American option pricing problem transformed on finite interval

Abstract

We study theAmerican option pricing linear complementarity problem (LCP), transformed on finite interval as it is initially defined on semi-infinite real axis. We aim to validate this transformation, investigating the well-posedness of the resulting problem in weighted Sobolev spaces. The monotonic penalty method reformulates the LCP as a semi-linear partial differential equation (PDE) and our analysis of the penalized problem results in uniform convergence estimates. The resulting PDE is further discretized by a fitted finite volume method since the transformed partial differential operator degenerates on the boundary. We show solvability of the semi-discrete and fully discrete problems. The Brennan-Schwarz algorithm is also presented for comparison of the numerical experiments, given in support to our theoretical considerations.

A simplified stochastic optimization model for logistic dynamics with control-dependent carrying capacity

“…with (0) = 0 and K max = aq max + b. Notice that the assumptions on the sign and Lipschitz continuity of the coefficients are not violated in the resulting equation (23).…”

“…As a result, the scheme can generate nodally-exact solutions to advection-decay equations having constant coefficients. Similar formalism has been applied to nonlinear advection-diffusion equations related to stochastic optimal control problems [23,57]. In this paper, solving the resulting discretized system is carried out with a standard policy iteration algorithm [2], which has effectively been applied to numerical resolution of discretized HJB equations [15,45].…”

“…The present finite difference discretization generates a unique numerical solution. In addition, numerical solutions with the finite difference discretization locally uniformly converge to the unique viscosity solution to the HJB equation (23) as l → +0.…”

“…The HJB equation (23) is numerically solved for a variety of parameter values, so that behaviour of the optimal control q * , the quantity to be found through solving the optimization problem, is explored more in detail. Figures 5-9 show the computed q * for different values of a in the carrying capacity, α in the decay term, ρ in the jump term, and w and δ in the performance index.…”

Novel methods in computational finance

“…Gyulov and Valkov [5] apply the monotonic penalty method to reformulate the option pricing linear complementarity problem, transformed on finite interval. A fitted finite volume method is proposed that supports their theoretical findings.…”