Risk-neutral Probability Density Functions 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.

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.

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.

Estimating Option-implied Risk Aversion for Indian Markets

What do nearly 1.5 lakh observations of options data say about risk preferences of Indian investors? This paper explores a nonparametric technique to compute probability density functions (PDFs) directly from NIFTY 50 option prices in India, based on the utility preferences of the representative investor. Use of probability density functions to estimate investor expectations of the distribution of future levels of the underlying assets has gained tremendous popularity over the last decade. Studying option prices provides information about the market participants’ probability assessment of the future outcome of the underlying asset. We compare the forecast ability of the risk-neutral PDF and risk-adjusted density functions to arrive at a unique index of relative risk aversion for Indian markets. Results indicate that risk-adjusted PDFs are reasonably better forecasts of investor expectations of future levels of the underlying assets. We find that Indian investors are not neutral to risk, contrary to the theoretical assumption of risk-neutrality among investors. The computed time-series of relative risk aversion overcomes the limitations of the VIX (implied volatility index) to yield a more reliable index, particularly useful for the Indian markets. Validity of the computed index is established by comparing with existing measures of risk and the relationships are found to be consistent with market 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.

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.

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.

Estimation of risk-neutral measures using quartic B-spline cumulative distribution functions with power tails

In this paper, we propose the B-spline (BSP) method, which overcomes problems with the smoothed implied volatility smile (SML) method for estimating option implied risk-neutral measures (RNMs). We model the risk-neutral cumulative distribution function (CDF) using quartic B-splines with power tails so that the resulting risk-neutral probability density function (PDF) has continuity and arbitrage-free properties. Since the number of knots is selected optimally in constructing the quartic B-spline risk-neutral CDF, our method avoids both overfitting and oversmoothing. To improve computational efficiency and accuracy, we introduce a three-step RNM estimation procedure that transforms a nonlinear optimization problem into a convex quadratic program. Monte-Carlo experiments and applications to S&P 500 index options suggest that the BSP method performs considerably better than the SML method. The BSP method always produces arbitrage-free RNM estimators and almost perfectly recovers the actual risk-neutral PDFs for various hypothetical distributions.

Inversion of option prices for implied risk-neutral probability density functions: general theory and its applications to the natural gas market

This paper applies inverse theory to the estimation of the implied risk-neutral probability density function (PDF) from option prices. A general framework of inverting option prices for the implied risk-neutral PDF is formulated from the option pricing formula of Cox and Ross [J. Financial Econ., 1976, 3, 145–166]. To overcome the non-uniqueness and instability inherent in the option inverse problem, the smoothness requirement for the shape of the PDF and a prior model are introduced by a penalty function. Positivity constraints are included as a hard bond on the PDF values. The option inverse problem then becomes a non-negative least-squares problem that can be solved by classic methods such as the non-negative least-squares program of Lawson and Hanson [Solving Least Squares Problems, 1974 (Prentice-Hall: Englewood Cliffs, NJ)]. The best solution is not the one that gives the best fit to the observed option prices or provides the smoothest PDF, but the one that gives the optimal trade-off between the goodness-of-fit and smoothness of the estimated risk-natural PDF. The proposed inversion technique is compared with the models of Black–Scholes (BS), a mixture of two lognormals (MLN), Jarrow and Rudd's Edgeworth expansion (JR), and jump diffusion (JD) for the estimation of the PDF from the option prices associated with the September 2007 NYMEX natural gas futures. It is found that the inversion technique not only gives the best goodness-of-fit, but also a significantly better model resolution. An empirical study for the last three months of the September 2007 futures contract shows that the shapes of the estimated PDFs become more symmetric as the futures contract becomes closer to the expiration date. The dispersion of the estimated PDFs decreases with decreasing time to expiry, indicating the resolution of uncertainty with passing time.

A New Framework to Estimate the Risk-Neutral Probability Density Functions Embedded in Options Prices

Building on the widely-used double-lognormal approach by Bahra (1997), this paper presents a multi-lognormal approach with restrictions to extract risk-neutral probability density functions (RNPs) for various asset classes. The contributions are twofold: first, on the technical side, the paper proposes useful transformation/restrictions to Bahra’s original formulation for achieving economically sensible outcomes. In addition, the paper compares the statistical properties of the estimated RNPs among major asset classes, including commodities, the S&P 500, the dollar/euro exchange rate, and the US 10-year Treasury Note. Finally, a Monte Carlo study suggests that the multi-lognormal approach outperforms the double-lognormal approach.

A New Framework to Estimate the Risk-Neutral Probability Density Functions Embedded in Options Prices

"Volume 2010 (2010): Issue 181 (Aug 2010): A New Framework to Estimate the Risk-Neutral Probability Density Functions Embedded in Options Prices" published on 01 Aug 2010 by International Monetary Fund.

Mathematical Properties of Option Pricing Under Gamma Distribution Process

A problem of risk neutral probability density function estimation for prices of risky assets is discussed when the asset pricing model uses exponential random process with independent increments. The structure of increments consists of two components: systematic drift and a random gamma distributed component. To determine the risk neutral probability density Escher transform is used. Obtained results help to define a formula for European call option pricing. Also we compare the found risk neutral density with the density, when the random component is defined by a Wiener process, driven by gamma process.

Revealing the implied risk-neutral MGF from options: The wavelet method

Options are believed to contain unique information on the risk-neutral moment generating function (MGF) or the risk-neutral probability density function (PDF) of the underlying asset. This paper applies the wavelet method to approximate the implied risk-neutral MGF from option prices. Monte Carlo simulations are carried out to show how the risk-neutral MGF can be obtained using the wavelet method. With the Black–Scholes model as the benchmark, we offer a novel method to reveal the implied MGF, and to price in-sample options and forecast out-of-sample option prices with the estimated MGF.

Recovering risk-neutral probability density functions from options prices using cubic splines and ensuring nonnegativity

We present a new approach to estimate the risk-neutral probability density function (pdf) of the future prices of an underlying asset from the prices of options written on the asset. The estimation is carried out in the space of cubic spline functions, yielding appropriate smoothness. The resulting optimization problem, used to invert the data and determine the corresponding density function, is a convex quadratic or semidefinite programming problem, depending on the formulation. Both of these problems can be efficiently solved by numerical optimization software. In the quadratic programming formulation the positivity of the risk-neutral pdf is heuristically handled by posing linear inequality constraints at the spline nodes. In the other approach, this property of the risk-neutral pdf is rigorously ensured by using a semidefinite programming characterization of nonnegativity for polynomial functions. We tested our approach using data simulated from Black–Scholes option prices and using market data for options on the S&P 500 Index. The numerical results we present show the effectiveness of our methodology for estimating the risk-neutral probability density function.

Inversion of Option Prices for Implied Risk Neutral Probability Density Functions: General Theory and its Applications to the Natural Gas Market

This paper applies classic linear inverse theory to the estimation of the implied risk neutral probability density function (PDF) from option prices. To overcome non-uniqueness and instability inherent in the option inverse problem, smoothness requirement for the shape of a PDF and an initial model are introduced by a penalty function. Positivity constraints are included as a hard bond on the PDF values. Then the option inverse problem becomes a non-negative least-squares problem which can be solved by the classic methods such as the non-negative least squares program of Lawson and Hanson (1974). The best solution is not the solution that gives best fit, but the solution that gives the optimal trade-off between the goodness of fit and smoothness of the estimated risk natural PDF. The proposed inversion technique is compared to the models of Black-Scholes (BS), a mixture of two lognormals (MLN), Jarrow and Rudd's Edgeworth expansion (JR), and jump diffusion (JD) for the estimation of the PDF from the option prices associated with the September 2007 NYMEX natural gas futures. It is found that the inversion technique not only gives best goodness of fit, but also the significantly better model resolution. BS, JD and MLN models basically cannot resolve the densities far away from the strikes where option prices are observed and can resolve long wavelength features of the densities inside the strikes where option prices are observed. On the other hand, the inversion model can resolve not only the significant details of the densities inside the strikes where option prices are observed, but also the long wavelength features of the densities away from the strikes where option prices are observed. The empirical study for the last three months of the September 2007 futures contract shows that the shapes of the estimated PDFs become more symmetric as the futures contract is closer to the expiration date. The dispersion of the estimated PDFs decreases with decreasing the time to expire, indicating the resolution of uncertainty with time.

Empirical risk aversion functions-estimates and assessment of their reliability

In this paper, we examine investor's risk preferences implied by option prices. In order to derive these preferences, we specify the functional form of a pricing kernel and then shift its parameters until realized returns are best explained by the subjective probability density function, which consists of the ratio of the risk-neutral probability density function and the pricing kernel. We examine, alternatively, pricing kernels of power, exponential, and higher order polynomial forms. Using S&P 500 index options, we find surprising evidence of risk neutrality, instead of risk aversion, in both the power and exponential cases. When extending the underlying assumption on the specification of the pricing kernel to one of higher order polynomial functions, we obtain functions exhibiting ‘monotonically decreasing’ relative risk aversion (DRRA) and anomalous ‘inverted U-shaped’ relative risk aversion. We find, however, that only the DRRA function is robust to variation in sample characteristics, and is statistically significant. Finally, we also find that most of our empirical results are consistent, even when taking into account market imperfections such as illiquidity.

Retrieving risk neutral densities based on risk neutral moments through a Gram–Charlier series expansion

In this paper we employ a new method to retrieve the risk neutral probability density function of future asset prices, or their implied log-returns, based on an exponential form of a Gram–Charlier series expansion, known as C-type. This type of expansion guarantees that the values of the risk neutral density will be always positive and it can account for strong deviations of the stock price distributions from the Gaussian. In a set of numerical and empirical applications, the paper shows the accuracy and versatility of the method for recovering the true risk neutral density.