Risk-neutral Moments Research Articles

Estimation of expected return integrating real-time asset prices implied information and historical data

In this paper, we develop a novel estimation for expected stock returns combining forward-looking information implied by real-time asset prices and backward-looking information implied by historical data. Considering a general heterogeneous market composed of both informed investors and noise investors, we investigate the market equilibrium characterized by the expected returns, risk-neutral moments and market portfolio. To mitigate the negative impact of the market noise on the forward-looking information implied in market equilibrium, we then incorporate historical data and propose the combined estimation for expected return within a Bayesian framework. The combined estimation is adaptive to the market composition and adjustable to changes in market states. Monte Carlo simulations and empirical studies are performed to validate the merits of the proposed approach.

Explaining intraday crude oil returns with higher order risk-neutral moments

High frequency crude oil option data is used to extract the higher order risk-neutral moments from the crude oil market. These risk-neutral moments include the variance, third central moment and the recently developed tail risk variation measures. We find it is beneficial to disaggregate these risk-neutral moments into their semi-moments, and to work with their log differences instead of the level. The log differences of the second and third semi-moments, and to a lesser extent, the log differences of the tail risk measures, are found to explain returns in the crude oil and S&P 500 futures at high frequency. We also provide evidence that the efficient market hypothesis holds at high frequency in these markets.

Bakshi, Kapadia, and Madan (2003) risk-neutral moment estimators: A Gram–Charlier density approach

This paper is a sequel to Aschakulporn and Zhang (J Futures Mark 42(3):365–388, 2022). The errors of the Bakshi et al. (Rev Financ Stud 16(1):101–143, 2003) risk-neutral moment estimators is studied using the Gram–Charlier density—with the skewness and excess kurtosis specified. To obtain skewness with (absolute) errors less than 10^{-3}, the range of strikes (K_{min }, K_{max }) must contain at least 3/4 to 4/3 of the forward price and have a step size (Delta K) of no more than 0.1% of the forward price. The range of strikes and step size corresponds to truncation and discretization errors, respectively. This is consistent to Aschakulporn and Zhang (2022) for non-volatile market periods.

Predicting Intraday Crude Oil Returns with Higher Order Risk-Neutral Moments

Predicting Intraday Crude Oil Returns with Higher Order Risk-Neutral Moments

A unified entropic pricing framework of option: Using Cressie-Read family of divergences

The entropy valuation of option (Stutzer, 1996) provides a risk-neutral probability distribution (RND) as the pricing measure by minimizing the Kullback–Leibler (KL) divergence between the empirical probability distribution and its risk-neutral counterpart. This article establishes a unified entropic framework by developing a class of generalized entropy pricing models based upon Cressie-Read (CR) family of divergences. The main contributions of this study are: (1) this unified framework can readily incorporate a set of informative risk-neutral moments (RNMs) of underlying return extracted from the option market which accurately captures the characteristics of the underlying distribution; (2) the classical KL-based entropy pricing model is extended to a unified entropic pricing framework upon a family of CR divergences. For each of the proposed models under the unified framework, the optimal RND is derived by employing the dual method. Simulations show that, compared to the true price, each model of the proposed family can produce high accuracy for option pricing. Meanwhile, the pricing biases among the models are different, and we hence conduct theoretical analysis and experimental investigations to explore the driving causes.

The limitations of estimating implied densities from option prices

Bakshi et al. (Stock Return Characteristics, Skew Laws, and the Differential Pricing of Individual Equity Options. Rev. Financ. Stud., 2003, 16(1), 101–143) develop a novel method for estimating a stocks's τ-period risk-neutral return moments from option prices. Their result relies on the integration of all OTM calls and puts to find the prices of a quadratic, cubic, and quartic contract; from which risk-neutral estimates of the 2nd, 3rd, and 4th moments are derived. Dennis and Mayhew (Risk-neutral skewness: Evidence from stock options. J. Financ. Quant. Anal., 2002, 37(3), 471–493) point out that error is introduced into the BKM estimation in practice since options do not span a continuum and the range of moneyness and distance between strikes varies across firms and maturities. We use Monte Carlo simulation to show that the standard approach of Dennis and Mayhew (Risk-neutral skewness: Evidence from stock options. J. Financ. Quant. Anal., 2002, 37(3), 471–493) used within Conrad et al. (Ex ante skewness and expected stock returns. J. Finance., 2013, 68(1), 85–124) and other leading empirical works leads to very biased, and in some cases inconsistent, estimates of BKM implied moments. More so, we prove the particularly concerning result that estimation error using the Dennis and Mayhew (Risk-neutral skewness: Evidence from stock options. J. Financ. Quant. Anal., 2002, 37(3), 471–493) methodology increases dramatically when a stock's actual return distribution is non-normal or the firm's stock price is not exactly at-the-money. Fortunately, the 2-step interpolation method developed by DeMiguel et al. (Improving portfolio selection using option-implied volatility and skewness. J. Financ. Quant. Anal., 2013, 48(6), 1813–1845) is largely robust to the estimation errors we document. Based on our simulation results, we recommend the use of the DeMiguel et al. (Improving portfolio selection using option-implied volatility and skewness. J. Financ. Quant. Anal., 2013, 48(6), 1813–1845) methodology to estimate the 3rd and 4th implied moments, and either the DeMiguel et al. (Improving portfolio selection using option-implied volatility and skewness. J. Financ. Quant. Anal., 2013, 48(6), 1813–1845) methodology or the methodology of Fukasawa et al. (Model-free implied volatility: From Surface to Index. Int. J. Theor. Appl. Finance, 2011, 14(4), 433–463) to estimate the 2nd implied moment (implied volatility). In addition, we argue that the economically large and robust results of our MC simulations and brief empirical study indicate that the findings of prior studies which use the estimation methodology of Dennis and Mayhew (Risk-neutral skewness: Evidence from stock options. J. Financ. Quant. Anal., 2002, 37(3), 471–493), followed by Conrad et al. (Ex ante skewness and expected stock returns. J. Finance., 2013, 68(1), 85–124), should be reevaluated.

Informative option portfolios in filter design for option pricing models

Option pricing models are tools for pricing and hedging derivatives. Good models are complex and the econometrician faces many design decisions when bringing them to the data. I show that strategically constructed low-dimensional filter designs match and often outperform those that try to use all the available option data, in terms of state recovery, pricing, and hedging. The filters are built around option portfolios that aggregate option data, and track changes in risk-neutral variance and skewness. They also explicitly account for difficulties in the recovery of risk-neutral moments from option prices. The performance advantage is greatest in empirically relevant settings: in models with strongly skewed jump components that are not driven by Brownian volatility.

Venturing into Uncharted Territory: An Extensible Parametric Implied Volatility Surface Model

A new parametric representation of implied volatility surfaces is proposed. The factors adequately capture the moneyness and maturity slopes, the smile attenuation, and the smirk. Furthermore, the implied volatility specification is twice continuously differentiable and well behaved asymptotically, allowing for clean interpolation and extrapolation over a wide range of moneyness and maturity. Fitting performance on S&P 500 options compares favourably with existing benchmarks. The benefits of a smoothed implied volatility surface are illustrated through the valuation of illiquid index derivatives, the extraction of the risk-neutral density and risk-neutral moments, the calculation of option price sensitivities, and the calculation of SVIX for the equity risk premium lower bound.

Can risk-neutral skewness and kurtosis subsume the information content of historical jumps?

We examine the relation between jump variations and risk-neutral moments in volatility forecasting. We propose a method that involves no extrapolation in computing the risk-neutral moments of Bakshi et al. (2003) and document that risk-neutral skewness and kurtosis subsume the information content of historical jumps. While historical jumps have significant explanatory power for future volatility and such power is actually not weakened by the inclusion of risk-neutral volatility in models, their predictability does disappear when risk-neutral skewness and kurtosis are included.

A Theory of Dissimilarity Between Stochastic Discount Factors

This paper proposes a measure of dissimilarity between stochastic discount factors (SDFs) in different economies. The SDFs are made comparable using the respective bond prices as the numeraire. The measure is dimensionless, synthesizes features of the risk-neutral moments of excess currency returns, and can be extracted from currency option prices. Linking theory to data, we provide evidence gathered from (i) the cross section of 45 currency option prices, (ii) the time series of currency returns, (iii) estimated SDFs using model-free restrictions, and (iv) structural models in international finance.This paper was accepted by David Simchi-Levi, finance.

Risk-Neutrality of RND and Option Pricing within an Entropy Framework.

This article constructs an entropy pricing framework by incorporating a set of informative risk-neutral moments (RNMs) extracted from the market-available options as constraints. Within the RNM-constrained entropic framework, a unique distribution close enough to the correct one is obtained, and its risk-neutrality is deeply verified based on simulations. Using this resultant risk-neutral distribution (RND), a sample of risk-neutral paths of the underlying price is generated and ultimately the European option’s prices are computed. The pricing performance and analysis in simulations demonstrate that this proposed valuation is comparable to the benchmarks and can produce fairly accurate prices for options.

Distorting Arrow-Debreu Securities: New Entropy Restrictions Implied by the Option Cross Section

Replacing equity return (as in the equity risk premium) with returns on an arbitrary contingent claim, we obtain a new class of economic risk premiums to impose upon candidate models. These risk premiums reflect the distance between the physical and risk-neutral moments for asset returns, can be estimated in a model-free fashion from the option cross section, and provide sharp information in distinguishing alternative models. Confronting leading macro-finance models with our risk premiums, we uncover a wide dispersion in performance across candidate models. Our evidence points to the importance of incorporating persistent stochastic volatilities and/or higher moments in fundamentals to reconcile with the option data, as exemplified by Bansal and Yaron (2004) and Bekaert and Engstrom (2017).

Unraveling Momentum's Moments

I examine the momentum anomaly through the prism of higher order risk neutral moments of high and low momentum stocks. I use option prices from 2002 to 2013 to extract time varying ex ante estimates of variance, skewness and kurtosis of high and low momentum stocks. High momentum stocks have lower variance, less negative skewness and lower kurtosis. This is in contrast to studies that use static, ex post estimates of skewness to explain momentum returns as a function of exposure to the higher order moments of the underlying return distribution. These results highlight that momentum returns can not be explained by ex ante estimates of higher order moments and remain dicult to resolve empirically in a rational expectations framework.

Forward-looking information on growth and uncertainty implied by derivative securities: Evidence from an emerging market

In this paper, we investigate investors' expectations on economic growth and uncertainty risk implied by derivative securities. Empirical evidence on investors’ expectations implied by derivative securities has been intensively studied in the U.S. and other developed markets, however, such evidence still seems to be rare for emerging markets. Using high frequency data of the Taiwan Stock Exchange (TAIEX) weighted index and its derivatives from Jan-02-2003 to Dec-31-2014, we construct time series of implied dividends, variance risk premium and higher risk-neutral moments. We find that term structure of the implied dividend yield and variance risk premium have some abilities on predicting the excess return of the TAIEX weighted index and growth of industrial production index of Taiwan. We also demonstrate that there is a strong and positive relation between the risk-neutral skewness and options slope, which is in line with what previous literature found in the U.S. stock market.

Robust Estimation of Risk-Neutral Moments

This study provides an in-depth analysis of how to estimate risk-neutral moments robustly. A simulation and an empirical study show that estimating risk- neutral moments presents a trade-off between (1) the bias of estimates caused by a limited strike price domain and (2) the variance of estimates induced by micro-structural noise. The best trade-off is offered by option-implied quantile moments estimated from a volatility surface interpolated with a local-linear kernel regression and extrapolated linearly. A similarly good trade-off is achieved by estimating regular central option-implied moments from a volatility surface interpolated with a cubic smoothing spline and flat extrapolation.

An Empirical Implementation of the Ross Recovery Theorem as a Prediction Device

Abstract Building on the method of Ludwig (2015) to construct robust state price density surfaces from snapshots of option prices, we develop a nonparametric estimation strategy based on the recovery theorem of Ross (2015). Using options on the S&P 500, we then investigate whether or not recovery yields predictive information beyond what can be gleaned from risk-neutral densities. Over the 13 year period from 2000 to 2012, we find that market timing strategies based on recovered moments outperform those based on risk-neutral moments.

It only takes a few moments to hedge options

We propose a novel non-structural method for hedging European options, relying on two model-independent results: First, under suitable regularity conditions, an option price can be disentangled into a linear combination of risk-neutral moments. Second, there exists an explicit approximate functional form linking the risk-neutral moments to the futures price of the underlying asset and the related variance swap contracts. We show that S&P 500 call prices are mainly explained by two factors that are related to level and volatility of the underlying index. We empirically compare the performance of two strategies where the vega exposure is adjusted either by a direct position in a variance swap contract or, indirectly, through an at-the-money call. While both strategies ensure effective immunization in periods of market turmoil, taking direct exposure on variance swaps is not optimal during extended periods of subdued volatility.

Risk-Neutrality of RND and Option Pricing within an Entropy Framework

This article constructs an entropy pricing framework by incorporating a set of informative risk-neutral moments (RNMs) extracted from the market-available options as constraints. Within the RNM-constrained entropic framework, a unique distribution close enough to the correct one is obtained, and its risk-neutrality is deeply verified based on simulations. Using this resultant risk-neutral distribution (RND), a sample of risk-neutral paths of the underlying price is generated and ultimately the European option’s prices are computed. The pricing performance and analysis in simulations demonstrate that this proposed valuation is comparable to the benchmarks and can produce fairly accurate prices for options.

A non-structural investigation of VIX risk neutral density

Abstract We propose a non-structural method to retrieve the risk-neutral density (RND) implied by options on the CBOE Volatility Index (VIX). The methodology is based on orthogonal polynomial expansions around a kernel density and yields the RND of the underlying asset without the need for a parametric specification. The classic family of Laguerre expansions is extended to include the GIG and the generalized Weibull kernels. We show that orthogonal polynomial expansions yield accurate approximations of the RND of VIX and they generally outperform commonly used non-parametric methods when controlling for accuracy. Based on a panel of observed VIX options, we retrieve the variance swap term structure, the time series of VVIX, the VIX risk-neutral moments and the Volatility-at-Risk, which reveal a number of stylized facts on the RND of VIX.

Risk-neutral moments in the crude oil market

In this paper, we provide a comprehensive study on the higher-order risk-neutral moments (RNMs) and differences in RNMs (DRNMs) in the crude oil market, implied by options written on the United States Oil Fund (USO). Based on the t-statistic, the in-sample and the out-of-sample R2 statistics, we compare the USO return predictability and USO option return predictability by using RNMs and DRNMs from May 2007 to April 2016. We find that (i) all RNMs have a poor out-of-sample performance of predicting USO returns and simple option returns, while the risk-neutral volatility (VOL) outperforms in terms of both in-sample and out-of-sample predicting delta-hedged option returns; (ii) most of the DRNMs can significantly predict the future USO returns, and (iii) differences in the risk-neutral third cumulant (DTC) and differences in the risk-neutral fourth cumulant (DFC) are two important predictors for the future USO option returns.