July 17, 2024

Financial Mathematics Seminars - January 30, 2002


Audio of Lectures

Dynamic Mixture Models of Option Pricing
Eric Renault, Université de Montréal
The valuation of security prices and sensitivities lends to the evaluation of expectations with respect to the probability distribution of a number of state variables. Like for conditional Monte-Carlo, conditioning on some state variables is a fruitful technique if it allows to do part of the integration analytically. Option prices and sensitivities then appear as mixtures of popular closed-from formulas. This paper makes this approach systematic in the context of arbitrage pricing. It characterizes the specific distributional features of the joint stochastic process of state variables and assets dividends which make possible risk-neutral valuation as typically attributed to complete markets. In this case, option pricing formulas do not explicitly depend upon the risk premium for the underlying asset. Besides, the paper stresses how some instantaneous correlation between asset returns and mixing state variables precludes preference-free option pricing. This effect is also characterized in terms of volatility smile skewness and shown to be empirically relevant for S&P 500 call option pricing.

Preferences, State Variables and Option Pricing
René Garcia, Université de Montréal

This paper assesses the empirical performance of an intertemporal option pricing model with latent variables which generalizes the Black-Scholes and the stochastic volatility formulas. We derive a closed-form formula for an equilibrium model with recursive preferences where the fundamentals follow a Markov switching process. In a simulation experiment based on the model, we show that option prices are more informative about preference parameters than stock returns. When we estimate the preference parameters implicit in S&P 500 call option prices given our model, we find quite reasonable values for the coefficient of relative risk aversion and the intertemporal elasticity of substitution. Finally, when we calibrate the model to minimize out-of-sample pricing errors, we are in the same order of magnitude as the ad hoc BS model of Dumas, Fleming and Whaley (1998).