Une approche unifiee pour une forme exacte dU prix d'une option dans les differents modeles a volatilite stochastique

Brownian Motion on the Hyperbolic Plane and Selberg Trace Formula

“…However, what we would like to emphasize is that our arguments can be traced in the reverse way and that we can also show the results of Yor [50] and Leblanc [31] mentioned above. In this sense we will study one object from different points of view.…”

“…[5,13,20]). This method is essentially the same as that of Yor [50] and Leblanc [31], who have used random time change in the theory of diffusion processes. We have learned some methods of calculations from both of them.…”

“…Section 3). Following the idea of [50], Leblanc [31] has calculated the Laplace transform of the joint distribution of (A S , a S , w S ) in the study of the Hull White model. We can obtain the explicit form of the Green function for H M *, B by a slight modification of their arguments.…”

Path dependent options on yields in the affine term structure model

“…Thanks to the Girsanov Theorem, we can pass from the process (15) to a Brownian Motion (see Leblanc 1994) thanks to the change of probability : …”

The α-Hypergeometric Stochastic Volatility Model

“…A closed-form solution (for vanilla options) for this model was proposed much later in Leblanc (1996) but it was also shown in Jourdain (2004) that the spot loses its martingale property for some parameter values. From a modelling point of view the model of Chesney and Scott (1989) is certainly the most natural one as it specifies for the volatility the exponential of a stationary Ornstein-Uhlenbeck process.…”