Option Pricing and Bayesian Learning

Abstract: This thesis consists of three chapters devoted to both empirical and theoretical aspects of option pricing. The first chapter investigates the market for European options on the Swedish OMX index using daily data for the period 1993-2000. The assumption of constant volatility of the returns underlying the Black and Scholes option pricing formula is assessed by extracting the volatilities implied by the observed call prices and put prices. These are then related to moneyness and time to expiration. It is found that for both calls and puts the implied volatility is decreasing in time to expiration., while the largest implied volatilities were exhibited for contracts with extreme values of moneyness. The arbitrage relation Put-Call Parity is tested. The second chapter presents the theory of how Bayesian learning affects asset pricing. The model is due to Timmerman and Guidolin (2003) and is an equilibrium, representative agent model in the vein of Lucas (1978). Their results on asset prices are derived envoking the urn model due to Polya, and it is shown that Bayesian learning implies a recombining tree. Using the moments of the Polya distribution, a closed form approximation is derived for the stock and call price with Bayesian learning. Moreover, the relationship between the Lucas model without learning and the (discrete time) Black and scholes is described explicitly. The third chapter applies the closed form approximation of the call price with Bayesian learning, as well as alternative models, to data on the Swedish OMX index. Bayesian learning is found to produce superior in-sample fit. Out-of-sample predictions are performed both one-day ahead and one week ahead. The model with Bayesian learning outperforms the other models in terms of the percentage of dates it gives the lowest out-of-sample error, for both prediction lengths. It also gives the lowest average out-of-sample error for the one-day ahead predictions.