Delay geometric Brownian motion in financial option valuation

Abstract

Motivated by influential work on complete stochastic volatility models, such as Hobson and Rogers [11], we introduce a model driven by a delay geometric Brownian motion (DGBM) which is described by the stochastic delay differential equation . We show that the equation has a unique positive solution under a very general condition, namely that the volatility function V is a continuous mapping from to itself. Moreover, we show that the delay effect is not too sensitive to time lag changes. The desirable robustness of the delay effect is demonstrated on several important financial derivatives as well as on the value process of the underlying asset. Finally, we introduce an Euler–Maruyama numerical scheme for our proposed model and show that this numerical method approximates option prices very well. All these features show that the proposed DGBM serves as a rich alternative in modelling financial instruments in a complete market framework.

Keywords:

• stochastic delay differential equations
• derivative pricing
• Euler–Maruyama
• local Lipschitz condition
• strong convergence

Keywords:

• 60G44
• 91G20
• 91G80

Notes

^1. These figures are courtesy of Nairn McWilliams.

^2. This figure is courtesy of Nairn McWilliams.

^3. This figure is courtesy of Nairn McWilliams.