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

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 optimization 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.

[1]  Nilay Shah,et al.  The Importance of being Global . Application of Global Sensitivity Analysis in Monte Carlo Option Pricing , 2007 .

[2]  Eric Fournié,et al.  Monte Carlo Methods in Finance , 2002 .

[3]  B. Øksendal.  Stochastic differential equations : an introduction with applications , 1987 .

[4]  Paola Annoni,et al.  Estimation of global sensitivity indices for models with dependent variables , 2012, Comput. Phys. Commun..

[5]  D. Shahsavani,et al.  Variance-based sensitivity analysis of model outputs using surrogate models , 2011, Environ. Model. Softw..

[6]  H. Niederreiter.  Low-discrepancy and low-dispersion sequences , 1988 .

[7]  Paola Annoni,et al.  Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index , 2010, Comput. Phys. Commun..

[8]  A. Saltelli,et al.  Reliability Engineering and System Safety , 2008 .

[9]  Paul Wilmott,et al.  Paul Wilmott on Quantitative Finance , 2010 .

[10]  Paul Glasserman,et al.  Monte Carlo Methods in Financial Engineering , 2003 .

[11]  P. Kloeden,et al.  Numerical Solutions of Stochastic Differential Equations , 1995 .

[12]  A. Owen.  THE DIMENSION DISTRIBUTION AND QUADRATURE TEST FUNCTIONS , 2003 .

[13]  Anargyros Papageorgiou,et al.  Deterministic Simulation for Risk Management , 1999 .

[14]  Sergei S. Kucherenko,et al.  On global sensitivity analysis of quasi-Monte Carlo algorithms , 2005, Monte Carlo Methods Appl..

[15]  P. Boyle.  Options: A Monte Carlo approach , 1977 .

[16]  I. Sobol,et al.  Global sensitivity indices for nonlinear mathematical models. Review , 2005 .

[17]  I. Sobol,et al.  Construction and Comparison of High-Dimensional Sobol' Generators , 2011 .

[18]  Xiaoqun Wang,et al.  Dimension Reduction Techniques in Quasi-Monte Carlo Methods for Option Pricing , 2009, INFORMS J. Comput..

[19]  A. Owen,et al.  Estimating Mean Dimensionality of Analysis of Variance Decompositions , 2006 .

[20]  A. Saltelli,et al.  Making best use of model evaluations to compute sensitivity indices , 2002 .

[21]  W. Horowitz,et al.  , 1 ' 2 , 2009 .

[22]  D. Duffie.  Dynamic Asset Pricing Theory , 1992 .

[23]  A. Owen,et al.  Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension , 1997 .

[24]  Michael Zinganel.  1:1 , 2014, Wohnen Zeigen.

[25]  I. Sobola,et al.  Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates , 2001 .

[26]  Anargyros Papageorgiou,et al.  The Brownian Bridge Does Not Offer a Consistent Advantage in Quasi-Monte Carlo Integration , 2002, J. Complex..

[27]  Oren Cheyette.  Interest Rate Models , 2002 .

[28]  A. Owen,et al.  Quasi-Regression and the Relative Importance of the ANOVA Components of a Function , 2002 .

[29]  PRINCETON UNIVERSITY PRESS , 2005 .

[30]  D. Brigo,et al.  Interest Rate Models , 2001 .

[31]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[32]  Anargyros Papageorgiou,et al.  New Results on Deterministic Pricing of Financial Derivatives , 1996 .

[33]  S. Rahman.  Reliability Engineering and System Safety , 2011 .

[34]  I. Sobol.  On the distribution of points in a cube and the approximate evaluation of integrals , 1967 .

[35]  Marco Bianchetti,et al.  Pricing and Risk Management with High-Dimensional Quasi-Monte Carlo and Global Sensitivity Analysis: Technical Paper , 2015 .

[36]  Joseph F. Traub,et al.  Faster Valuation of Financial Derivatives , 1995 .

[37]  I. Sobol.  Global Sensitivity Indices for Nonlinear Mathematical Models , 2004 .

[38]  Peter Jaeckel,et al.  Monte Carlo methods in finance , 2002 .

[39]  Nilay Shah,et al.  The identification of model effective dimensions using global sensitivity analysis , 2011, Reliab. Eng. Syst. Saf..

[40]  Art B. Owen,et al.  Variance and discrepancy with alternative scramblings , 2002 .

[41]  Ilya M. Sobol,et al.  Quasi-Monte Carlo: A high-dimensional experiment , 2014, Monte Carlo Methods Appl..

[42]  N. Metropolis.  THE BEGINNING of the MONTE CARLO METHOD , 2022 .

[43]  Liguo Wang,et al.  Numerical Solutions of Stochastic Differential Equations , 2016 .

[44]  P. Glasserman,et al.  Monte Carlo methods for security pricing , 1997 .

[45]  Dan Rosen,et al.  Measuring Portfolio Risk Using Quasi Monte Carlo Methods , 1998 .

[46]  Takuji Nishimura,et al.  Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator , 1998, TOMC.