A relationship exists between aggregate risk-neutral and subjective probability distributions and risk aversion functions. Using a variation of the method developed by Jackwerth and Rubinstein (1996), we estimate risk-neutral probabilities reliably from option prices. Subjective probabilities are estimated from realized returns. This paper then introduces a technique to empirically derive risk aversion functions implied by option prices and realized returns simultaneously. These risk aversion functions dramatically change shapes around the 1987 crash: Precrash, they are positive and decreasing in wealth and thus consistent with standard economic theory. Postcrash, they are partially negative and increasing and irreconcilable with the theory. Overpricing of out-of-the-money puts is the most likely cause. A simulated trading strategy exploiting this overpricing shows excess returns even after accounting for the possibility of further crashes and transaction costs. * Jens Carsten Jackwerth is a visiting assistant professor at the London Business School. For helpful discussions I

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In this survey we discuss models with level-dependent and stochastic volatility from the viewpoint of derivative asset analysis. Both classes of models are generalisations of the classical Black-Scholes model; they have been developed in an effort to build models that are flexible enough to cope with the known deficits of the classical BlackScholes model. We start by briefly recalling the standard theory for pricing and hedging derivatives in complete frictionless markets and the classical Black-Scholes model. After a review of the known empirical contradictions to the classical Black-Scholes model we consider models with level-dependent volatility. Most of this survey is devoted to derivative asset analysis in stochastic volatility models. We discuss several recent developments in the theory of derivative pricing under incompleteness in the context of stochastic volatility models and review analytical and numerical approaches to the actual computation of option values.

We present a framework for calibrating a pricing model to a prescribed set of option prices quoted...

Using market European option prices, a method for computing a smooth local volatility function in a 1-factor continuous diffusion model is proposed. Smoothness is introduced to facilitate accurate approximation of the true local volatility function from a finite set of observation data. It is emphasized that accurately approximating the true local volatility function is crucial in hedging even simple European options, and pricing exotic options. A spline functional approach is used: the local volatility function is represented by a spline whose values at chosen knots are determined by solving a constrained nonlinear optimization problem. The optimization formulation is amenable to various option evaluation methods; a partial differential equation implementation is discussed. Using a synthetic European call option example, we illustrate the capability of the proposed method in reconstructing the unknown local volatility function. Accuracy of pricing and hedging is also illustrated. Moreover, it is demonstrated that, using a different constant implied volatility for an option with different strike/maturity can produce erroneous hedge factors. In addition, real market European call option data on the S&P 500 stock index is used to compute the local volatility function; stability of the approach is demonstrated.

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In this paper a stochastic volatility model is presented that directly prescribes the stochastic development of the implied Black-Scholes volatilities of a set of given standard options. Thus the model is able to capture the stochastic movements of a full term structure of implied volatilities. The conditions are derived that have to be satisfied to ensure absence of arbitrage in the model and its numerical implementation is discussed.

A general approach for calibrating Monte Carlo models to the market prices of benchmark securities is presented. Starting from a given model for market dynamics (price diffusion, rate diffusion, etc.), the algorithm corrects for price-misspecifications and finite-sample effects in the simulation by assigning "probability weights" to the simulated paths. The choice of weights is done by minimizing the Kullback-Leibler relative entropy distance of the posterior measure to the empirical measure. The resulting ensemble prices the given set of benchmark instruments exactly or in the sense of least-squares. We discuss pricing and hedging in the context of these weighted Monte Carlo models. A significant reduction of variance is demonstrated theoretically as well as numerically. Concrete applications to the calibration of stochastic volatility models and term-structure models with up to forty benchmark instruments are presented. The construction of implied volatility surfaces and forward-rate...

The `volatility smile' is one of the well-known biases of Black-Scholes models for pricing options. In this paper, we introduce a robust method of reducing this bias by pricing subject to a deterministic functional volatility oe = oe(S; t). This instantaneous volatility is chosen as a spline whose weights are determined by a regularised numerical strategy that approximately minimises the difference between Black-Scholes vanilla prices and known market vanilla prices over a range of strikes and maturities; these Black-Scholes prices are calculated by solving the relevant partial differential equation numerically using finite element methods. The instantaneous volatility generated from vanilla options can be used to price exotic options where the skew and termstructure of volatility are important, and we illustrate the application to barrier options. Contents 1 Introduction 4 2 Space-time spline representation of oe(S; t) 7 3 Regularised optimisation strategy 9 3.1 The cost-functiona...

This paper derives and implements a nonparametric, arbitrage-free technique for multivariate contingent claims (MVCC) pricing. This technique is based on nonparametric estimation of a multivariate risk-neutral density function using data from traded options markets and historical asset returns. “New ” multivariate claims are priced using expectations under this measure. An appealing feature of nonparametric arbitrage-free derivative pricing is that fitted prices are obtained that are consistent with traded option prices and are not based on specific restrictions on the underlying asset price process or the functional form of the risk-neutral density. Nonparametric MVCC pricing utilizes the method of copulas to combine nonparametrically estimated marginal risk-neutral densities (based on options data) into a joint density using a separately estimated nonparametric dependence function (based on historical returns data). This paper provides theory linking objective and risk-neutral dependence functions, and empirically testable conditions that justify the use of historical data for estimation of the risk-neutral dependence function. The nonparametric MVCC pricing technique is implemented for the valuation of bivariate underperformance and outperformance options on the S&P500 and DAX index. Price deviations are