Pricing and Risk Management with High-Dimensional Quasi-Monte Carlo and Global Sensitivity Analysis

Abstract

We review and apply Quasi Monte Carlo (QMC) and Global Sensitivity Analysis (GSA) techniques to pricing and risk management (greeks) of representative financial instruments of increasing complexity. We compare QMC vs standard Monte Carlo (MC) results in great detail, using high-dimensional Sobol' low discrepancy sequences, different discretization methods, and specific analyses of convergence, performance, speed up, stability, and error optimisation for finite differences greeks. We find that our QMC outperforms MC in most cases, including the highest-dimensional simulations and greeks calculations, showing faster and more stable convergence to exact or almost exact results. Using GSA, we are able to fully explain our findings in terms of reduced effective dimension of our QMC simulation, allowed in most cases, but not always, by Brownian bridge discretization. We conclude that, beyond pricing, QMC is a very promising technique also for computing risk figures, greeks in particular, as it allows to reduce the computational effort of high-dimensional Monte Carlo simulations typical of modern risk management.

Application of Quasi Monte Carlo and Global Sensitivity Analysis to Option Pricing and Greeks

“…We argue that, using QMC sampling (instead of MC) to generate the scenarios of the underlying risk factors and to price exotic trades may significantly improve the accuracy, the performance and the stability of such monster-simulations, as shown by preliminary results on real portfolios in [BKS14]. Furthermore, GSA should suggest how to order the risk factors according to their relative importance, thus reducing the effective dimensionality.…”

“…Then, N (A) * (a) can be estimated through linear regression results of Section 4.3. Some concern should be given to extrapolation for finite differences, as discussed in [BKS15]. Table 10: Speed-Up S * (a) of optimal QMC vs MC techniques, in order to achieve 1% and 0.1% accuracy in the computation of prices and FD greeks.…”

“…Secondly, effectiveness of QMC depends on the integrand function, and, most importantly, the convergence rate can depend on the dimensionality of the problem. However, it's been shown that in many financial applications QMC outperforms standard Monte Carlo even in the presence of very high dimensions [SAKK12, PT95, PP99, CMO97, KMRZ98a, KMRZ98b, KS07,BKS15]. This fact is usually explained by a reduced effective dimension of the problem, with respect to its nominal dimension.…”

Pricing and Rick Analysis in Hyperbolic Local Volatility Model with Quasi‐Monte Carlo

“…Many practical studies have proven that the Sobol LDS (Sobol, 1967), is in many aspects superior to other LDS (see e.g. Bianchetti et al, 2015;Glasserman, 2013;Jäckel, 2002;Kucherenko & Shah, 2007;Wilmott, 2007). For this reason it was used in this work.…”

“…This combination of Sobol, points with the Brownian bridge construction was proposed by Moskowitz and Caflisch (1996) and has been found to be highly effective in finance applications (see e.g. Acworth et al, 1998;Åkesson & Lehoczky, 2000;Bianchetti et al, 2015;Caflisch et al, 1997;Kucherenko & Shah, 2007).…”