Derivation and analysis of partial integro-differential inequality on
shout options with its underlying asset subject to jump-diffusion model
Abstract
Up to present, research on shout options remains only on the assumption
that the underlying asset follows either Brownian motion or geometric
Brownian motion. But it can not be evaluated accurately by PDE on
geometric Brownian motion. To solve this problem this paper derives a
new partial integro-differential inequality (PIDI) for shout options
pricing on the assumption that the price of the underlying asset follows
the jump-diffusion model and constructs the mathematical model by
combining specific features and terminal conditions. On the basis of
this model we obtain some results about shout options pricing. For this
mathematical model this paper proposes a new competitive algorithm to
choose two aspects. One is employing high-order difference for integral
and partial derivative terms, the other is using Howard algorithm (also
called policy iteration) for the complementarity problem. Numerical
examples show that this algorithm yields an accurate technique and is
more efficient than the traditional approaches in the case of geometric
Brownian motion and jump-diffusion model, respectively.