Multilevel Monte Carlo Path Simulation

Abstract

We show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using Monte Carlo path simulations. In the simplest case of a Lipschitz payoff and an Euler discretisation, the computational cost to achieve an accuracy of O( ) is reduced from O( −3 ) to O( −2 (log ) 2 ). The analysis is supported by numerical results showing significant computational savings.

On irregular functionals of SDEs and the Euler scheme

“…The multilevel Monte Carlo method introduced by Giles [7] requires a variance estimate for the difference of the payoff and its approximation. Corollary 3.2 gives the parameter β = γ−ε in [7, Theorem 3.1 iii)] in the case of a payoff of bounded variation, especially for the binary option, and any approximation scheme satisfying the moment estimate (3.2).…”

“…The approximation of solutions of SDEs is related to the multilevel Monte Carlo method introduced by M. Giles [7], [8]. One purpose of the multilevel Monte Carlo method is to approximate the expected payoff of an option with a small computational cost.…”

Vibrato Monte Carlo Sensitivities

“…Another direction for future research is the use of the vibrato idea for multilevel Monte Carlo analysis [12]. Analytic conditional expectation is currently used to treat discontinuous payoffs to obtain improved convergence rates with the Milstein scheme [10].…”

Monte Carlo Computation in Finance

“…Because of bias cancelation, the estimator 2μ(2m)−μ(m) can have lower bias and a better rate of convergence. This idea is extended by [44] to multiple grids of different fineness, instead of just two. The estimator given L grids, with N paths simulated on the th grid which has m steps, is ∑ L =1 ∑ N i=1 (μ (i) (m ) −μ (i) (m −1 ))/N , wherê µ (i) (m ) involves simulating the same Wiener process sample path {W (i) (t)} 0≤t≤T for all grids.…”