Risk-neutral Return Distribution Research Articles

Estimation of rare disaster concerns from option prices—An arbitrage‐free RND‐based smile construction approach

AbstractThis research addresses the estimation of measures of rare disaster concerns from option prices. We propose a new smile construction approach to obtain the required continuum of implied volatilities from discretely sampled observations that are affected by microstructure noise. We extrapolate implied volatilities of far out‐of‐the‐money options by modeling the tails of the risk‐neutral return distribution (RND) ensuring that option prices do not admit arbitrage. Our numerical analysis and empirical application show that the RND‐based approach consistently outperforms standard techniques. It substantially reduces estimation errors resulting in considerably higher estimates of the rare disaster concern index () when event risk is high.

Non-equilibrium skewness, market crises, and option pricing: Non-linear Langevin model of markets with supersymmetry

This paper presents a tractable model of non-linear dynamics of market returns using a Langevin approach. Due to non-linearity of an interaction potential, the model admits regimes of both small and large return fluctuations. Langevin dynamics are mapped onto an equivalent quantum mechanical (QM) system. Borrowing ideas from supersymmetric quantum mechanics (SUSY QM), a parameterized ground state wave function (WF) of this QM system is used as a direct input to the model, which also fixes a non-linear Langevin potential. Using a two-component Gaussian mixture as a ground state WF with an asymmetric double well potential produces a tractable low-parametric model with interpretable parameters, referred to as the NES (Non-Equilibrium Skew) model. Supersymmetry (SUSY) is then used to find time-dependent solutions of the model in an analytically tractable way. Additional approximations give rise to a final practical version of the NES model, where real-measure and risk-neutral return distributions are given by three component Gaussian mixtures. This produces a closed-form approximation for option pricing in the NES model by a mixture of three Black–Scholes prices, providing accurate calibration to option prices for either benign or distressed market environments, while using only a single volatility parameter. These results stand in stark contrast to the most of other option pricing models such as local, stochastic, or rough volatility models that need more complex specifications of noise to fit the market data.

Empirical performance of stochastic volatility option pricing models

This paper examines the empirical performance of four stochastic volatility option pricing models: Heston, Heston[Formula: see text], Bates and Heston–Hull–White. To compare these models, we use individual stock options data from January 1996 to August 2014. The comparison is made with respect to pricing and hedging performance, implied volatility surface and risk-neutral return distribution characteristics, as well as performance across industries and time. We find that the Heston model outperforms the other models in terms of in-sample pricing, whereas Heston[Formula: see text] model outperforms the other models in terms of out-of-sample hedging. This suggests that taking jumps or stochastic interest rates into account does not improve the model performance after accounting for stochastic volatility. We also find that the model performance deteriorates during the crises as well as when the implied volatility surface is steep in the maturity or strike dimension.

Non-Equilibrium Skewness, Market Crises, and Option Pricing: Non-Linear Langevin Model of Markets with Supersymmetry

This paper presents a tractable model of non-linear dynamics of market returns using a Langevin approach. Due to non-linearity of an interaction potential, the model admits regimes of both small and large return fluctuations. Langevin dynamics are mapped onto an equivalent quantum mechanical (QM) system. Borrowing ideas from supersymmetric quantum mechanics (SUSY QM), a parameterized ground state wave function (WF) of this QM system is used as a direct input to the model, which also fixes a non-linear Langevin potential. Using a two-component Gaussian mixture as a ground state WF with an asymmetric double well potential produces a tractable low-parametric model with interpretable parameters, referred to as the NES (Non-Equilibrium Skew) model. Supersymmetry (SUSY) is then used to find time-dependent solutions of the model in an analytically tractable way. Additional approximations give rise to a final practical version of the NES model, where real-measure and risk-neutral return distributions are given by three component Gaussian mixtures. This produces a closed-form approximation for option pricing in the NES model by a mixture of three Black-Scholes prices, providing accurate calibration to option prices for either benign or distressed market environments, while using only a single volatility parameter. These results stand in stark contrast to the most of other option pricing models such as local, stochastic, or rough volatility models that need more complex specifications of noise to fit the market data.

Asset prices and “the devil(s) you know”

In this paper, we study the asset pricing implications of persistence in the risk-neutral return distribution’s central moments. We detect a both economically and statistically significant premium of stocks with low over stocks with high such persistence. Annual value-weighted excess (risk-adjusted) returns are 4.38% (3.06%). These results cannot be explained by factors and characteristics documented in the previous literature. Furthermore, it is not the persistence of only one of the individual distributional moments but rather the joint persistence in all central moments of the risk-neutral distribution that is priced.

A dynamic equilibrium model for U-shaped pricing kernels

This paper proposes a dynamic equilibrium model that can provide a unified explanation for the stylized facts observed in stock index markets such as the fat tails of the risk-neutral return distribution relative to the physical distribution, negative expected returns on deep OTM call options and negative realized variance risk premiums. In particular, we focus on the U-shaped pricing kernel against the stock index return, which is closely related to the negative call returns. We assume that the stock index return follows a time-changed Lévy process and that a representative investor has power utility over the aggregate consumption that forms a linear regression of the stock index return and its stochastic activity rate. This model offers a macroeconomic interpretation of the stylized facts from the perspective of the sensitivity of the activity rate and stock index return on aggregate consumption as well as the investor’s risk aversion.

A Recovery that We Can Trust? Deducing and Testing the Restrictions of the Recovery Theorem

How reliable is the recovery theorem of Ross (2015)? We explore this question in the context of options on the 30-year Treasury bond futures, allowing us to deduce restrictions that link the physical and risk-neutral return distributions. Our empirical results undermine the implications of the recovery theorem. First, we reject an implicit assumption of the recovery theorem that the martingale component of the stochastic discount factor is identical to unity. Second, we consider the restrictions between the physical and risk-neutral return moments when the recovery theorem holds, and reject them in both forecasting regressions and generalized method of moments estimations. Received November 7, 2016; editorial decision July 24, 2017 by Editor Stijn Van Nieuwerburgh. Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online.

Firm-Specific Risk-Neutral Distributions: The Role of CDS Spreads

We propose a method to extract individual firms' risk-neutral return distributions by combining options and credit default swaps (CDS). Options provide information about the central part of the distribution, and CDS anchor the left tail. Jointly, options and CDS span the intermediate part of the distribution, which is driven by moderate-sized jump risk. We study the returns on a trading strategy that buys (sells) stocks exposed to positive (negative) moderate-sized jump risk unspanned by options or CDS individually. Controlling for many known factors, this strategy earns a 0.5% premium per month, highlighting the economic value of combining options and CDS.

A Dynamic Equilibrium Model for U-Shaped Pricing Kernels

This paper proposes a dynamic equilibrium model that can provide a unified explanation for the stylized facts observed in stock index markets such as the fat tails of risk-neutral return distribution relative to physical distribution, negative expected returns on deep OTM call options, and negative realized variance risk premiums. In particular, we focus on the U-shaped pricing kernel against the stock index return, which is closely related to the negative call returns. We assume that the stock index return follows the time-changed Levy process and that a representative investor has power utility over the aggregate consumption that forms a linear regression of the stock index return and its stochastic activity rate. This model offers a macroeconomic interpretation of the stylized facts from the perspective of the sensitivity of the activity rate and stock index return on the aggregate consumption as well as the investor's risk aversion.

Momentum and reversion in risk neutral martingale probabilities

A time homogeneous, purely discontinuous, parsimonous Markov martingale model is proposed for the risk neutral dynamics of equity forward prices. Transition probabilities are in the variance gamma class with spot dependent parameters. Markov chain approximations give access to option prices. The model is estimated on option prices across strike and maturity for five days at a time. Properties of the estimated processes are described via an analysis of return quantiles, momentum functions that measure the response of tail probabilities to such moves. Momentum and reversion are also addressed via the construction of reverse conditional expectations. Term structures for the moments of marginal distributions support a decay in skewness and excess kurtosis with maturity at rates slower than those implied by Lévy processes. Out of sample performance is additionally reported. It is observed that risk neutral dynamics by and large reflect the presence of momentum in numerous probabilities. However, there is some reversion in the upper quantiles of risk neutral return distributions.

Option pricing under jump-diffusion models with mean-reverting bivariate jumps

We propose a jump-diffusion model where the bivariate jumps are serially correlated with a mean-reverting structure. Mathematical analysis of the jump accumulation process is given, and the European call option price is derived in analytical form. The model and analysis are further extended to allow for more general jump sizes. Numerical examples are provided to investigate the effects of mean-reversion in jumps on the risk-neutral return distributions, option prices, hedging parameters, and implied volatility smiles.

Capturing Option Anomalies with a Variance-Dependent Pricing Kernel

We develop a GARCH option model with a variance premium by combining the Heston-Nandi (2000) dynamic with a new pricing kernel that nests Rubinstein (1976) and Brennan (1979). While the pricing kernel is monotonic in the stock return and in variance, its projection onto the stock return is nonmonotonic. A negative variance premium makes it U-shaped. We present new semi-parametric evidence to con…firm this U-shaped relationship between the risk-neutral and physical probability densities. The new pricing kernel substantially improves our ability to reconcile the time series properties of stock returns with the cross-section of option prices. It provides a unified explanation for the implied volatility puzzle, the overreaction of long-term options to changes in short-term variance, and the fat tails of the risk-neutral return distribution relative to the physical distribution.

Momentum and Reversion in Risk Neutral Martingale Probabilities

A time homogeneous, purely discontinuous, parsimonous Markov martingale model is proposed for the risk neutral dynamics of equity forward prices. Transition probabilities are in the variance gamma class with spot dependent parameters. Markov chain approximations give access to option prices. The model is estimated on option prices across strike and maturity for five days at a time. Properties of the estimated processes are described via an analysis of return quantiles, momentum functions that measure the response of tail probabilities to such moves. Momentum and reversion are also addressed via the construction of reverse conditional expectations. Term structures for the moments of marginal distributions support a decay in skewenss and excess kurtosis with maturity at rates slower than those implied by Levy processes. Out of sample performance is additionally reported. It is observed that risk neutral dynamics by and large reflect the presence of momentum in numerous probabilities. However, there is some reversion in the upper quantiles of risk neutral return distributions.

Risk-Neutral Systemic Risk Indicators

This paper describes a set of indicators of systemic risk computed from current market prices of equity and equity index options. It displays results from a prototype version, computed daily from January 2006 to January 2013. The indicators represent a systemic risk event as the realization of an extreme loss on a portfolio of large-intermediary equities. The technique for computing them combines risk-neutral return distributions with implied return correlations drawn from option prices, tying together the single-firm return distributions via a copula to simulate the joint distribution and thus the financial-sector portfolio return distribution. The indicators can be computed daily using only current market prices; no historical data are involved. They are therefore forward-looking and can exploit all the information impounded in current prices. However, the indicators blend both market expectations and the market's desire to protect itself against volatility and tail risk, so they cannot be readily decomposed into these two elements. The paper presents evidence that the indicators have some predictive power for systemic risk events and that they can serve as a meaningful market-adjusted point of comparison for fundamentals-based systemic risk indicators.

A Generalized Measure of Riskiness

This paper proposes a generalized measure of riskiness that nests the original measures pioneered by Aumann and Serrano (Aumann, R. J., R. Serrano. 2008. An economic index of riskiness. J. Political Econom. 116(5) 810–836) and Foster and Hart (Foster, D. P., S. Hart. 2009. An operational measure of riskiness. J. Political Econom. 117(5) 785–814). The paper introduces the generalized options' implied measure of riskiness based on the risk-neutral return distribution of financial securities. It also provides asset allocation implications and shows that the forward-looking measures of riskiness successfully predict the cross section of 1-, 3-, 6-, and 12-month-ahead risk-adjusted returns of individual stocks. The empirical results indicate that the generalized measure of riskiness is able to rank equity portfolios based on their expected returns per unit of risk and hence yields a more efficient strategy for maximizing expected return of the portfolio while minimizing its risk. This paper was accepted by Wei Xiong, finance.

Management Insights

Management Insights

Too Good to Be True? An Analysis of the Options Market's Reactions to Earnings Releases

Using option implied risk neutral return distributions before and after earnings announcements, we study the option market's reaction to extreme events over earnings announcements. While earnings announcements generally reduce short term uncertainty about the stock price, very good news does not reduce uncertainty and slightly bad news actually increases uncertainty. Cross-sectional tests of realized volatility reductions are largely in line with option implied findings, except for cases where very good news is released, which suggests an irrational level of perceived uncertainty in options markets following very good news. We also find that left tail probabilities decrease over earnings releases while right tail probabilities increase. We interpret these findings as evidence of maintained investor expectations that very good news is generally not released during earnings announcements combined with unwarranted skepticism at the release of such news.

Capturing Option Anomalies with a Variance-Dependent Pricing Kernel

We develop a GARCH option model with a variance premium by combining the Heston-Nandi (2000) dynamic with a new pricing kernel that nests Rubinstein (1976) and Brennan (1979). While the pricing kernel is monotonic in the stock return and in variance, its projection onto the stock return is nonmonotonic. A negative variance premium makes it U-shaped. We present new semi-parametric evidence to con firm this U-shaped relationship between the risk-neutral and physical probability densities. The new pricing kernel substantially improves our ability to reconcile the time series properties of stock returns with the cross-section of option prices. It provides a unified explanation for the implied volatility puzzle, the overreaction of long-term options to changes in short-term variance, and the fat tails of the risk-neutral return distribution relative to the physical distribution.

Nonparametric Risk-Neutral Return and Volatility Distributions

Using a no-arbitrage condition we develop a nonparametric technique to extract the risk-neutral distribution of both asset returns and instantaneous volatilities from plain vanilla option prices. Our technique extends existing approaches that lead to riskneutral return distributions only. In order to estimate the risk-neutral volatility distribution, we do not need to that derivatives on volatility are traded. More generally, as our method yields a nonparametric estimate of the joint risk-neutral return/volatility distribution, we can also estimate conditional distributions of returns given future volatility levels. This opens the possibility to answer several important questions on risk-neutral volatility distributions and, thus, volatility risk premiums. Using S&P500 data, we conflrm negative volatility risk premiums and a right-shift in the future volatility distribution for higher initial volatility levels, but flnd additionally positive risk-neutral volatility skewness. Moreover, volatility skewness is more pronounced in low volatility periods. This is consistent with a large aversion towards unexpected positive volatility shocks. With respect to the risk-neutral return distribution, we conflrm overall negative skewness, but flnd that conditionally on decreasing volatility levels, the negative return skewness disappears. Concerning the risk-neutral dependence between return and volatility, we conflrm that this dependence is negative. Compared to parametric models, we flnd that risk-neutral volatility of volatility is much smaller than predicted by the popular Heston (1993) model. This indicates the necessity of a jump component in the risk-neutral return process. Furthermore, the risk-neutral volatility of volatility cannot be described by a single difiusion risk-neutral volatility process.

Crash-O-Phobia: A Domestic Fear or A Worldwide Concern?

From a large options data set on major equity indexes across the world, we find that worldwide, implied volatilities of options on equity indexes exhibit strikingly similar behaviors. When plotted against moneyness, implied volatilities show a heavily skewed smirk pattern, implying that out-of-the-money put options are more expensive than the corresponding out-of-the-money call options, and that the risk-neutral return distribution for these indexes is heavily negatively skewed. Furthermore, as the option maturity increases from one month to five years, the implied volatility smirk does not flatten out, but steepens, indicating that the risk-neutral distribution of equity index returns becomes even more negatively skewed at longer horizons. The average term structure of the implied volatility level is relatively flat, and the standard deviation of the implied volatility declines as maturity increases. Although fairly persistent, the implied volatility series are stationary. Finally, principal component analysis reveals that equity index volatility movements across the world share one global component.