q-fin updates on arXiv.org
Tue, 10 Mar 2020 06:01:47 GMT language
In this paper we focus on qualitative properties of solutions to a nonlocal
nonlinear partial integro-differential equation (PIDE). Using the theory of
abstract semilinear parabolic equations we prove existence and uniqueness of a
solution in the scale of Bessel potential spaces. Our aim is to generalize
known existence results for a wide class of L'evy measures including with a
strong singular kernel.
As an application we consider a class of PIDEs arising in the financial
mathematics. The classical linear Black-Scholes model relies on several
restrictive assumptions such as liquidity and completeness of the market.
Relaxing the complete market hypothesis and assuming a Levy stochastic process
dynamics for the underlying stock price process we obtain a model for pricing
options by means of a PIDE. We investigate a model for pricing call and put
options on underlying assets following a Levy stochastic process with jumps. We
prove existence and uniqueness of solutions to the penalized PIDE representing
approximation of the linear complementarity problem arising in pricing American
style of options under Levy stochastic processes. We also present numerical
results and comparison of option prices for various Levy stochastic processes
modelling underlying asset dynamics.