Simulation of the drawdown and its duration in Lévy models via stick-breaking Gaussian approximation

Overview

Abstract

In this talk, I will describe a computational method for expected functionals of the drawdown and its duration in exponential Lévy models. It is based on a novel simulation algorithm for the joint law of the state, supremum and time the supremum is attained of the Gaussian approximation for a general Lévy process. We bound the bias for various locally Lipschitz and discontinuous payoffs arising in applications and analyse the computational complexities of the corresponding Monte Carlo and multilevel Monte Carlo estimators. Monte Carlo methods for Lévy processes (using Gaussian approximation) have been analysed for Lipschitz payoffs, in which case the computational complexity of our algorithm is up to two orders of magnitude smaller when the jump activity is high. At the core of our approach are bounds on certain Wasserstein distances, obtained via the novel stick-breaking Gaussian (SBG) coupling between a Lévy process and its Gaussian approximation. Numerical performance (based on the implementation in the GitHub repository https://github.com/jorgeignaciogc/SBG.jl) exhibits a good agreement with our theoretical bounds. Numerical evidence suggests that our algorithm remains stable and accurate when estimating Greeks for barrier options and outperforms the “obvious” algorithm for finite-jump-activity Lévy processes. This is joint work with J. Gonzalez-Cázares.

Brief Biography

Aleks is a Professor of Probability at the Department of Statistics at the University of Warwick and a Fellow of the The Alan Turing Institute. Aleks was previously a Chair in Probability at the Department of Mathematics of King's College London, and before that a  Reader in Probability at the Mathematics Department of Imperial College London. Aleks obtained his Ph.D. in low-dimensional topology at the DPMMS and Trinity College Cambridge, before working in the City of London as a front-office quantitative analyst in Foreign Exchange derivatives markets. Aleks’ research interests include:

Probability: stability of stochastic systems; invariance principles; local time; coupling; stochastic processes on manifolds; stochastic analysis for processes with and without jumps; Levy processes; random walks; Markov chains; branching; stochastic control & optimal stopping

Numerical stochastics: simulation of processes with and without jumps; weak and strong approximations; Markov chain Monte Carlo; exact simulation; stochastic gradient descent

Mathematical finance: hedging, risk management and price prediction; implied volatility surface; stochastic volatility models with jumps; arbitrage

Statistics: calibration and parameter estimation algorithms for continuous-time models using discrete observations; option price prediction; regularisation

For further information on the research interests and work of Aleks’ research group at Warwick see the Webpage and the YouTube channel Prob-AM.