Nested multilevel Monte Carlo methods and a modified Euler-Maruyama scheme utilising approximate Gaussian random variables suitable for vectorised hardware and low-precisions

Thesis

We present a modified Euler-Maruyama scheme using approximate random variables, produced by the inverse transform method, using cheap approximations to the inverse Gaussian cumulative distribution function. We analyse the error for two approximations: a piecewise constant approximation on equally spaced intervals, and a piecewise linear approximation using geometric intervals dense at the singularities. High speed implementations faster than Intel's MKL are provided, suitable for modern vector hardware. The error between the approximations from the exact and modified Euler-Maruyama schemes is bounded by the error from the approximate random variables.

We incorporate this scheme into a multilevel Monte Carlo framework producing a nested scheme, and show that the discretisation error couples to the random variables' approximation error. The result directly extends to Lipschitz and differentiable payoff functions. For Lipschitz and non-differentiable payoff functions simulated using a time step delta, there is a transition from a variance decay of O(delta) to order(delta^1/2) as the discretisation error becomes dominant. These variance bounds are demonstrated numerically for geometric Brownian motion and a variety of payoff functions of varying smoothness.

For approximate random variables computed in low-precision, a model for the accumulated rounding error is developed and assessed. Half-precision is viable for a range of coarse path simulations, and can be extended further by incorporating a Kahan compensated summation. We empirically demonstrate these ideas are transferable to the Milstein scheme, and the more difficult Cox-Ingersoll-Ross process and its non-central chi-squared distribution. We estimate that under the Black-Scholes model, options can be priced using path simulations with approximate Gaussian random variables, obtaining a five-times or more speed improvement without losing accuracy.

Authors: Sheridan-Methven, O

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Approximations; Monte Carlo; Random Numbers; Computing; Analysis
• Subjects: Numerical Analysis; Mathematics; Computer Science