Risk-neutral Probability 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

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.

The discontinuation of the EUR/CHF minimum exchange rate: information from option-implied break probabilities

Abstract We derive risk-neutral probability densities for future euro/Swiss franc exchange rates as implied by option prices. We find that the credibility of the Swiss franc floor decreased somewhat as the spot exchange rate approached the lower bound of 1.20 CHF per euro. We also compare the forecasting performance of a random walk benchmark model with an error-correction model (ECM) augmented with option-implied break probabilities of breaching the currency floor. We find some evidence that the augmented ECM has an informational advantage over the random walk when using one-month break probabilities. But we find that one-month option-implied densities cannot predict the entire range of exchange rate realizations.

Analytically Deriving Risk-Neutral Densities from Volatility Smiles in Delta

This article develops a method for analytically deriving RNDs (risk-neutral densities) of future asset prices from volatility smiles. It extends an existing analytical method, which is for volatility smiles with respect to the strike price, to cover smiles with respect to option delta. A worked-out example on currency options shows that the analytically derived RNDs are free of approximation errors that would arise with numerical methods. The proposed method should be useful to practitioners. <b>TOPICS:</b>Options, futures and forward contracts <b>Key Findings</b> • This article develops an analytical method for deriving RNDs (risk-neutral probability densities) of future asset prices from volatility smiles with respect to option delta. • A worked-out example shows that exact RNDs computed by the method are free of approximation errors associated with numerical derivatives. • The proposed method should be useful to practitioners.

Calibration of local volatility model with stochastic interest rates by efficient numerical PDE methods

Long-maturity options or a wide class of hybrid products are evaluated using a local volatility-type modelling for the asset price S(t) with a stochastic interest rate r(t). The calibration of the local volatility function is challenging and time-consuming because of the multi-dimensional nature of the problem. A key requirement of any equity hybrid derivatives pricing model is the ability to rapidly and accurately calibrate to vanilla option prices. In this paper, we develop a calibration technique based on a partial differential equation (PDE) approach which allows an accurate calibration and provides an efficient implementation algorithm. The essential idea is based on solving the derived forward equation satisfied by $$P(t, S, r) \mathcal {Z}(t, S, r)$$, where P(t, S, r) represents the risk-neutral probability density of (S(t), r(t)) and $$\mathcal {Z}(t, S, r)$$ the projection of the stochastic discounting factor in the state variables (S(t), r(t)). The solution provides effective and sufficient information for the calibration and pricing. The PDE solver is constructed by using ADI (alternative direction implicit) method based on an extension of the Peaceman–Rachford scheme. Furthermore, an efficient algorithm to compute all the corrective terms in the local volatility function due to the stochastic interest rates is proposed by using the PDE solutions and grid points. It reduces by one order the computations costs and then allows to speed up significantly the calibration procedure. Different numerical experiments are examined and compared to demonstrate the results of our theoretical analysis.

Calibration of the risk-neutral density function by maximization of a two-parameter entropy

In the present work, a two-parameter entropy is maximized to calibrate the risk-neutral probability density function of the future asset price using options data subject to the expectation and the variance constraint. In the variance constraint, the volatility is assumed to be mean-reverting and following a quadratic path. The desired power law distribution is verified for the density function obtained, and it contains both the entropy parameters giving an additional degree of freedom. The calibrated density function is used to price the European call options for different strikes. The results thus obtained are discussed for the one-parameter Renyi and Tsallis entropies.

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.

The Information Content in the Offshore Renminbi Foreign-Exchange Option Market: Analytics and Implied USD/CNH Densities

In line with the deepening of the derivative foreign-exchange market in Hong Kong, we recover risk-neutral probability densities for future US dollar/offshore renminbi exchange rates as implied by exchange rate option prices. The risk-neutral densities (RND) approach is shown to be useful in analyzing market sentiment and risk aversion in the renminbi market. We include a forecasting exercise that confirms market participants were able to forecast the shape of the actual densities correctly for short horizons, even if their exact location could not be determined.

Option-Implied Libor Rate Expectations across Currencies

In this paper, I study risk-neutral probability densities regarding future Libor rates denominated in British pounds, euros, and US dollars as implied by option prices. I apply Breeden and Litzenberger’s (1978) result regarding the relationship between option prices and implied probabilities for the underlying to estimate full probability density functions for future Libor rates. I use these estimates in case studies, detailing the evolution of probabalistic expectations for future Libor rates over the course of several important market events. Next, I compute distributional moments from density functions estimated for fixed horizons and test for Granger causality across the three Libor rate distributions considering their mean, standard deviation, skewness, and kurtosis. I further break these relationships down by various fixed horizon lengths, as well as the slope and curvature in the term structure of moments over different horizons. The results show a rich interconnectedness among these three Libor rates that extends well beyond levels of future mean expectations.

Recovery of asset information from options prices

ABSTRACTPast data on an asset is of limited value to market agents seeking information about an asset’s future behaviour. Options, on the other hand, are forward-looking instruments: their payoff is based on the distribution of the underlying asset’s future price at the time of maturity of the option. Current prices of traded options therefore contain information about the market’s view on future asset prices and allow us to find implied risk-neutral probability distributions for such assets. David Shimko developed a method which gives an analytic expression for implied risk-neutral probability density functions, and a procedure for generating lognormal tails for the distribution (Shimko, 1993).Researchers have noted that the distributions generated by his formula are not quite satisfactory in terms of area under the density function, skewness and kurtosis. In this paper, we prove that the ‘Shimko formula’ (Shimko, 1993, p. 36) contains errors, and we derive the correct formula. This new expression for the density function now gives an area below the curve equal to one, and improved values for the kurtosis and skewness of distributions. We also provide a new way of generating non-lognormal tails for the density function.Shimko, D. C. (1993). Bounds of probability. Risk (Concord, NH), 6(4), 33–37.

B-spline techniques for volatility modeling

This paper is devoted to the application of B-splines to volatility modeling, specifically the calibration of the leverage function in stochastic local volatility (SLV) models and the parameterization of an arbitrage-free implied volatility surface calibrated to sparse option data. We use an extension of classical B-splines obtained by including basis functions with infinite support. We first discuss the application of shape-constrained B-splines to the estimation of conditional expectations, not merely from a scatter plot but also from the given marginal distributions. One application is the Monte Carlo calibration of SLV models by Markov projection. We then present a new technique for the calibration of an implied volatility surface to sparse option data. We use a B-spline parameterization of the Radon–Nikodym derivative of the underlying’s risk-neutral probability density with respect to a roughly calibrated base model. We show that this method provides smooth arbitrage-free implied volatility surfaces. Finally, we sketch a Galerkin method with B-spline finite elements to the solution of the partial differential equation satisfied by the Radon–Nikodym derivative.

A robust method to retrieve option implied risk neutral densities for defaultable assets

Risk neutral densities recovered from option prices can be used to infer market participants' expectations of future stock returns and are a vital tool for pricing illiquid exotic options. Although there is a broad literature on the subject, most studies do not address the likelihood of default. To fill this gap, in this paper we develop a novel method to retrieve the risk neutral probability density function from call options written on a defaultable asset. The primary advantage of the method is that default probabilities inferred by the model can be analytically expressed and, if available, can be incorporated as an input in a flexible, robust and easily implementable manner.

A Generalized Bachelier Formula for Pricing Basket and Spread Options

In this paper we propose a closed-form pricing formula for European basket and spread options. Our approach is based on approximating the risk-neutral probability density function of the terminal value of the basket using a Gauss-Hermite series expansion around the Gaussian density. The new method is quite general as it can be applied for a basket with a large number of assets and for all dynamics where the joint characteristic function of log-returns is known in closed form. We provide a simulation study to show the accuracy and the speed of our methodology.

Determining and benchmarking risk neutral distributions implied from option prices

Risk neutral probability density functions (RNDs) play a central role in assessing models for stock market behavior. However, it remains challenging to distill a realistic estimate for the RND from empirical data. In this work we introduce a novel method to infer a RND estimate from observed option prices. Our method efficiently yields a realistic rational function approximation to the RND, it is flexible w.r.t. the shape of the underlying distribution and robust in the presence of noise. To show this, we first investigate how well a method can actually retrieve a known distribution from noisy option prices. Then we consider real market data and show how our method can be used to derive a single continuously differentiable RND estimate from empirical call and put option price data.

Robust Estimation of Shape-Constrained State Price Density Surfaces

Given a theoretical pricing model, an implied volatility can be extracted from an option’s market price. Given a set of options with the same maturity and a range of strike prices, it is possible to extract (an approximation to) the entire risk-neutral probability density without having to assume a theoretical pricing model. There are a variety of related methods to do this, but all are subject to certain problems, including the fact that the data never exist to allow full estimation of the tails. Some methods produce improper densities with negative portions. In this article, Ludwig introduces a neural network approach to extract risk-neutral densities from option prices, imposing only a small number of constraints, such as probabilities must be nonnegative and an option’s price must be above intrinsic value. The resulting densities are smooth and sensible, even for days that other approaches find extremely difficult to handle.

Implied Binomial Trees with Cubic Spline Smoothing

The binomial model is a workhorse of numerical option pricing. But the plain vanilla model produces a risk-neutral probability density for the stock price at expiration that becomes lognormal in the limit. This is consistent with Black–Scholes’ assumptions but not with actual option prices in the market. An alternative is an implied tree, which begins with the risk-neutral density extracted from the market prices for options with the desired maturity. The procedure constructs a lattice that is consistent with the desired distribution. An N-step tree has N + 1 terminal nodes, so fitting the tree to the market prices can be quite time consuming. The author proposes a simplification that greatly increases efficiency with negligible cost in accuracy. The trick is to calibrate only a subset of the terminal nodes to prices in the options market and fill in the other nodes by cubic spline interpolation. <b>TOPICS:</b>Options, quantitative methods

A Robust Method to Retrieve Option Implied Risk Neutral Densities for Defaultable Assets

Risk neutral densities recovered from option prices can be used to infer market participants’ expectations of future stock returns and are a vital tool for pricing illiquid exotic options. Although there is a broad literature on the subject, most studies do not address the likelihood of default. To fill this gap, in this paper we develop a novel method to retrieve the risk neutral probability density function from call options written on a defaultable asset. The primary advantage of the method is that default probabilities inferred by the model can be analytically expressed and, if available, can be incorporated as an input in a ‡flexible, robust and easily implementable manner.

Maximum Entropy Estimates for Risk-Neutral Probability Measures with Non-Strictly-Convex Data

This article investigates use of the Principle of Maximum Entropy for approximation of the risk-neutral probability density on the price of a financial asset as inferred from market prices on associated options. The usual strict convexity assumption on the market-price to strike-price function is relaxed, provided one is willing to accept a partially supported risk-neutral density. This provides a natural and useful extension of the standard theory. We present a rigorous analysis of the related optimization problem via convex duality and constraint qualification on both bounded and unbounded price domains. The relevance of this work for applications is in explaining precisely the consequences of any gap between convexity and strict convexity in the price function. The computational feasibility of the method and analytic consequences arising from non-strictly-convex price functions are illustrated with a numerical example.