Options depending on the forward skew are very popular. One such option is the forward starting call option—the basic building block of a cliquet option. The models which are widely applied to account for the forward skew dynamics, to price such options include the Heston model, Heston-Hull-White model and Bates model. Within these models, solutions for options including forward start features are available using (semi) analytical formulas. Now, exponential (subordinated) Lévy models have become increasingly popular for modeling the asset dynamics. While the simple exponential Lévy models imply the same forward volatility surface for all future times, the subordinated models do not. Depending on the subordinator the dynamic of the forward volatility surface and therefore stochastic volatility can be modeled.

This time-change accounts for the concept of stochastic volatility. Several authors use such models to price exotic options, (Carr et al., 2003; and Schoutens et al., 2003). Recently, we have also shown how to use transition probabilities (Kienitz, 2008a) and characteristic functions (Kienitz, 2008b), to compute stable Greeks for path-dependent options with discontinuous payoff functions using Monte Carlo methods for a processes of the form as Equation (1.3). Furthermore, analytic formulas based on the characteristic function and the Fourier transform of the option's payoff function have been introduced by Lewis (2001), to efficiently price the options.

Analytical pricing formulas based on the characteristic function and Fourier transform methods are available for this class of models. This paper extends the applicability of analytical pricing to options including forward start features. To this end, it derives the forward characteristic functions which can be used in Fourier transform-based methods. As examples, the paper considers the Variance Gamma (VG) model and the Normal Inverse Gaussian (NIG) model subordinated by a Gamma-Ornstein-Uhlenbeck process and respectively by a Cox-Ingersoll-Ross process. The analytical results obtained are also checked by applying the Monte Carlo methods. These results can, for instance, be applied for calibration of the forward volatility surface.

Derivatives Market Journal, exponential Lévy models, Monte Carlo methods, Normal Inverse Gaussian Model, Cadlag Stochastic Process, Independent Increments, Stationary Increments, Fast Fourier Transform Techniques, Background Driving Lévy Process, Financial Derivatives, Financial Modelling, Risk Management.