We show that several advanced equity option models incorporating stochastic volatility can be calibrated very nicely to a realistic option surface. More specifically, we focus on the Heston Stochastic Volatility model (with and without jumps in the stock price process), the BarndorffNielsen-Shephard model and Levy models with stochastic time. All these models are capable of accurately describing the marginal distribution of stock prices or indices and hence lead to almost identical European vanilla option prices. As such, we can hardly discriminate between the different processes on the basis of their smile-conform pricing characteristics. We therefore are tempted applying them to a range of exotics. However, due to the different structure in path-behaviour between these models, the resulting exotics prices can vary significantly. It motivates a further study on how to model the fine stochastic behaviour of assets over time.

Abstract We show that several advanced equity option models incorporating stochastic volatility can be calibrated very nicely to a realistic implied volatility surface. More specically , we focus on the Heston Stochastic Volatility model (with and without jumps in the stock price process), the Barndor-Nielsen-Shephard model and L evy models with stochastic time. All these models are capable of accurately describing the marginal distrib- ution of stock prices or indices and hence lead to almost identical European vanilla option prices. As such, we can hardly discriminate between the dif- ferent processes on the basis of their smile-conform pricing characteristics. We therefore are tempted applying them to a range of exotics. However, due to the dieren t structure in path-behaviour between these models, we nd,that the resulting exotics prices can vary signican tly. We subse- quently introduce moment,derivatives. These are derivatives that depend on the realized moments of (daily) logreturns. An already traded example of these derivatives is the Variance Swap. We show how to hedge these options and calculate their prices by Monte-Carlo simulation. A compar- ison of these moment derivatives premiums,demonstrates an even bigger discrepancy between the aforementioned models. It motivates a further study on how to model the ne,stochastic behaviour of assets over time.