreciation) of the U.S. dollar is increasing in the domestic (foreign) interest rate. The probability of a jump in variance is increasing in the variance only. Jumps in exchange rates are associated with announcements; jumps in variance are not. On average, jumps account for 25% of currency risk. The dollar carry index retains these features. Options suggest that jump risk is priced.

Implied Volatilities as Forecasts of Future Volatility, Time-Varying Risk Premia, and Returns Variability

The unbiasedness tests of implied volatility as a forecast of future realized volatility have found implied volatility to be a biased predictor. We explain this puzzle by recognizing that option prices contain a market risk premium not only on the asset itself, but also on its volatility. Hull and White (1987) show using a stochastic volatility model that a call option price can be represented as an expected value of the Black-Scholes formula evaluated at the average integrated volatility. If we allow volatility risk to be priced, this expectation should be taken under the risk-neutral probability measure, and can be decomposed into the expectation with respect to the physical measure and the risk-premium term. This term is just a linear function of the unobservable spot volatility. The decomposition explains the bias documented in the empirical literature and shows that the realized and historical volatility, which are used in the tests, are in fact the estimates of the unobserved quadratic variation and spot volatility of the stock-return generating process. Therefore, the use of these estimates generates the error-in-the-variables problem. We generalize the above results from a stochastic volatility model to a model with multiple volatility and jump factors. We provide an empirical illustration based on two US equity indices and three foreign currency rates. We find, that when we take into an account the risk-premium and use efficient methods to estimate volatility, the unbiasedness hypothesis can not be rejected, and the point estimate of the loading on the implied volatility in the traditional regression is equal to 1.

Alternative Models of Stock Prices Dynamics

The purpose of this paper is to shed further light on the tensions that exist between the empirical fit of stochastic volatility (SV) models and their linkage to option pricing. A number of recent papers have investigated several specifications of one-factor SV diffusion models associated with option pricing models. The empirical failure of one-factor affine, Constant Elasticity of Variance (CEV), and one-factor log-linear SV models leaves us with two strategies to explore: (1) add a jump component to better fit the tail behavior or (2) add an additional (continuous path) factor where one factor controls the persistence in volatility and the second determines the tail behavior. Both have been partially pursued and our paper embarks on a more comprehensive examination which yields some rather surprising results. Adding a jump component to the basic Heston affine model is known to be a successful strategy as demonstrated by Andersen et al. (1999), Eraker et al. (1999), Chernov et al. (1999), and Pan (1999). Unfortunately, the presence of a jump component introduces quite a few unpleasant econometric issues. In addition, several financial issues, like hedging and risk factors become more complex. In this paper we show that a two-factor log-linear SV diffusion model (without jumps) appears to yield a remarkably good empirical fit. We estimate the model via the EMM procedure of Gallant and Tauchen (1996) which allows us to compare the non-nested log-linear SV diffusion with the affine jump specification. Obviously, there is one drawback to the log-linear SV models when it comes to pricing derivatives since no closed-form solutions are available. Against this cost weights the advantage of avoiding all the complexities involved with jump processes.

A Case of Empirical Reverse Engineering: Estimation of the Pricing Kernel

We revisit the Roll (1977) critique regarding the unobservability of the market portfolio in the framework of the CAPM. It is equivalent to the unobservability of the pricing kernel (also known as the stochastic discount factor) in the language of the modern asset pricing theory. We advocate an econometric procedure which allows the estimation of the pricing kernel without either any assumptions about the investors preferences or the use of the consumption data. We propose a model of equity price dynamics, which allows for (i) simultaneous consideration of multiple stock prices, (ii) analytical formulas for derivatives such as futures, options and bonds, and (iii) a realistic description of all of these assets. The analytical specification of the model allows us to infer the dynamics of the pricing kernel. The model, calibrated to a comprehensive dataset including the S&P 500 index, individual equities, T-bills and gold futures, yields the conditional filter of the unobservable pricing kernel. As a result we obtain the estimate of the kernel, which is positive almost surely (i.e. precludes arbitrage), consistent with the equity risk premium, the risk-free discounting, and with the observed asset prices by construction. We find that the S&P 500 index contains virtually zero idiosyncratic volatility. The pricing kernel filter involves a highly nonlinear function of the contemporaneous and lagged returns on the index. This contradicts typical implementations of CAPM, which use a linear function of the market proxy return as the pricing kernel. Hence, the S&P 500 index does not have to coincide with the market portfolio if it is used in conjuction with nonlinear asset pricing models.