Risk-neutral Density Research Articles

Parametric risk-neutral density estimation via finite lognormal-Weibull mixtures

This paper proposes a new parametric risk-neutral density (RND) estimator based on a finite lognormal-Weibull mixture (LWM) density. We establish the consistency and asymptotic normality of the LWM method in a general misspecified parametric framework. Based on the theoretical results, we propose a sequential test procedure to evaluate the goodness-of-fit of the LWM model, which leads to an adaptive choice for the number and type of mixture components. Our simulation results show that, in finite samples with various observation error specifications, the LWM method can approximate complex RNDs generated by state-of-the-art multi-factor stochastic volatility models with a few (typically less than 4) mixtures. Application of the LWM model on index options confirms its reliability in recovering empirical RNDs with a heavy left tail or bimodality, which can be incorrectly identified as bimodality or a heavy left tail by existing (semi)-nonparametric methods if the goodness-of-fit to the observed data is ignored.

Derivation of discrete analog of Breeden–Litzenberger relation for risk-neutral density

This short paper derives a formula for the risk-neutral density function at discrete points over the strike range using Abel’s summation formula in terms of the arbitrage-free call price function in a European option contingent. The estimate of risk-neutral density is crucial from a risk management perspective and other crucial assessments.

Volatility-dependent probability weighting and the dynamics of the pricing kernel puzzle

In order to estimate volatility-dependent probability weighting functions, we obtain risk neutral and physical densities from the Pan (J Financ Econ 63(1):3–50, 2002. https://doi.org/10.1016/S0304-405X(01)00088-5) stochastic volatility and jumps model. Across volatility levels, we find pronounced inverse S-shapes, i.e. small probabilities are overweighted, and probability weighting almost monotonically increases in volatility, indicating higher skewness preferences and crash aversion in volatile market environments. Moreover, by estimating probabilistic risk attitudes, equivalent to the share of risk aversion related to probability weighting, we shed further light on the pricing kernel puzzle. While pricing kernels estimated from the Pan (J Financ Econ 63(1):3–50, 2002. https://doi.org/10.1016/S0304-405X(01)00088-5) model display the typical U-shape as documented in the literature, pricing kernels—net of probability weighting—are strictly monotonically decreasing and thus in line with economic theory. Equivalently, we find risk aversion to be positive across wealth levels. Our results are robust to alternative maturities, wealth percentiles, alternative functional forms, a nonparametric empirical setting and variations of the Pan (J Financ Econ 63(1):3–50, 2002. https://doi.org/10.1016/S0304-405X(01)00088-5) coefficient estimates.

Pricing European Options Using Burr-XII Distribution: Simulations and Risk Neutral Density

Pricing European Options Using Burr-XII Distribution: Simulations and Risk Neutral Density

Implied volatility surfaces: a comprehensive analysis using half a billion option prices

AbstractThis study delves into the critical aspect of accurately estimating single stock volatility surfaces, a task indispensable for option pricing, risk management, and empirical asset pricing. Utilizing a comprehensive dataset consisting of half a billion daily price observations for options on 499 US individual stocks and the S&P 500, the research investigates the accuracy of diverse methods for constructing volatility surfaces. The comparative evaluation of the three-dimensional kernel smoother by OptionMetrics (IvyDB US file and data reference manual, version 5.2, Rev. 01/27/2022, Computer software manual, New York, 2022), the semi-parametric spline by Figlewski (in: Robert F. Engle (ed) Estimating the implied risk neutral density. Volatility and time series econometrics: Essays in honor, Oxford University Press, Oxford, 2008), and a refined one-dimensional kernel smoother reveals the distinct superiority of the latter. This method consistently outperforms its counterparts across all moneyness, maturity, and liquidity categories, with markedly lower error metrics. The study further uncovers significant distortions in the extraction of Bakshi et al. (Rev Financ Stud 16:101–143, 2003) moments and skewness spanning induced by the noise-infused three-dimensional kernel smoother, which could potentially mislead derivative pricing and trading decisions. The findings offer valuable insights to traders, risk managers, investors, and researchers, suggesting a robust, one-size-fits-all method for crafting more accurate and less noisy volatility predictions. The research advances our understanding of option-implied information, its extraction, and broader implications for financial markets.

The Effect of Exit Time and Entropy on Asset Performance Evaluation.

The objective of this study is to evaluate assets' performance by considering the exit time within the risk measurement framework alongside Shannon entropy and, alternatively, excluding these factors, which can be used to create a portfolio aligned with short- or long-term objectives. This portfolio effectively balances the potential risks and returns, guiding investors to make decisions that are in line with their financial goals. To assess the performance, we used data envelopment analysis (DEA), whereby we utilized the risk measure as an input and the mean return as an output. The stop point probability-CVaR (SPP-CVaR) was the risk measurement used when considering the exit time. We calculated the SPP-CVaR by converting the risk-neutral density to the real-world density, calibrating the parameters, running simulations for price paths, setting the stop-profit points, determining the exit times, and calculating the SPP-CVaR for each stop-profit point. To account for negative data and to incorporate the exit time, we have proposed a model that integrates the mean return and SPP-CVaR, utilizing DEA. The resulting inefficiency scores of this model were compared with those of the mean-CVaR model, which calculates the risk across the entire time horizon and does not take the exit time and Shannon entropy into account. To accomplish this, an analysis was conducted on a portfolio that included a variety of stocks, cryptocurrencies, commodities, and precious metals. The empirical application demonstrated the enhancement of asset selection for both short-term and long-term investments through the combined use of Shannon entropy and the exit time.

Novel computational technique for the direct estimation of risk-neutral density using call price data quotes

Novel computational technique for the direct estimation of risk-neutral density using call price data quotes

Belief distortion near 52W high and low: Evidence from Indian equity options market

AbstractWe examine investors' behavioral biases and preferences in the options market near 52‐week high and low (52W‐H/L) using Indian options market data. We document that as the stock price approaches 52W high (low), the skewness of risk‐neutral density (RND), and out‐of‐the‐money (OTM) call volume decreases (increases), while OTM put volume increases (decreases). After crossing the 52W high (low), the skewness of RND and OTM call volume increases (decreases), while OTM put volume decreases (increases). The effects are economically large and significant. Our findings provide evidence consistent with the anchoring theory of belief distortion near 52W‐H/L. There is no evidence of preference distortion, contrary to what prospect theory predicts.

On the Class of Risk Neutral Densities under Heston’s Stochastic Volatility Model for Option Valuation

The celebrated Heston’s stochastic volatility (SV) model for the valuation of European options provides closed form solutions that are given in terms of characteristic functions. However, the numerical calibration of this five-parameter model, which is based on market option data, often remains a daunting task. In this paper, we provide a theoretical solution to the long-standing ‘open problem’ of characterizing the class of risk neutral distributions (RNDs), if any, that satisfy Heston’s SV for option valuation. 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 option prices under Heston’s stochastic volatility model. In fact, we also show that any RND with mean being the forward spot price that satisfies Heston’s option valuation solution must also be a member of the scale family of distributions in that mean. As particular examples, we show that under a certain re-parametrization, the one-parameter versions of the log-normal (i.e., Black–Scholes), gamma, and Weibull distributions, along with their respective inverses, are all members of this class and thus, provide explicit RNDs for direct option pricing under Heston’s SV model. We demonstrate the applicability and suitability of these explicit RNDs via exact calculations and Monte Carlo simulations, using already published index data and a calibrated Heston’s model (S&P500, ODAX), as well as an illustration based on recent option market data (AMD).

A Probabilistic Approach for Denoising Option Prices

This paper aims to directly denoise option price while adhering to the no–arbitrageconditions. To achieve our goal, we propose the Gaussian Process (GP) method thatentails training the GP on noisy data of option prices as a linear function of the pairof maturity and strike. Utilizing the GP approach not only allows for removing noiseson the option price surface by verifying the no–arbitrage conditions but also is a proba-bilistic approach that allows quantifying the uncertainty on the quantity of interest byconstructing confidence bands around the estimate. The GP further permits forecastingout-of-the-sample prices without needing to compute the risk-neutral density of the op-tion price surface. To investigate the efficiency of GP in removing the noise from optionprices, we tested it on a simulated dataset. The overall MSE between the computedBlack–Scholes prices and the GP denoised is 0.10, and between the Black–Scholes pricesand the noisy prices is 2.21 - a 95.33% noise removal. The curves of the graphs for thedenoised prices are all convex and non–increasing in strikes, upholding the no–arbitrageconditions. To our best knowledge, the challenge of directly denoising option prices hasled to little interest in this area, and our work is the first to undertake this task

Intra-Day Risk-Neutral Densities and Macroeconomic Announcements

Intra-Day Risk-Neutral Densities and Macroeconomic Announcements

Optionality as a binary operation

In finance, optionality is a possible property of a financial contract giving the owner a choice between two or more assets. For example, a convertible bond has optionality because its owner must choose between having a bond or having some shares of stock. In mathematics, a binary operation acts on two elements in a set to produce a third element in that set. When a financial contract such as a convertible bond enjoys optionality between exactly two assets, then the arbitrage-free current value of the contract can potentially be treated as the outcome of a binary operation acting on the two current asset values. In this paper, we treat one of the two assets as riskless and demand that the binary operation linking the two current asset values always produces an arbitrage-free option price. In this context, we focus on the interplay between the properties of the risk-neutral density of the risky asset and the algebraic properties of the binary operation.

Parallel computing in finance for estimating risk-neutral densities through option prices

Option pricing is one of the most active Financial Economics research fields. Black-Scholes-Merton option pricing theory states that risk-neutral density is lognormal. However, markets' pieces of evidence do not support that assumption. More realistic assumptions impose substantial computational burdens to calculate option pricing functions. Risk-neutral density is a pivotal element to price derivative assets, which can be estimated through nonparametric kernel methods. A significant computational challenge exists for determining optimal kernel bandwidths, addressed in this study through a parallel computing algorithm performed using Graphical Processing Units. The paper proposes a tailor-made Cross-Validation criterion function used to define optimal bandwidths. The selection of optimal bandwidths is crucial for nonparametric estimation and is also the most computationally intensive. We tested the developed algorithms through two data sets related to intraday data for VIX and S&P500 indexes.

VOLATILITY SMILE INTERPOLATION WITH RADIAL BASIS FUNCTIONS

The Radial Basis Functions (RBF) interpolation is a popular approximation technique used to smooth scattered data in various dimensions. This study uses RBF interpolation to interpolate the volatility skew of the S&P500 index options. The interpolated skews are used to construct the risk-neutral densities of the index and its local volatility surface. The RBF interpolation is contrasted throughout the study with the cubic spline interpolation. An analysis of the densities and the local volatility shows that RBF are an effective and practical tool for interpolating the implied volatility surface.

Estimating time-varying risk aversion from option prices and realized returns

Risk aversion is estimated from risk-neutral densities and realized index returns

On the utility maximization of the discrepancy between a perceived and market implied risk neutral distribution

A method is developed to determine the portfolio that maximizes the expected utility of an agent that trades the difference between a perceived future price distribution of an asset and the associated market implied risk neutral density. Exact results to construct and price such a portfolio are presented under the assumption that the underlying asset price evolves according to a geometric Brownian motion. Integer programming optimization techniques are applied to the general case where one first calibrates the asset price risk neutral density directly from option market data using Gatheral’s SVI parameterization. Several numerical examples approximating the optimal payoff function with liquid securities are given.

A new representation of the risk-neutral distribution and its applications

This paper establishes a novel model-free representation of the risk-neutral density in terms of market-observed options prices by combining exact series representations of the Dirac Delta function and the Carr-Madan asset spanning formula. Compared to the widely used method for obtaining the risk-neutral densities via the Breeden–Litzenberger device, our method yields estimates of risk-neutral densities that are model-free, automatically smooth, and in closed-form. The closed-form feature of our new representation makes it ideal for many potential applications including a new model-free representation of the local volatility function in the Dupire's local volatility model. The validity of our method is demonstrated through simulation studies as well as an empirical application using S&P 500 index option data. Extension of the method to higher dimensions is also obtained by extending the spanning formula.

Risk Neutral Density Estimation of Option Price from the Perspective of Empirical Analysis

Risk Neutral Density Estimation of Option Price from the Perspective of Empirical Analysis

Implied price processes anchored in statistical realizations

<p style='text-indent:20px;'>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.</p>

Option Pricing under Randomised GBM Models

By employing a randomisation procedure on the variance parameter of the standard geometric Brownian motion (GBM) model, we construct new families of analytically tractable asset pricing models. In particular, we develop two explicit families of processes that are respectively referred to as the randomised gamma (G) and randomised inverse gamma (IG) models, both characterised by a shape and scale parameter. Both models admit relatively simple closed-form analytical expressions for the transition density and the no-arbitrage prices of standard European-style options whose Black-Scholes implied volatilities exhibit symmetric smiles in the log-forward moneyness. Surprisingly, for integer-valued shape parameter and arbitrary positive real scale parameter, the analytical option pricing formulas involve only elementary functions and are even more straightforward than the standard (constant volatility) Black-Scholes (GBM) pricing formulas. Moreover, we show some interesting characteristics of the risk-neutral transition densities of the randomised G and IG models, both exhibiting fat tails. In fact, the randomised IG density only has finite moments of the order less than or equal to one. In contrast, the randomised G density has a finite first moment with finite higher moments depending on the time-to-maturity and its scale parameter. We show how the randomised G and IG models are efficiently and accurately calibrated to market equity option data, having pronounced implied volatility smiles across several strikes and maturities. We also calibrate the same option data to the wellknown SABR (Stochastic Alpha Beta Rho) model.