Implied Binomial Trees

This article develops a new method for inferring risk-neutral probabilities (or state-contingent prices) from the simultaneously observed prices of European options. These probabilities are then used to infer a unique fully specified recombining binomial tree that is consistent with these probabilities (and, hence, consistent with all the observed option prices). A simple backwards recursive procedure solves for the entire tree. From the standpoint of the standard binomial option pricing model, which implies a limiting risk-neutral lognormal distribution for the underlying asset, the approach here provides the natural (and probably the simplest) way to generalize to arbitrary ending risk-neutral probability distributions.     

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.     

A Generalization of the Brennan–Rubinstein Approach for the Pricing of Derivatives

This paper derives preference-free option pricing equations in a discrete time economy where asset returns have continuous distributions. There is a representative agent who has risk preferences with an exponential representation. Aggregate wealth and the underlying asset price have transformed normal distributions which may or may not belong to the same family of distributions. Those pricing results are particularly valuable (a) to show new sufficient conditions for existing risk-neutral option pricing equations (e.g., the Black–Scholes model), and (b) to obtain new analytical solutions for the price of European-style contingent claims when the underlying asset has a transformed normal distribution (e.g., a negatively skew lognormal distribution).     

Two-dimensional risk-neutral valuation relationships for the pricing of options

The Black–Scholes model is based on a one-parameter pricing kernel with constant elasticity. Theoretical and empirical results suggest declining elasticity and, hence, a pricing kernel with at least two parameters. We price European-style options on assets whose probability distributions have two unknown parameters. We assume a pricing kernel which also has two unknown parameters. When certain conditions are met, a two-dimensional risk-neutral valuation relationship exists for the pricing of these options: i.e. the relationship between the price of the option and the prices of the underlying asset and one other option on the asset is the same as it would be under risk neutrality. In this class of models, the price of the underlying asset and that of one other option take the place of the unknown parameters.      

CAPM and Option Pricing With Elliptically Contoured Distributions

This article offers an alternative proof of the capital asset pricing model (CAPM) when asset returns follow a multivariate elliptical distribution. Empirical studies continue to demonstrate the inappropriateness of the normality assumption for modeling asset returns. The class of elliptically contoured distributions, which includes the more familiar Normal distribution, provides flexibility in modeling the thickness of tails associated with the possibility that asset returns take extreme values with nonnegligible probabilities. As summarized in this article, this class preserves several properties of the Normal distribution. Within this framework, we prove a new version of Stein's lemma for this class of distributions and use this result to derive the CAPM when returns are elliptical. Furthermore, using the probability distortion function approach based on the dual utility theory of choice under uncertainty, we also derive an explicit form solution to call option prices when the underlying is log‐elliptically distributed. The Black–Scholes call option price is a special case of this general result when the underlying is log‐normally distributed.     

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.     

Local volatility function models under a benchmark approach

Without requiring the existence of an equivalent risk-neutral probability measure this paper studies a class of one-factor local volatility function models for stock indices under a benchmark approach. It is assumed that the dynamics for a large diversified index approximates that of the growth optimal portfolio. Fair prices for derivatives when expressed in units of the index are martingales under the real-world probability measure. Different to the classical approach that derives risk-neutral probabilities the paper obtains the transition density for the index with respect to the real-world probability measure. Furthermore, the Dupire formula for the underlying local volatility function is recovered without assuming the existence of an equivalent risk-neutral probability measure. A modification of the constant elasticity of variance model and a version of the minimal market model are discussed as specific examples together with a smoothed local volatility function model that fits a snapshot of S&P500 index options data.     

Gain,Loss, and Two-State Modeling

Gain and loss, calculated from the upside and downside portions of return distributions, play a pivotal role in the two-state model. A two-state economy possesses a universal gain-loss ratio (G/L) for all assets that is related to the ratio of state prices and to the familiar risk-neutral probabilities. This paper derives many asset pricing properties in a two-state context and shows the role of gain and loss. Applied to bonds, for example, risky debt yields depend directly on both G/L and a bond's potential loss. Using S&P 500 data over a 72-year period, the market has priced an Arrow-Debreu security in the gain state at approximately $0.36, while the Arrow-Debreu security in the loss state has been priced at $0.61. Historically, the S&P 500's expected gain is about three times its expected loss.     

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.     

Detection of arbitrage in a market with multi-asset derivatives and known risk-neutral marginals

In this paper we study the existence of arbitrage opportunities in a multi-asset market when risk-neutral marginal distributions of asset prices are known. We first propose an intuitive characterization of the absence of arbitrage opportunities in terms of copula functions. We then address the problem of detecting the presence of arbitrage by formalizing its resolution in two distinct ways that are both suitable for the use of optimization algorithms. The first method is valid in the general multivariate case and is based on Bernstein copulas that are dense in the set of all copula functions. The second one is easier to work with but is only valid in the bivariate case. It relies on results about improved Fréchet–Hoeffding bounds in presence of additional information. For both methods, details of implementation steps and empirical applications are provided.     

Pricing options on scenario trees

We examine valuation procedures that can be applied to incorporate options in scenario-based portfolio optimization models. Stochastic programming models use discrete scenarios to represent the stochastic evolution of asset prices. At issue is the adoption of suitable procedures to price options on the basis of the postulated discrete distributions of asset prices so as to ensure internally consistent portfolio optimization models. We adapt and implement two methods to price European options in accordance with discrete distributions represented by scenario trees and assess their performance with numerical tests. We consider features of option prices that are observed in practice. We find that asymmetries and/or leptokurtic features in the distribution of the underlying materially affect option prices; we quantify the impact of higher moments (skewness and excess kurtosis) on option prices. We demonstrate through empirical tests using market prices of the S&P500 stock index and options on the index that the proposed procedures consistently approximate the observed prices of options under different market regimes, especially for deep out-of-the-money options.     

Option Pricing with Threshold Diffusion Processes

The threshold diffusion (TD) model assumes a piecewise linear drift term and piecewise smooth diffusion term, which can capture many nonlinear features and volatility clustering often observed in financial time series data. We solve the problem of option pricing with a TD asset pricing process by deriving the minimum entropy martingale measure, which is the risk-neutral measure closest to the underlying TD probability measure in terms of Kullback-Leibler divergence, given the historical regime-switching pattern. The proposed valuation model is illustrated with a numerical example.     

Risk Preferences Heterogeneity: Evidence from Asset Markets

Using asset market data, as well as theoretical relations between investors' preferences,option-implied, risk-neutral, probability distribution functions (PDFs,) and index-implied,actual, PDFs, this paper extracts a time-series of investors' relative risk aversion (RRA)functions. Based on results recently derived by Benninga and Mayshar (2000), thesefunctions are used to recover the evolution of risk preferences heterogeneity. Applyingnon-parametric estimation on European call options written on the S & P500 index, wefind that: (i) the RRA functions are decreasing; and (ii) the constructed risk preferencesheterogeneity series is positively correlated in a static, as well as a dynamic, setup witha prevalent proxy for investors heterogeneity, namely, the spread between auction- andmarket-yields of Treasury bills.     

Recovering the probability density function of asset prices using garch as diffusion approximations

This paper uses Garch models to estimate the objective and risk-neutral density functions of financial asset prices and by comparing their shapes, recover detailed information on economic agents' attitudes toward risk. It differs from recent papers investigating analogous issues because it uses Nelson's result that Garch schemes are approximations of the kind of differential equations typically employed in finance to describe the evolution of asset prices. This feature of Garch schemes usually has been overshadowed by their well-known role as simple econometric tools providing reliable estimates of unobserved conditional variances. We show instead that the diffusion approximation property of Garch gives good results and can be extended to situations with (i) non-standard distributions for the innovations of a conditional mean equation of asset price changes and (ii) volatility concepts different from the variance. The objective PDF of the asset price is recovered from the estimation of a nonlinear Garch fitted to the historical path of the asset price. The risk-neutral PDF is extracted from cross-sections of bond option prices, after introducing a volatility risk premium function. The direct comparison of the shapes of the two PDF_s reveals the price attached by economic agents to the different states of nature. Applications are carried out with regard to the futures written on the Italian 10-year bond.     

Disagreement,tastes, and asset prices

Standard asset pricing models assume that: (i) there is complete agreement among investors about probability distributions of future payoffs on assets; and (ii) investors choose asset holdings based solely on anticipated payoffs; that is, investment assets are not also consumption goods. Both assumptions are unrealistic. We provide a simple framework for studying how disagreement and tastes for assets as consumption goods can affect asset prices.     

Interpreting Implied Risk-Neutral Densities: The Role of Risk Premia

This paper examines differences between risk-neutral and objective probability densities of future interest rates. The identification and quantification of these differences are important when risk-neutral densities (RNDs), such as option-implied RNDs, are used as indicators of actual beliefs of investors. We employ a multi-factor essentially affine modeling framework applied to German time-series and cross-section term structure data in order to identify both the risk-neutral and the objective term structure dynamics. We find important differences between risk-neutral and objective distributions due to risk premia in bond prices. Moreover, the estimated premia vary over time in a quantitatively significant way, which implies that the differences between the objective and the risk-neutral distributions also vary over time. We therefore conclude that one should be cautious in interpreting RNDs in terms of expectations. The method used in this paper provides an alternative approach to identifying objective probabilities of future interest rates.     

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.     

Alternative Defaultable Term Structure Models

The objective of this paper is to consider defaultable term structure models in a general setting beyond standard risk-neutral models. Using as numeraire the growth optimal portfolio, defaultable interest rate derivatives are priced under the real-world probability measure. Therefore, the existence of an equivalent risk-neutral probability measure is not required. In particular, the real-world dynamics of the instantaneous defaultable forward rates under a jump-diffusion extension of a HJM type framework are derived. Thus, by establishing a modelling framework fully under the real-world probability measure, the challenge of reconciling real-world and risk-neutral probabilities of default is deliberately avoided, which provides significant extra modelling freedom. In addition, for certain volatility specifications, finite dimensional Markovian defaultable term structure models are derived. The paper also demonstrates an alternative defaultable term structure model. It provides tractable expressions for the prices of defaultable derivatives under the assumption of independence between the discounted growth optimal portfolio and the default-adjusted short rate. These expressions are then used in a more general model as control variates for Monte Carlo simulations of credit derivatives. Nicola Bruti-Liberati: In memory of our beloved friend and colleague.     

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.