Computing Functionals of Multidimensional Diffusions via Monte Carlo Methods

We discuss suitable classes of diffusion processes, for which functionals relevant to finance can be computed via Monte Carlo methods. In particular, we construct exact simulation schemes for processes from this class. However, should the finance problem under consideration require e.g. continuous monitoring of the processes, the simulation algorithm can easily be embedded in a multilevel Monte Carlo scheme. We choose to introduce the finance problems under the benchmark approach, and find that this approach allows us to exploit conveniently the analytical tractability of these diffusion processes.

[1]  Alessandro Gnoatto,et al.  The Explicit Laplace Transform for the Wishart Process , 2011, Journal of Applied Probability.

[2]  E. Platen,et al.  A tractable model for indices approximating the growth optimal portfolio , 2012 .

[3]  J. D. da Fonseca,et al.  Estimating the Wishart Affine Stochastic Correlation Model using the empirical characteristic function , 2012 .

[4]  A. Itkin.  New solvable stochastic volatility models for pricing volatility derivatives , 2012, 1205.3550.

[5]  Jan Baldeaux,et al.  Quasi-Monte Carlo methods for the Heston model , 2012 .

[6]  Oliver Pfaffel.  Wishart Processes , 2012, 1201.3256.

[7]  Quasi-Monte Carlo methods for derivatives on realised variance of an index under the benchmark approach , 2011 .

[8]  E. Thorp,et al.  The Kelly Capital Growth Investment Criterion: Theory and Practice , 2011 .

[9]  C. Gouriéroux,et al.  International Money and Stock Market Contingent Claims , 2010 .

[10]  Nicola Bruti-Liberati.  Numerical Solution of Stochastic Differential Equations with Jumps in Finance , 2010 .

[11]  M. Craddock,et al.  ON THE EQUIVALENCE OF LIE SYMMETRIES AND GROUP REPRESENTATIONS , 2010 .

[12]  M. Craddock,et al.  Lie symmetry methods for multidimensional linear parabolic PDEs and diffusions , 2010 .

[13]  Oliver Pfaffel,et al.  On strong solutions for positive definite jump diffusions , 2009, 0910.1784.

[14]  Damir Filipović,et al.  Affine Processes on Positive Semidefinite Matrices , 2009, 0910.0137.

[15]  M. Craddock.  Fundamental solutions, transition densities and the integration of Lie symmetries , 2009 .

[16]  The calculation of expectations for classes of diffusion processes by Lie symmetry methods , 2009, 0902.4806.

[17]  J. D. da Fonseca,et al.  Hedging (Co)Variance Risk with Variance Swaps , 2008 .

[18]  Claudio Tebaldi,et al.  A multifactor volatility Heston model , 2008 .

[19]  Nicole El Karoui,et al.  Wishart Stochastic Volatility: Asymptotic Smile and Numerical Framework , 2008 .

[20]  Michael B. Giles,et al.  Multilevel Monte Carlo Path Simulation , 2008, Oper. Res..

[21]  Alexandros Beskos,et al.  A Factorisation of Diffusion Measure and Finite Sample Path Constructions , 2008 .

[22]  Andrea Buraschi,et al.  Correlation Risk and the Term Structure of Interest Rates , 2008 .

[23]  Markus Leippold,et al.  Asset Pricing with Matrix Affine Jump Diffusions , 2008 .

[24]  C. Tebaldi,et al.  SOLVABLE AFFINE TERM STRUCTURE MODELS , 2007 .

[25]  O. Barndorff-Nielsen,et al.  Positive-definite matrix processes of finite variation , 2007 .

[26]  Mark Craddock,et al.  Lie Group Symmetries as Integral Transforms of Fundamental Solutions , 2006 .

[27]  G. Roberts,et al.  Retrospective exact simulation of diffusion sample paths with applications , 2006 .

[28]  D. Heath,et al.  A Benchmark Approach to Quantitative Finance , 2006 .

[29]  Claudio Tebaldi,et al.  Option pricing when correlations are stochastic: an analytical framework , 2006 .

[30]  Andrea Buraschi,et al.  Correlation Risk and Optimal Portfolio Choice , 2006 .

[31]  G. Roberts,et al.  Exact simulation of diffusions , 2005, math/0602523.

[32]  Currency Derivatives under a Minimal Market Model with Random Scaling , 2005 .

[33]  Eckhard Platen,et al.  Symmetry group methods for fundamental solutions , 2004 .

[34]  C. Gouriéroux,et al.  Derivative Pricing with Multivariate Stochastic Volatility: Application to Credit Risk , 2004 .

[35]  C. Gouriéroux,et al.  Wishart Quadratic Term Structure Models , 2003 .

[36]  Symmetry Group Methods for Fundamental Solutions and Characteristic Functions , 2003 .

[37]  D. Duffie,et al.  Affine Processes and Application in Finance , 2002 .

[38]  M. Musiela,et al.  Martingale Methods in Financial Modelling , 2002 .

[39]  Stefan Heinrich,et al.  Multilevel Monte Carlo Methods , 2001, LSSC.

[40]  Gregory A. Willard,et al.  Local martingales, arbitrage, and viability Free snacks and cheap thrills , 2000 .

[41]  Stefan Heinrich,et al.  Monte Carlo Complexity of Global Solution of Integral Equations , 1998, J. Complex..

[42]  John B. Long.  The numeraire portfolio , 1990 .

[43]  P. Olver.  Applications of Lie Groups to Differential Equations , 1986 .

[44]  J. Kent.  Aspects of Multivariate Statistical Theory , 1984 .

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

[46]  H. Markowitz.  Investment for the Long Run: New Evidence for an Old Rule , 1976 .

[47]  E Thorp,et al.  A FAVORABLE STRATEGY FOR TWENTY-ONE. , 1961, Proceedings of the National Academy of Sciences of the United States of America.

[48]  L. Breiman.  INVESTMENT POLICIES FOR EXPANDING BUSINESSES OPTIMAL IN A LONG-RUN SENSE , 1960 .

[49]  H. Latané.  Criteria for Choice Among Risky Ventures , 1959, Journal of Political Economy.

[50]  John L. Kelly,et al.  A new interpretation of information rate , 1956, IRE Trans. Inf. Theory.