Calibrating Arbitrage-Free Stochastic Volatility Models by the Relative Entropy Method

We develop a new framework to calibrate stochastic volatility option pricing models to an arbitrary prescribed set of prices of liquidly traded options. Our approach produces an arbitrage-free stochastic volatility diusion process that minimizes the distance to a prior diusion model. We use the notion of relative entropy (also known under the name of Kullback-Leibler distance) to quantify the distance between the two diusions. The problem is formulated as a stochastic control problem. We also show that, in a very natural limiting regime, it results in a calibrating method for complete models. Implementation issues are discussed in details for calibrating both the stochastic volatility and the complete models.