Risk-neutral Density Research Articles

Principled pasting: attaching tails to risk-neutral probability density functions recovered from option prices

The popular ‘curve-fitting’ method of using option prices to construct an underlying asset's risk neutral probability density function (RND) first recovers the interior of the density and then attaches left and right tails. Typically, the tails are constructed so that values of the RND and risk neutral cumulative distribution function (RNCDF) from the interior and the tails match at the attachment points. We propose and demonstrate the feasibility of also requiring that the left and right tails accurately price the options with strikes at the attachment points. Our methodology produces a RND that provides superior pricing performance than earlier curve-fitting methods for both those options used in the construction of the RND and those that were not. We also demonstrate that Put-Call Parity complicates the classification of in and out of sample options.

Localized radial basis functions for no-arbitrage pricing of options under stochastic alpha–beta–rho dynamics

Closed-form explicit formulas for implied Black–Scholes volatilities provide a rapid evaluation method for European options under the popular stochastic alpha–beta–rho (SABR) model. However, it is well known that computed prices using the implied volatilities are only accurate for short-term maturities, but, for longer maturities, a more accurate method is required. This work addresses this accuracy problem for long-term maturities by numerically solving the no-arbitrage partial differential equation with an absorbing boundary condition at zero. Localized radial basis functions in a finite-difference mode are employed for the development of a computational method for solving the resulting two-dimensional pricing equation. The proposed method can use either multiquadrics or inverse multiquadrics, which are shown to have comparable performances. Numerical results illustrate the accuracy of the proposed method and, more importantly, that the computed risk-neutral probability densities are nonnegative. These two key properties indicate that the method of solution using localized meshless methods is a viable and efficient means for price computations under SABR dynamics. doi:10.1017/S1446181121000237

State price density estimation with an application to the recovery theorem

Abstract This article introduces a model to estimate the risk-neutral density of stock prices derived from option prices. To estimate a complete risk-neutral density, current estimation techniques use a single mathematical model to interpolate option prices on two dimensions: strike price and time-to-maturity. Instead, this model uses B-splines with at-the-money knots for the strike price interpolation and a mixed lognormal function that depends on the option expiration horizon for the time-to-maturity interpolation. The results of this “hybrid” methodology are significantly better than other risk-neutral density extrapolation methods when applied to the recovery theorem.

A Model-Free Fourier Cosine Method for Estimating the Risk-Neutral Density

In this article, we present a new nonparametric method to extract the risk-neutral density from market-observed options prices. The method is based on novelly combining the Fourier cosine series method and the Carr-Madan spanning formula. In contrast to the seminal Breeden-Litzenberger formula, which is based on twice differentiating the options prices with respect to the strikes, our method is based on integrating the options prices with respect to available strikes at a given maturity. We employ the Black-Scholes model, constant elasticity of variance model, and the Heston model as data-generating processes in the numerical experiments, and real market data in the empirical experiments. These models demonstrate that the proposed method is accurate, is highly efficient to evaluate, and compares favorably with existing methods in the literature.

European option pricing using Gumbel distribution

A risk-neutral-density (RND) function plays a key role in determining option prices accurately. The RND function is useful in understanding the negative skewness and the excess kurtosis of the stock returns. RND models bases on the log-normal distribution (for example, Black–Scholes (BS) model (Black, F and M Scholes (1973). The pricing of options and corporate liabilities, The Journal of Political Economy 3, 637–654)) are less accurate to capture the negative skewness and the excess kurtosis of stock returns (Markose, S and A Alentorn (2011). The generalized extreme value distribution, implied tail index, and option pricing, The Journal of Derivatives 18(3), 35–60). The Extreme Value Theory (EVT) is useful in addressing the tail behavior of stock returns. The tail shape parameter in a Generalized Extreme Value (GEV) distribution class helps control tails’ size and shape. This paper model, the RND function in pricing the European call option, is modeled using a GEV distribution class, namely, a Gumbel distribution. It then compares GEV distribution option values with the BS values. It observes that the BS model underprices the call option values near maturity compared to those obtained by the GEV model. The Gumbel distribution for stock returns is useful in reducing the pricing bias of the BS model.

Risk Neutral Jump Arrival Rates Implied in Option Prices and Their Models

ABSTRACT Characteristic functions of risk neutral densities are constructed from the prices of options at a fixed maturity using well-known procedures. The logarithm of these characteristic functions are shown to synthesize the Fourier transform of jump arrival tails. The formal arrival rate tails are actual arrival rates if their derivatives have an appropriate sign. The derivatives of formal arrival rate tails embedded in option prices are observed on occasion to be negative, reflecting signed jump arrival rates. Although puzzling at first, we further observe that simple analytical cosine perturbations of the symmetric variance gamma Lévy density provides theoretical examples of such signed arrival rates consistent with a probability density. Additionally signed arrival rates also arise when models of signals perturbed by independent noise yield examples of characteristic functions for signal densities that are ratios of pure jump infinitely divisible characteristic functions. Such ratio characteristic functions can reflect signed arrival rates. Specific models using ratios of bilateral gamma and CGMY models are developed and calibrated to short maturity option prices. The ratio models provide significant improvements over their non-ratio counterparts. The models fall in the class of what have recently been termed to be quasi-infinitely divisible distributions.

The impact of COVID-19 on tail risk: Evidence from Nifty index options

We investigate the impact of COVID-19 using multiple forward-looking measures of uncertainty in Indian stock markets using liquid Nifty index options. The WHO declaration of COVID-19 as a pandemic coincides with a sharp rise in all measures of uncertainty considered, including option-implied volatility smiles, risk-neutral density, skewness, and kurtosis. We find that while subsequent government-imposed lockdowns and monetary easing induced a near-normalization of skewness and kurtosis, the volatility level remained elevated, demonstrating the importance of higher moments in capturing uncertainty during a pandemic. Structural breaks identified using the Bai–Perron methodology closely track the dates of significant announcements or interventions.

Risk premia in electricity derivatives markets

This study examines the prices of options contingent on electricity futures traded on the European Energy Exchange, with the aim to recover the probability density functions and risk premia. After we extract the risk-neutral probability density functions from prices of such options, we transform the risk-neutral densities into real-world densities using both parametric and non-parametric statistical calibration methods and investigate the evolution of risk premia and pricing kernels. We find that both risk-neutral and real-world option-implied densities accurately forecast realized futures electricity prices. Positively skewed densities suggest that there is an inverse (or positive) leverage effect in the electricity market, meaning that a higher probability of large price increases in electricity has been incorporated in the traded option prices. In addition, we find that the state price densities are mostly increasing, implying that investors are more risk-averse to high electricity prices. Over a period of 15 years, our results provide evidence of negative market price of risk and risk premia in this new market.

LOCALIZED RADIAL BASIS FUNCTIONS FOR NO-ARBITRAGE PRICING OF OPTIONS UNDER STOCHASTIC ALPHA–BETA–RHO DYNAMICS

AbstractClosed-form explicit formulas for implied Black–Scholes volatilities provide a rapid evaluation method for European options under the popular stochastic alpha–beta–rho (SABR) model. However, it is well known that computed prices using the implied volatilities are only accurate for short-term maturities, but, for longer maturities, a more accurate method is required. This work addresses this accuracy problem for long-term maturities by numerically solving the no-arbitrage partial differential equation with an absorbing boundary condition at zero. Localized radial basis functions in a finite-difference mode are employed for the development of a computational method for solving the resulting two-dimensional pricing equation. The proposed method can use either multiquadrics or inverse multiquadrics, which are shown to have comparable performances. Numerical results illustrate the accuracy of the proposed method and, more importantly, that the computed risk-neutral probability densities are nonnegative. These two key properties indicate that the method of solution using localized meshless methods is a viable and efficient means for price computations under SABR dynamics.

A New Nonparametric Estimate of the Risk-Neutral Density with Applications to Variance Swaps

Estimates of risk-neutral densities of future asset returns have been commonly used for pricing new financial derivatives, detecting profitable opportunities, and measuring central bank policy impacts. We develop a new nonparametric approach for estimating the risk-neutral density of asset prices and reformulate its estimation into a double-constrained optimization problem. We evaluate our approach using the S&P 500 market option prices from 1996 to 2015. A comprehensive cross-validation study shows that our approach outperforms the existing nonparametric quartic B-spline and cubic spline methods, as well as the parametric method based on the normal inverse Gaussian distribution. As an application, we use the proposed density estimator to price long-term variance swaps, and the model-implied prices match reasonably well with those of the variance future downloaded from the Chicago Board Options Exchange website.

W-shaped implied volatility curves and the Gaussian mixture model

The number of crossings of the implied volatility function with a fixed level is bounded above by the number of crossings of the risk-neutral density with the density of a log-normal distribution with the same mean as the forward price. It is bounded below by the number of convex payoffs priced equally by the two densities. We discuss the implications of these bounds for the implied volatility in the N-component Gaussian mixture model, with particular attention to the possibility of W-shaped smiles. We show that the implied volatility in this model crosses any level at most times. We show that a bimodal density need not produce a W-shaped smile, and a unimodal density can produce an oscillatory smile. We give monotonicity properties of the implied volatility in Gaussian mixtures under stochastic orderings of the location parameters and volatilities of the mixture components. For some of these results we make use of a novel convexity property of the Black-Scholes price at one strike with respect to the price at another strike. The combined constraints from density crossings and extreme strike asymptotics restrict the allowed shapes of the implied volatility. As an application we discuss a symmetric N = 3 Gaussian mixture model which generates three possible smile shapes: U-shaped, W-shaped and an oscillatory shape with two minima and two maxima.

On the RND under Heston's Stochastic Volatility Model

We consider Heston's (1993) stochastic volatility model for valuation of European options to which (semi) closed form solutions are available and are given in terms of characteristic functions. We prove that the class of scale-parameter distributions with mean being the forward spot price satisfies Heston's solution. Thus, we show that any member of this class could be used for the direct risk-neutral valuation of the option price under Heston's SV model. In fact, we also show that any RND with mean being the forward spot price that satisfies Hestons' option valuation solution, must be a member of a scale-family of distributions in that mean. As particular examples, we show that one-parameter versions of the {\it Log-Normal, Inverse-Gaussian, Gamma, Weibull} and the {\it Inverse-Weibull} distributions are all members of this class and thus provide explicit risk-neutral densities (RND) for Heston's pricing model. We demonstrate, via exact calculations and Monte-Carlo simulations, the applicability and suitability of these explicit RNDs using already published Index data with a calibrated Heston model (S&P500, Bakshi, Cao and Chen (1997), and ODAX, Mrazek and Pospisil (2017)), as well as current option market data (AMD).

A Descriptive Study of High-Frequency Trade and Quote Option Data*

Abstract This paper provides a guide to high-frequency option trade and quote data disseminated by the Options Price Reporting Authority (OPRA). We present a comprehensive overview of the U.S. option market, including details on market regulation and the trading processes for all 16 constituent option exchanges. We review the existing literature that utilizes high-frequency options data, summarizes the general structure of the OPRA dataset, and presents a thorough empirical description of the observed option trades and quotes for a selected sample of underlying assets that contains more than 25 billion records. We outline several types of irregular observations and provide recommendations for data filtering and cleaning. Finally, we illustrate the usefulness of the high-frequency option data with two empirical applications: option-implied variance estimation and risk-neutral density estimation. Both applications highlight the rich information content of the high-frequency OPRA data.

A Closed-form Pricing Solution for Options on Assets with Pricing Errors

This paper examines the effects that pricing errors in the underlying asset have on options prices, their Greeks, and their implied risk neutral densities. Pricing errors can be viewed as a random proportional transaction cost. When pricing errors are information-unrelated, options prices are unambiguously higher than the Black-Scholes case and increasing in the pricing error variance. Hedging volatility is higher and the optimal exercise price for American put options is decreased. The option implied risk-neutral density and option Greeks are materially affected, which leads to suboptimal risk management and hedging when pricing errors are not accounted for.

Implied Price Processes Anchored in Statistical Realizations

It is observed that statistical and risk neutral densities of compound Poisson processes are unconstrained relative to each other. Continuous processes are too constrained and generally not consistent with market data. Pure jump limit laws deliver operational models simultaneously consistent with both data sets with the additional imposition of no measure change on the arbitrarily small moves. The measure change density must have a finite Hellinger distance from unity linking the two worlds. Models are constructed using the bilateral gamma and the CGMY models for the risk neutral specification. They are linked to the physical process by measure change models. The resulting models simultaneously calibrate statistical tail probabilities and option prices. The resulting models have up to eight or ten parameters permitting the study of risk reward relations at a finer level. Rewards measured by power variations of the up and down moves are observed to value negatively(positively) the even(odd) variations of their own side with the converse holding for the opposite side.

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.

Deep Hedging: Learning Risk-Neutral Implied Volatility Dynamics

We present a numerically efficient approach for machine-learning a risk-neutral measure for paths of simulated spot and option prices up to a finite horizon under convex transaction costs and convex trading constraints. This approach can then be used to implement a stochastic implied volatility model in the following two steps: 1) Train a market simulator for option prices, for example as discussed in our recent work here; 2) Find a risk-neutral density, specifically in our approach the minimal entropy martingale measure. The resulting model can be used for risk-neutral pricing, or for Deep Hedging in the case of transaction costs or trading constraints. To motivate the proposed approach, we also show that market dynamics are free from in the absence of transaction costs if and only if they follow a risk-neutral measure. We additionally provide a more general characterization in the presence of convex transaction costs and trading constraints. These results can be seen as an analogue of the fundamental theorem of asset pricing for statistical arbitrage under trading frictions and are of independent interest.

Measuring investors’ risk aversion in China’s stock market

Our paper measures investors’ aversion to downside risk in China’s stock market by comparing two estimates of probability density functions for asset prices. The dynamics of the risk aversion indicator justifies the theoretical argument that the option-implied risk-neutral density incorporates information about investors’ risk preferences which is not captured in the historical data. Our risk aversion indicator sheds light on the investment timing of market practitioners: when the degree of risk aversion is particularly low but about to rise, there will be positive abnormal returns for most market sectors in the near future.

Detecting and Repairing Arbitrage in Traded Option Prices

ABSTRACT Option price data are used as inputs for model calibration, risk-neutral density estimation and many other financial applications. The presence of arbitrage in option price data can lead to poor performance or even failure of these tasks, making pre-processing of the data to eliminate arbitrage necessary. Most attention in the relevant literature has been devoted to arbitrage-free smoothing and filtering (i.e., removing) of data. In contrast to smoothing, which typically changes nearly all data, or filtering, which truncates data, we propose to repair data by only necessary and minimal changes. We formulate the data repair as a linear programming (LP) problem, where the no-arbitrage relations are constraints, and the objective is to minimize prices’ changes within their bid and ask price bounds. Through empirical studies, we show that the proposed arbitrage repair method gives sparse perturbations on data, and is fast when applied to real-world large-scale problems due to the LP formulation. In addition, we show that removing arbitrage from prices data by our repair method can improve model calibration with enhanced robustness and reduced calibration error.

COVID-19 and market expectations: Evidence from option-implied densities

We compare risk-neutral densities from equity index options across several countries during the early phase of the COVID-19 pandemic. The initial reaction in all analyzed markets was late, abrupt and simultaneous. Only a few weeks later, densities started to differ across markets.