Efficient exponential timestepping algorithm using control variate technique for simulating a functional of exit time of
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Arabian Journal of Mathematics
Hasan Alzubaidi
Efficient exponential timestepping algorithm using control variate technique for simulating a functional of exit time of one-dimensional Brownian diffusion with applications in finance Received: 26 August 2019 / Accepted: 28 June 2020 © The Author(s) 2020
Abstract The exponential timestepping Euler algorithm with a boundary test is adapted to simulate an expected of a function of exit time, such as the expected payoff of barrier options under the constant elasticity of variance (CEV) model. However, this method suffers from a high Monte Carlo (MC) statistical error due to its exponentially large exit times with unbounded samples. To reduce this kind of error efficiently and to speed up the MC simulation, we combine such an algorithm with an effective variance reduction technique called the control variate method. We call the resulting algorithm the improved Exp algorithm for abbreviation. In regard to the examples we consider in this paper for the restricted CEV process, we found that the variance of the improved Exp algorithm is about six times smaller than that of the Jansons and Lythe original method for the down-and-out call barrier option. It is also about eight times smaller for the up-and-out put barrier option, indicating that the gain in efficiency is significant without significant increase in simulation time. Mathematics Subject Classification
60H35 · 65C05 · 91B25
1 Introduction The measurement of exit times or functionals of exit times using the existing numerical methods is considered to be one of the most difficult tasks, even if updates of the process are generated with high accuracy. The resulting errors are generally large due to the possibility that the exit event may have occurred between the computational nodes as it is discussed in [16,22,23]. Our goal is to estimate the expectation of f (Sτ H , τ H ) for some given function f which represents the discounted payoff of a barrier option in finance. The process St , t ≥ 0 is a one-dimensional diffusion process that satisfies the stochastic differential equation of the form dSt = μ(St )dt + σ (St )dWt , S0 = s ∈ R,
(1)
where Wt is a standard one-dimensional Brownian motion defined on a filtered probability space (Ω, F, Ft , P). The functions μ, σ : R → R, with σ 2 (s) > 0, represent the drift and diffusion parameters of the diffusion process St , respectively, and are assumed to satisfy regularity conditions, which are sufficient to guarantee the existence and uniqueness of the solution to Eq. (1) (see, e.g. [26, Theorem 4.5.3] for sufficient conditions). Additionally, τ H is the first time for the diffusion process St to hit or cross a barrier H and can be written as inf{t ≥ 0 : St ≤ H |S0 = s} if s > H, (2) τH = inf{t ≥ 0 : St ≥ H |S0 = s} if s < H. A common application for this computational problem in financial mathematics is a pricing of barrier option, whose payoff in maturity depends on whether the underlying asset price crosses some predefined H. Alzubaidi (B) Department of Mathematic
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