HIGH PERFORMANCE COMPUTING IN THE LEAST SQUARES MONTE CARLO APPROACH GILLES DESVILLES Consultant, Rationnel Maître de Conférences, CNAM
Introduction Valuation of American options on several assets requires numerical procedures Finite Difference & Binomial Tree barred by Curse of Dimensionality 2
Introduction Since Longstaff Schwartz in 2001 Least Square Monte Carlo (LSMC) is expected to break the curse but No theoretical proof supporting the conjecture Virtually no test on multiple asset options 3
In search of Convergence Longstaff Schwartz 2001 estimate US put 100,000 paths MC standard error $0.015 for prices [$1;$9] Nothing reported on response time Can we do better? 4
In search of Convergence Suppose standard error is $0.001 If LSMC price is $4.835 then market maker cannot split between $4.83 or $4.84 $0.01 uncertainty LSMC converges when standard error = $0.000 called Financial Convergence 5
In search of Convergence Highly sequential C++ program Initial Program First Designed for 256MB RAM and 1 Single Core Now Run on 24GB RAM and 2 Dual Cores L Ecuyer MRG32k3a Generator Variance Reduction S = 36 σ = 0.20 T = 2 Graph 2 8M paths hit hardware limits Financial convergence to $0.000 not reached 6
In search of Convergence Highly parallel C++ program with OpenMp New Program Designed for and Run on 24GB RAM and 2 Dual Cores L Ecuyer MRG32k3a Generator Variance Reduction S = 36 σ = 0.20 T = 2 Graph 4 Extends paths to 26M! But still financial convergence not reached!! 7
In search of Convergence Highest known precision in LSMC US put pricing 8
In search of Convergence Can we reach financial convergence with GPUs? 1 Nvidia Tesla C2050 graphical card C++ program with Nvidia Cuda 9
In search of Convergence Highly parallel processing with Cuda American Put Pricing Response Times of Basic Full GPU versus OpenMp Full CPU 4,194,304 paths 64 dates per year L Ecuyer MRG32k3a Generator Variance Reduction S = 36, 38, 40 or 44 σ = 0.20 or 0.40 T =1 or 2 years Graph 6 Restricts paths to 4M! error up to $0.03 But response time divided by 2 10
No Curse of Dimensionality Conjecture about LSMC Response time does not explode when number of assets increases Contrary to trees and finite differences 11
No Curse of Dimensionality Longstaff Schwartz 2001 estimate US 5-asset maxcall 1-2 min response 50,000 paths How response time behaves when 50,000 becomes millions? 12
No Curse of Dimensionality 5-asset option versus single asset option American 5-Asset MaxCall versus American Single Asset Put Pricing Response Time In the Money Underliers L Ecuyer MRG32k3a Generator Variance Reduction K = 40 σ = 0.20 T = 2 years 5 regressors 50 dates per year Graph 7 Response time ratio constant as paths increase non exploding response time 13
GPU speed 5-asset option CPU-GPU mix versus CPU sequential American 5-Asset MaxCall CPU-GPU Mix versus One Core CPU Program Response Time S = 110 L Ecuyer MRG32k3a Generator Variance Reduction K = 100 σ = 0.20 T = 3 years 19 regressors 3 dates per year Graph 9 Response time divided by 14! 14
GPU speed 5-asset option CPU-GPU mix enhanced with GPU tips American 5-Asset MaxCall CPU-GPU Mix versus One Core CPU Effect of 6 GPU Programming Tips on Program Response Time S = 110 L Ecuyer MRG32k3a Generator Variance Reduction K = 100 σ = 0.20 T = 3 years 19 regressors 3 dates per year Graph 10 Response time now divided by 22! 15
Conclusion Thanks to computer power and parallel processing Lowest known standard errors of 2 seminal benchmarks: US vanilla put and US call on max of 5 assets LSMC not barred by infamous curse of dimensionality GPU faster but too short 16