The integrated volatility implied by option prices, a Bayesian approach

Abstract

In this thesis, we present the new concept of implied integrated volatility. When the stock price volatility is stochastic, the integrated volatility is the time-average of the stock price variance. This volatility is a fundamental quantity in option theory, as the stock price returns depend on the stock price volatility only via the integrated volatility. The implied integrated volatility is the integrated volatility implied by option Hull-White prices. It is a stochastic extension of the Black-Scholes implied volatility. Unlike the latter, however, it is independent of the strike price of options. We suggest that this volatility can be used in volatility estimation, in pricing illiquid options consistently with corresponding liquid ones, and in hedging options. Estimating the implied integrated volatility is an ill-posed inverse problem. We present methods to estimate it within a Bayesian framework. This approach provides us with not only a point estimate, but also the possibility to gauge the reliability of this estimate.