Risk-neutral Density Research Articles

Crash risk and risk neutral densities

“Crash risk” has been one of the major focuses in the recent asset pricing literature. Motivated by the recent literature that suggests an increase in crash risk since Fall 2008 and the recent troubles in the Euro zone, we use EUR/USD FX options for January 2, 2008 to March 18, 2015 to study option-implied risk-neutral densities (RND). We find that RND, especially higher moments, has superior explanatory power in predicting and explaining crash risk and its risk premiums. Furthermore, the higher moments of RND co-move closely with macroeconomic variables. Consistently, we find RND moments outperform the implied volatility from the Black–Scholes model.

The eurozone (expected) inflation: An option's eyes view

We estimate inflation risk-neutral densities (RNDs) in the euro area, using inflation swaps and caps/floors options, and introduce a parsimonious approach to jointly estimate the RNDs across horizons. Thus, we obtain daily implicit RNDs for forward measures, such as the 5-year-on-5-year inflation rate, which, although not directly traded in the market, is a key reference for monetary policy. Then, we discuss several indicators derived from the information content of those historical RNDs that are relevant for policy makers, and compare them in light of the ECB's decisions and communication since 2009. Specifically, we study the evolution of tail risks (associated with deflation and high inflation), the balance of inflation risks, risk aversion, and how forward inflation rates react to the ECB's non-conventional monetary policies (Longer Term Refinancing Operations, LTRO; Securities Market Programme, SMP; Asset Purchase Programme, APP; and its variants and extensions).

Analyzing the Risks Embedded in Option Prices with rndfittool

This paper introduces a new computational tool for the analysis of the risks embedded in a set of prices of European-style options. The software enables the estimation of the risk-neutral density (RND) from the observed option prices by means of orthogonal polynomial expansions. Orthogonal polynomials offer a viable alternative to more standard techniques based on interpolation and estimation of the second-order derivatives of option prices. The app rndfittool is available on GitHub and its usage is illustrated with examples based on real data.

Asymptotic Expansion of Risk-Neutral Pricing Density

A new method for pricing contingent claims based on an asymptotic expansion of the dynamics of the pricing density is introduced. The expansion is conducted in a preferred coordinate frame, in which the pricing density looks stationary. The resulting asymptotic Kolmogorov-backward-equation is approximated by using a complete set of orthogonal Hermite-polynomials. The derived model is calibrated and tested on a collection of 1075 European-style ‘Deutscher Aktienindex’ (DAX) index options and is shown to generate very precise option prices and a more accurate implied volatility surface than conventional methods.

The Second Partial Derivative of Option Price with Respect to the Strike: A Historical Reminiscence

An option’s market price reflects the risk-neutral probability that it will end up in the money. Research has been increasing in recent years that shows how, given a set of market prices for options covering a range of strikes, an estimate of the entire risk-neutral probability distribution can be obtained. The technique is based on the fact that the second partial derivative of the option pricing function with respect to the strike price is the risk-neutral density (discounted from option expiration). This idea is generally attributed to Breeden and Litzenberger’s 1978 paper. In this article, Zimmermann shows that the connection between the second partial derivative of the option price with respect to the exercise price and risk-neutral probabilities has a much longer history, including a little-known 1974 note by Fischer Black, and going all the way back to Bachelier in 1900.

Risk-Neutral Densities: A Review

Trading in options with a wide range of exercise prices and a single maturity allows a researcher to extract the market's risk-neutral density (RND) over the underlying price at expiration. The RND contains investors’ beliefs about the true probabilities blended with their risk preferences, both of which are of great interest to academics and practitioners alike. With a particular focus on US equity options, I review the historical development of this powerful concept, practical details of fitting an RND to options market prices, and the many ways in which investigators have tried to distill true expectations and risk premia from observed RNDs. I briefly discuss areas of active current research including the pricing kernel puzzle and the volatility surface, and offer thoughts on what has been learned about RNDs so far and fruitful directions for future research.

Risk Neutral Densities: A Review

Risk Neutral Densities: A Review

Model Independent Multi-Asset Volatility Smile with Empirical Confirmation

This paper presents a new formalism to price European options in all asset classes that fits the market data remarkably well. We use a model-independent representation of European Option prices as path integrals over all of the underlying asset price from inception to maturity. The no arbitrage condition of the path integral representation generates a consistency condition upon the risk neutral probability density function to maturity. When solving for the risk neutral density function to maturity that satisfies the consistency condition, the European volatility smile is obtained. As an illustration of the formalism, we show that when the underlying asset price changes at constant volatility (standard deviation), the option price satisfies the Black-Scholes-Merton model without the pre-assumption that the underlying asset price satisfies a Brownian motion. More generally, without any pre-assumption on the underlying asset’s stochastic process, the risk neutral density function for a given expiration date is determined by the option prices of three strikes with the same expiration. We offer a parameterization of the volatility smile with a closed-form expression using pre-calculated tables. Comprehensive empirical evidence is provided, confirming that the new model accurately matches the market option prices in equities, commodities, currencies and interest rates. In addition, a formalism to obtain the conditional probability transfer density function from the vanilla options term structure is suggested.

A General Theory of Option Pricing

We present a new formalism for option pricing that does not require an assumption on the stochastic process of the underlying asset price and yet produces remarkably accurate results versus the market. The new formalism applies for general Markovian stochastic behavior including continuous and discontinuous (jump) processes and in its broadest scheme contains all known models for Markovian option pricing and some new ones. The method is based on obtaining the risk neutral density function that satisfies a consistency condition, guaranteeing no arbitrage. For example, we show that when the underlying asset undergoes a continuous stochastic process with deterministic time dependent standard deviation the formalism produces the Black-Scholes-Merton formula without using a Wiener process. We show that in the general case the price of European options depends only on all the moments of the price return of the underlying asset. We offer a method to calculate the prices of European options when the volatility smile at maturity is independent of the term structure prior to the maturity, as observed in options markets. In the continuous case where only moments up to second order contribute to the price then any set of three option prices with the same maturity contains the information to determine the whole volatility smile for this maturity. In all the many examples we examined our method generates option prices that match the option markets prices very accurately in all asset classes. This confirms that the options market exhibits no-arbitrage. Moreover, using bootstrapping we demonstrate how to determine the conditional density function from inception to maturity, thus allowing the calculation of path dependent options. The new formalism also allows for the replication of ‘W-shape’ volatility smile that infrequently appears in some equity markets.

Implied Volatility Sentiment: A Tale of Two Tails

Low probability events are overweighted in the pricing of out-of-the-money index puts and single stock calls. This behavioral bias is strongly time-varying, and is linked to equity market sentiment and higher moments of the risk-neutral density. We find that our implied volatility (IV) sentiment measure, jointly derived from index and single stock options, explains investors’ overweight of tail events well. When employed within a trading strategy, our IV-sentiment measure delivers economically significant results, which are more consistent than the ones produced by the market sentiment factor. Out-of-sample tests on reversal prediction show that our IV-sentiment measure adds value over and above traditional factors in the equity risk premium literature.

Option-implied objective measures of market risk

Foster and Hart (2009) introduce an objective measure of the riskiness of an asset that implies a bound on how much of one’s wealth is ‘safe’ to invest in the asset while (a.s.) guaranteeing no-bankruptcy in the long run. In this study, we translate the Foster-Hart measure from static and abstract gambles to dynamic and applied finance using nonparametric estimation of risk-neutral densities from S&P 500 call and put option prices covering 2003 to 2013. This exercise results in an option-implied market view of objective riskiness. The dynamics of the resulting ‘option-implied Foster-Hart bound’ are analyzed and assessed in light of well-known risk measures including value at risk, expected shortfall and risk-neutral volatility. The new measure is shown to be a significant predictor of ahead-return downturns. Furthermore, it is able to grasp more characteristics of the risk-neutral probability distributions than other measures, furthermore exhibiting predictive consistency. The robustness of the risk-neutral density estimation method is analyzed via a bootstrap.

Implicit Probability Distribution for WTI Options: The Black Scholes vs. the Semi-Nonparametric Approach

This paper contributes to the literature on the estimation of the Risk Neutral Density (RND) function by modeling the prices of options for West Texas Intermediate (WTI) crude oil that were traded in the period between January 2016 and January 2017. For these series we extract the implicit RND in the option prices by applying the traditional Black & Scholes (1973) model and the semi-nonparametric (SNP) model proposed by Backus, Foresi, Li, & Wu (1997). The results obtained show that when the average market price is compared to the average theoretical price, the lognormal specification tends to systematically undervalue the estimation. On the contrary, the SNP option pricing model, which explicitly adjust for negative skewness and excess kurtosis, results in markedly improved accuracy.

Option augmented density forecasts of market returns with monotone pricing kernel

Basic financial theory indicates that the ratio of the conditional density of the future value of a market index and the corresponding risk neutral density should be monotone, but a sizeable empirical literature finds otherwise. We therefore consider an option augmented density forecast of the market return obtained by transforming a baseline density forecast estimated from past excess returns so as to monotonize its ratio with a risk neutral density estimated from current option prices. To evaluate our procedure, we compare baseline and option augmented monthly density forecasts for the S&P 500 index over the period 1997–2013. We find that monotonizing the pricing kernel leads to a modest improvement in the calibration of density forecasts. Supplementary results supportive of this finding are given for market indices in France, Germany, Hong Kong, Japan and the UK.

Can We Really Discard the Forecasting Ability of Risk-Neutral Distributions?

This paper analyzes the forecasting ability of option-implied Risk-Neutral densities (RNDs) for three US indexes, S&P 500, Nasdaq 100 and Russell 2000 and for a long series (from 1996 to 2015) encompassing two major crisis. Traditional tests rely on restrictive assumptions (mainly normality and independence). In order to overcome these assumptions, we calculate block-bootstrap-based critical values. Different to existent literature, our results conclude failure to reject their forecasting ability, being these results consistent across different forecast horizons, methodologies and indexes considered. We also analyze the fit of the tails of the RNDs separately, finding that they tend to overestimate the frequency of occurrence of events in the left tail, providing a good fitting for the right tail.

Test of recent advances in extracting information from option prices

A large literature exists on techniques for extracting probability distributions for future asset prices from option prices. No definitive method has been developed however. The parametric ‘mixture of normals’, and non-parametric ‘smoothed implied volatility’ methods remain the most widespread approaches. These though are subject to estimation errors due to discretization, truncation, and noise. Recently, several authors have derived ‘model free’ formulae for computing the moments of the risk neutral density (RND) directly from option prices, without first estimating the full density. The accuracy of these formulae is studied here for the first time. The Black-Scholes formula is used to generate option prices, and error curves for the first 4 moments of the RND are computed using the ‘model-free’ formulae. It is found that, in practice, the formulae are prone to large and economically significant errors, because they contain definite integrals that can only be solved numerically. We show that without mathematically equivalent expressions with analytical solutions the formulae are difficult to deploy effectively in practice.

Subjective Probability Density Functions from FX Option Prices: Predictive Power and Performance on a Carry Trade Strategy

AbstractIn this article, we extracted the risk‐neutral densities (RNDs) and subjective probability density functions of the US Dollar/Brazilian Real (USD/BRL) exchange rate and evaluated its performance in predicting the future realizations of the USD/BRL exchange rate. The RNDs were estimated using two structural models and three nonstructural models. In the first category, we included the Variance Gamma‐OU model and the CGMY Gamma‐OU model. In the second category, we included the density functional based on confluent hypergeometric function model, the mixture of lognormal distributions model, and the smoothed implied volatility smile. The density functional based on confluent hypergeometric function and the CGMY Gamma‐OU produced 1‐month term densities (RND and subjective probability density function) with the highest forecasting power of the 1‐month USD/BRL exchange rate. Finally, we applied the CGMY Gamma‐OU model to extract a sample of subjective cumulative probabilities of 1‐month USD/BRL movements, and used them as explanatory variables in predictive time series models, whose dependent variable was the 1‐month carry trade return. Its predictive power was then tested and confirmed in three trading strategies that over performed the standard carry trade strategy in terms of annualized cumulative returns.

Financial Density Forecasts: A Comprehensive Comparison of Risk-Neutral and Historical Schemes

We investigate the forecasting ability of the most commonly used benchmarks in financial economics. We approach the main methodological caveats of probabilistic forecasts studies – small samples, limited models and non-holistic validations – by performing a comprehensive comparison of 15 predictive schemes during a time period of over 21 years. All densities are evaluated in terms of their statistical consistency, local accuracy and forecasting errors. Through the development of a new indicator, the Integrated Forecast Score (IFS), we show that risk-neutral densities outperform historical-based predictions in terms of information content. We find that the Variance Gamma model generates the highest out-of-sample likelihood of observed prices and the lowest predictive errors, whereas the ARCH-based GJR-FHS delivers the most consistent forecasts across the entire density range. In contrast, log-normal densities, the Heston model or the non-parametric Breeden-Litzenberger formula yield biased predictions and are rejected in statistical tests.

Asset allocation using option-implied moments

This study uses an option-implied distribution as the input in asset allocation. The computation of risk-neutral densities (RND) are based on the Dow Jones Industrial Average (DJIA) index option and its constituents. Since the RNDs estimation does not incorporate risk premium, the conversion of RND into risk-world density (RWD) is required. The RWD is obtained through parametric calibration using the beta distributions. The mean, volatility, and covariance are then calculated to construct the portfolio. The performance of the portfolio is evaluated by using portfolio volatility and Sharpe ratio.

SIEVE ESTIMATION OF THE MINIMAL ENTROPY MARTINGALE MARGINAL DENSITY WITH APPLICATION TO PRICING KERNEL ESTIMATION

We study the problem of nonparametric estimation of the risk-neutral densities from options data. The underlying statistical problem is known to be ill-posed and needs to be regularized. We propose a novel regularized empirical sieve approach for the estimation of the risk-neutral densities which relies on the notion of the minimal martingale entropy measure. The proposed approach can be used to estimate the so-called pricing kernels which play an important role in assessing the risk aversion over equity returns. The asymptotic properties of the resulting estimate are analyzed and its empirical performance is illustrated.

A Simple and Efficient Two-Factor Willow Tree Method for Convertible Bond Pricing with Stochastic Interest Rate and Default Risk

Most numerical methods for option pricing (lattice methods, Monte Carlo simulation) focus on the market’s short-run dynamics, in which instantaneous returns cumulate to a probability distribution at maturity, as the Black–Scholes model does. If the risk-neutral density is available, it is possible to jump immediately to maturity, but this only works for options without path-dependence and for a small number of returns processes. The willow tree method offers a solution between these two cases. The option’s life is split into a small number of discrete chunks, and the price dimension at each of these time steps is represented by a finite and relatively small set of possible values. Each node allows a transition to every price node at the next step; thus, the “tree” resembles a willow, with many long branches connecting the sets of prices separated by discrete time intervals. For the problems it can handle, the willow tree approach has been shown to be accurate and much faster than procedures that require modeling an asymptotically infinite number of instantaneous transitions. In this article, Lu and Xu show how to extend a willow tree to two stochastic factors, an underlying stock price and the interest rate, to apply the model to pricing convertible bonds. The procedure is amenable to many different returns processes. In a numerical illustration, the authors show that the two-factor willow tree is as accurate as a binomial or a Monte Carlo approach but requires much less computing time.