Recovering Probability Distributions from Option Prices

This article derives underlying asset risk-neutral probability distributions of European options on the S&P 500 index. Nonparametric methods are used to choose probabilities that minimize an objective function subject to requiring that the probabilities are consistent with observed option and underlying asset prices. Alternative optimization specifications produce approximately the same implied distributions. A new and fast optimization technique for estimating probability distributions based on maximizing the smoothness of the resulting distribution is proposed. Since the crash, the risk-neutral probability of a three (four) standard deviation decline in the index (about ?36 percent (?46 percent) over a year) is about 10 (100) times more likely than under the assumption of lognormality.     

Shedding light on a dark matter: Jump diffusion and option-implied investor preferences

We compare equilibrium jump diffusion option prices with endogenously determined stochastic dominance (SD) option bounds. We use model parameters from earlier studies and find that most equilibrium model prices consistent with SD bounds yield economically meaningless results. Further, the implied distributions of the SD bounds exhibit a tail risk comparable to that of the underlying return data, thus shedding light on the dark matter of the inconsistency of physical and risk-neutral tail probabilities. Since the SD bound assumptions are weaker, we conclude that these bounds should either replace or be used to verify the equilibrium model results.     

Option Bounds and the Pricing of the Volatility Smile

This paper presents a new approach forthe estimation of the risk-neutral probability distribution impliedby observed option prices in the presence of a non-horizontalvolatility smile. This approach is based on theoretical considerationsderived from option pricing in incomplete markets. Instead ofa single distribution, a pair of risk-neutral distributions areestimated, that bracket the option prices defined by the volatilitybid/ask midpoint. These distributions define upper and lowerbounds on option prices that are consistent with the observableoption parameters and are the tightest ones possible, in thesense of minimizing the distance between the option upper andlower bounds. The application of the new approach to a sampleof observations on the S&P 500 option market showsthat the bounds produces are quite tight, and also that theirderivation is robust to the presence of violations of arbitragerelations in option quotes, which cause many other methods tofail.     

Cross-sectional tests of deterministic volatility functions

We study the cross-sectional performance of option pricing models in which the volatility of the underlying stock is a deterministic function of the stock price and time. For each date in our sample of FTSE 100 index option prices, we fit an implied binomial tree to the panel of all European style options with different strike prices and maturities and then examine how well this model prices a corresponding panel of American style options. We find that the implied binomial tree model performs no better than an ad-hoc procedure of smoothing Black–Scholes implied volatilities across strike prices and maturities. Our cross-sectional results complement the time-series findings of Dumas et al. [J. Finance 53 (1998) 2059].     

Option Bounds with Finite Revision Opportunities

This article generalizes the single-period linear-programming bounds on option prices by allowing for a finite number of revision opportunities. It is shown that, in an incomplete market, the bounds on option prices can be derived using a modified binomial option-pricing model. Tighter bounds are developed under more restrictive assumptions on probabilities and risk aversion. For this case the upper bounds are shown to coincide with the upper bounds derived by Perrakis, while the lower bounds are shown to be tighter.     

A general closed form option pricing formula

A new method to retrieve the risk-neutral probability measure from observed option prices is developed and a closed form pricing formula for European options is obtained by employing a modified Gram–Charlier series expansion, known as the Gauss–Hermite expansion. This expansion converges for fat-tailed distributions commonly encountered in the study of financial returns. The expansion coefficients can be calibrated from observed option prices and can also be computed, for example, in models with the probability density function or the characteristic function known in closed form. We investigate the properties of the new option pricing model by calibrating it to both real-world and simulated option prices and find that the resulting implied volatility curves provide an accurate approximation for a wide range of strike prices. Based on an extensive empirical study, we conclude that the new approximation method outperforms other methods both in-sample and out-of-sample.

     

Binomial Option Pricing Biases and Inconsistent Implied Volatilities

We evaluate the binomial option pricing methodology (OPM) by examining simulated portfolio strategies. A key aspect of our study involves sampling from the empirical distribution of observed equity returns. Using a Monte Carlo simulation, we generate equity prices under known volatility and return parameters. We price American–style put options on the equity and evaluate the risk–adjusted performance of various strategies that require writing put options with different maturities and moneyness characteristics. The performance of these strategies is compared to an alternative strategy of investing in the underlying equity. The relative performance of the strategies allows us to identify biases in the binomial OPM leading to the well–known volatility smile . By adjusting option prices so as to rule out dominated option strategies in a mean–variance context, we are able to reduce the pricing errors of the OPM with respect to option prices obtained from the LIFFE. Our results suggest that a simple recalibration of inputs may improve binomial OPM performance.     

Option Prices and the Underlying Asset's Return Distribution

This work examines the relation between option prices and the true, as opposed to risk-neutral, distribution of the underlying asset. If the underlying asset follows a diffusion with an instantaneous expected return at least as large as the instantaneous risk-free rate, observed option prices can be used to place bounds on the moments of the true distribution. An illustration of the paper's results is provided by the analysis of the information concerning the mean and standard deviation of market returns contained in the prices of S&P 100 Index Options.     

Microstructural biases in empirical tests of option pricing models

This paper examines how noise in observed option prices arising from discrete prices and other microstructural frictions affects empirical tests of option pricing models and risk-neutral density estimation. The discrete tick size alone introduces enough noise to make model comparisons difficult, especially for lower-priced stocks. We demonstrate that microstructural noise can lead to incorrect inferences in the univariate diffusion test of Bakshi et al. (Rev Financ Stud 13:549–584, 2000), the transition density diffusion test of Aït-Sahalia (J Financ 57:2075–2112, 2002), and the speed-of-convergence test of Carr and Wu (J Financ 58:2581–2610, 2003). We also show that microstructural noise induces a bias into the implied risk-neutral moment estimators of Bakshi et al. (Rev Financ Stud 16:101–143, 2003). Even in active, liquid option markets, observation error is likely to reduce significantly the power of tests, and in some cases represents an important source of bias.     

Closed-form transformations from risk-neutral to real-world distributions

Risk-neutral and real-world densities are derived from option prices and risk assumptions, and are compared with historical densities obtained from time series. Two parametric risk-transformations are used to convert risk-neutral densities into real-world densities. Both transformations are estimated by maximizing the likelihood of observed index levels, for two parametric density families. Results for the FTSE-100 index show that parametric densities derived from option prices have more explanatory power than historical densities and higher likelihoods than densities estimated by spline methods. A combination of parametric real-world and historical densities provides the preferred predictive densities.     

An adjusted binomial model for pricing Asian options

We propose a model for pricing both European and American Asian options based on the arithmetic average of the underlying asset prices. Our approach relies on a binomial tree describing the underlying asset evolution. At each node of the tree we associate a set of representative averages chosen among all the effective averages realized at that node. Then, we use backward recursion and linear interpolation to compute the option price.     

Option Pricing with Heterogeneous Expectations

This paper re-derives the finite mixture option pricing model of Ritchey (1990), based on the assumption that the option investors hold heterogeneous expectations about the parameters of the lognormal process of the underlying asset price. By proving that the model admits no riskless arbitrage, this paper justifies that the entire family of finite mixture of lognormal distributions is a desirable candidate set for recovering the risk-neutral probability distributions from contemporaneous options quotes. The parametric method derived from the model is significantly simpler than the nonparametric method of Rubinstein (1994) for recovering the risk-neutral probability distributions from contemporaneous option prices.     

Effects of market default risk on index option risk-neutral moments

We investigate the relative importance of market default risk in explaining the time variation of the S&P 500 Index option-implied risk-neutral moments. The results demonstrate that market default risk is positively (negatively) related to the index risk-neutral volatility and skewness (kurtosis). These relations are robust in the presence of other factors relevant to the dynamics and microstructure nature of the spot and option markets. Overall, this study sheds light on a set of economic determinants which help to understand the daily evolution of the S&P 500 Index option-implied risk-neutral distributions. Our findings offer explanations of why theoretical predictions of option pricing models are not consistent with what is observed in practice and provide support that market default risk is important to asset pricing.     

Arbitrage-free credit pricing using default probabilities and risk sensitivities

The relation between physical probabilities (rating) and risk-neutral probabilities (pricing) is derived in a large market with a quasi-factor structure. Factor sensitivities and default probabilities are obtainable for all kinds of credits on historical rating data. Since factor prices can be backed out from market data, the model allows the pricing of non-marketable credits and structured products thereof. The model explains various empirical observations: credit spreads of equally rated borrowers differ, spreads are wider than implied by expected losses, and expected returns on CDOs must be greater than their rating matched, single-obligor securities due to the inherent systematic risk.     

American option valuation: new bounds, approximations, and a comparison of existing methods

We develop lower and upper bounds on the prices of Americancall and put options written on a dividend-paying asset. Weprovide two option price approximations one based on the lowerbound (termed LBA) and one based on both bounds (termed LUBA).The LUBA approximation has an average accuracy comparable toa l,000-step binomial tree. We introduce a modification of thebinomial method (termed BBSR) that is very simple to implementand performs remarkably well. We also conduct a careful large-scaleevaluation of many recent methods for computing American optionprices.     

Shorting in Speculative Markets

In models of trading with heterogeneous beliefs following Harrison-Kreps, short selling is prohibited and agents face constant marginal costs-of-carry. The resale option guarantees that prices exceed buy-and-hold prices and the difference is identified as a bubble. We propose a model where risk-neutral agents face asymmetric increasing marginal costs on long and short positions. Here, agents also value an option to delay, and a Hamilton-Jacobi-Bellman equation quantifies the influence of costs on prices. An unexpected decrease in shorting costs may deflate a bubble, linking financial innovations that facilitated shorting of mortgage-backed securities to the collapse of prices.     

A multi-horizon comparison of density forecasts for the S&P 500 using index returns and option prices

We compare density forecasts of the S&P 500 index from 1991 to 2004, obtained from option prices and daily and 5-min index returns. Risk-neutral densities are given by using option prices to estimate diffusion and jump-diffusion processes which incorporate stochastic volatility. Three transformations are then used to obtain real-world densities. These densities are compared with historical densities defined by ARCH models. For horizons of two and four weeks the best forecasts are obtained from risk-transformations of the risk-neutral densities, while the historical forecasts are superior for the one-day horizon; our ranking criterion is the out-of-sample likelihood of observed index levels. Mixtures of the real-world and historical densities have higher likelihoods than both components for short forecast horizons.     

Evaluating implied RNDs by some new confidence interval estimation techniques

This paper evaluates the precision of the parametric double lognormal and the non-parametric smoothing spline method for estimating risk-neutral distributions (RNDs) from observed option prices. By using a bootstrap technique, confidence bands are estimated for the risk-neutral distributions and the width of the confidence bands is used as a criterion when evaluating the precision of the two methods. Previous literature on estimating confidence bands has to a large extent been estimated using Monte Carlo methods. This paper argues that the bootstrap technique is to be preferred due to the non-normality feature of the error structure. Furthermore, it is shown that the inclusion of a heteroscedastic error structure improves the precision of the estimated RNDs. Our findings favour the smoothing spline method as it produces tighter confidence bands. In addition, an example of how to apply the estimated confidence bands in practice is also provided.     

VALUING REAL OPTIONS: CAN RISK‐ADJUSTED DISCOUNTING BE MADE TO WORK?

This paper examines three alternative approaches to valuing real options: (1) the standard option pricing technique using "risk-neutral" probabilities; (2) the use of risk-adjusted discount rates; and (3) discounting certainty-equivalent values with a riskless discount rate. As suggested by the title, a question of particular interest is whether an approach based on risk-adjusted discount rates can be "made to work" for valuing options. The answer is yes. Indeed, the authors show that any of the three approaches will provide a correct valuation if properly employed.
Nevertheless, there are important differences in the information requirements associated with each of the three methods. Another important issue is the relative degree of difficulty in calculating the correct option value. When these two considerations are taken into account, the risk-neutral option pricing procedure generally proves to be the preferred method. It tends to be computationally more convenient—often much more convenient—and to require less information than either the risk-adjusted discounting or certainty-equivalent procedures.     

A Simple Nonparametric Approach to Derivative Security Valuation

Canonical valuation uses historical time series to predict the probability distribution of the discounted value of primary assets' discounted prices plus accumulated dividends at any future date. Then the axiomatically-rationalized maximum entropy principle is used to estimate risk-neutral (equivalent martingale) probabilities that correctly price the primary assets, as well as any predesignated subset of derivative securities whose payoffs occur at this date. Valuation of other derivative securities proceeds by calculation of its discounted, risk-neutral expected value. Both simulation and empirical evidence suggest that canonical valuation has merit.