Calibration of a Stock’s Beta Using Option Prices

Abstract

In this paper, we present a continuous time Capital Asset Pricing Model where the volatilities of the market index and the stock are both stochastic. Using a singular perturbation technique, we provide approximations for the prices of European options on both the stock and the index. We derive then an estimator of the parameter beta under the risk-neutral pricing measure P∗ using option and underlying prices. Following that, we study empirically the discrepancy between the implied value of the parameter β under P∗ and its realized value under the real-world probability measure P. Finally, we show that the parameter β is crucial for the hedging of stock options using instruments on the index, and we study numerically the performance of the proposed hedging strategies.