Pricing Forward Start Options in Models Based on (Time-Changed) Levy Processes

14 Pages Posted: 14 Jan 2009 Last revised: 19 Jan 2009

See all articles by Philipp Beyer

Philipp Beyer

affiliation not provided to SSRN

Joerg Kienitz

University of Wuppertal - Applied Mathematics; University of Cape Town (UCT); Quaternion Risk Management

Date Written: December 16, 2008


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. Widely applied models to account for the forward skew dynamics to price such options include the Heston model, the Heston-Hull-White model and the Bates model. Within these models solutions for options including forward start features are available using (semi) analytical formulas.

Today exponential (subordinated) Levy models being increasingly popular for modelling the asset dynamics. While the simple exponential Levy models imply the same forward volatily 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 modelled.

Analytical pricing formulas based on the characteristic function and Fourier transform methods are available for the class of these models.

We extend the applicability of analytical pricing to options including forward start features. To this end we derive the forward characteristic functions which can be used in Fourier transform based methods.

As examples we consider the Variance Gamma model and the NIG model subordinated by a Gamma Ornstein Uhlenbeck process and respectively by an Cox-Ingersoll-Ross process. We check our analytical results by applying Monte Carlo methods. These results can for instance be applied to calibration of the forward volatility surface.

Keywords: Variance Gamma, Normal Inverse Gaussian, Gamma Ornstein Uhlenbeck, CIR, Subordinator, Time change, forward characteristic function, Option Pricing

JEL Classification: G10, G12, G13

Suggested Citation

Beyer, Philipp and Kienitz, Joerg, Pricing Forward Start Options in Models Based on (Time-Changed) Levy Processes (December 16, 2008). Available at SSRN: or

Philipp Beyer

affiliation not provided to SSRN ( email )

Joerg Kienitz (Contact Author)

University of Wuppertal - Applied Mathematics ( email )

Gaußstraße 20
42097 Wuppertal

University of Cape Town (UCT) ( email )

Private Bag X3
Rondebosch, Western Cape 7701
South Africa

Quaternion Risk Management ( email )

54 Fitzwilliam Square North
Dublin, D02X308

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