Option pricing in the presence of random arbitrage return

Abstract

We consider option pricing problems when we relax the condition of no arbitrage in the Black–Scholes model. Assuming random noise in the interest rate process, the derived pricing equation is in the form of stochastic partial differential equation. We used Karhunen–Loève expansion to approximate the stochastic term and a combined finite difference/finite element method to effect temporal and “spatial” discretization. Computational examples in which the noise is assumed to be a Ornstein–Uhlenbeck process are provided that illustrate not only the discretization methods used, but the type of results relevant to option pricing that can be obtained from the model.

Keywords:

• option pricing
• random arbitrage return
• stochastic PDEs
• finite element methods

2000 AMS Subject Classifications :

• 62P05
• 60H15
• 65N30

Notes

Because we have a stochastic term in Equation (13), using quadratic basis function does not improve the accuracy.