Monte Carlo on GPUs

Monte Carlo Simulation and its Efficient Implementation Robert Tong 28 January 2010

Outline Why use Monte Carlo simulation? Higher order methods and convergence GPU acceleration The need for numerical libraries

Why use Monte Carlo methods? Essential for high dimensional problems – many degrees of freedom For applications with uncertainty in inputs In finance Important in risk modelling Pricing/hedging derivatives

The elements of Monte Carlo simulation Derivative pricing Simulate a path of asset values Compute payoff from path Compute option value Numerical components Pseudo-random number generator Discretization scheme

In the past Faster solution has been provided by increasing processor speeds Want a quicker solution? Buy a new processor Present Multi-core/Many-core architectures, without increased processor clock speeds A major challenge for existing numerical algorithms The escalator has stopped... or gone into reverse! Existing codes may well run slower on multi-core The demand for ever increasing performance

Ways to improve performance in Monte Carlo simulation Use higher order discretization Keep low order (Euler) discretization – make use of multi-core potential e.g. GPU (Graphics Processing Unit) Use high order discretization on GPU Use quasi-random sequence (Sobol’, …) and Brownian Bridge Implement Sobol’ sequence and Brownian Bridge on GPU

Higher order methods – 1 (work by Kai Zhang, University of Warwick, UK)

GPU acceleration Retain low order Euler discretization Use multi-core GPU architecture to achieve speed-up

The Emergence of GPGPU Computing Initially – computation carried out by CPU (scalar, serial execution) CPU evolves to add cache, SSE instructions, ... GPU added to speed graphics display – driven by gaming needs multi-core, SIMT, limited flexibility CPU and GPU move closer CPU becomes multi-core GPU becomes General Purpose (GPGPU) – fully programmable

Current GPU architecture – e.g. NVIDIA Tesla

Tesla – processing power SM – Streaming Multiprocessor 8 X SP - Streaming Processor core 2 X Special Function Unit MT – Multithreaded instruction fetch and issue unit Instruction cache Constant cache (read only) Shared memory (16 Kb, read/write) C1060 – adds double precision 30 double precision cores 240 single precision cores

Tesla C1060 memory (from: M A Martinez-del-Amor et al. (2008) based on E Lindholm et al. (2008))

Programming GPUs – CUDA and OpenCL CUDA (Compute Unified Device Architecture, developed by NVIDIA) Extension of C to enable programming of GPU devices Allows easy management of parallel threads executing on GPU Handles communication with ‘host’ CPU OpenCL Standard language for multi-device programming Not tied to a particular company Will open up GPU computing Incorporates elements of CUDA

First step – obtaining and installing CUDA FREE download from http://www.nvidia.com/object/cuda_learn.html See: Quickstart Guide Require: CUDA capable GPU – GeForce 8, 9, 200, Tesla, many Quadro Recent version of NVIDIA driver CUDA Toolkit – essential components to compile and build applications CUDA SDK – example projects Update environment variables (Linux default shown) PATH /usr/local/cuda/bin LD_LIBRARY_PATH /usr/local/cuda/lib CUDA compiler nvcc works with gcc (Linux) MS VC++ (Windows)

Host (CPU) – Device (GPU) Relationship Application program initiated on Host (CPU) Device ‘kernels’ execute on GPU in SIMT (Single Instruction Multiple Thread) manner Host program Transfers data from Host memory to Device (GPU) memory Specifies number and organisation of threads on Device Calls Device ‘kernel’ as a C function, passing parameters Copies output from Device back to Host memory

Organisation of threads on GPU SM (Streaming Multiprocessor) manages up to 1024 threads Each thread is identified by an index Threads execute as Warps of 32 threads Threads are grouped in blocks (user specifies number of threads per block) Blocks make up a grid

Memory hierarchy On device can Read/write per-thread Registers Local memory Read/write per-block shared memory Read/write per-grid global memory Read only per-grid constant memory On host (CPU) can Read/write per-grid Global memory Constant memory

CUDA terminology ‘ kernel’ – C function executing on the GPU __global__ declares function as a kernel Executed on the Device Callable only from the Host void return type __device__ declares function that is Executed on the Device Callable only from the Device

Application to Monte Carlo simulation Monte Carlo paths lead to highly parallel algorithms Applications in finance e.g. simulation based on SDE (return on asset) drift + Brownian motion Requires fast pseudorandom or Quasi-random number generator Additional techniques improve efficiency: Brownian Bridge, stratified sampling, …

Random Number Generators: choice of algorithm Must be highly parallel Implementation must satisfy statistical tests of randomness Some common generators do not guarantee randomness properties when split into parallel streams A suitable choice: MRG32k3a (L’Ecuyer)

MRG32k3a: skip ahead Generator combines 2 recurrences: Each recurrence of form (M Giles, note on implementation) Precompute in operations on CPU,

MRG32k3a: modulus Combined and individual recurrences Can compute using double precision divide – slow Use 64 bit integers (supported on GPU) – avoid divide Bit shift – faster (used in CPU implementations) Note: speed of different possibilities subject to change as NVIDIA updates floating point capability

MRG32k3a: memory coalescence GPU performance limited by memory access Require memory coalescence for fast transfer of data Order RNG output to retain consecutive memory accesses is stored at sequential ordering (Implementation by M Giles)

MRG32k3a: single – double precision L’Ecuyer’s example implementation in double precision floating point Double precision on high end GPUs – but arithmetic operations much slower in execution than single precision GPU implementation in integers – final output cast to double Note: output to single precision gives sequence that does not pass randomness tests

MRG32k3a: GPU benchmarking – double precision GPU – NVIDIA Tesla C1060 CPU – serial version of integer implementation running on single core of quad-core Xeon VSL – Intel Library MRG32k3a ICC – Intel C/C++ compiler VC++ – Microsoft Visual C++ GPU CPU-ICC CPU-VC++ VSL-ICC VSL-VC++ Samples/ sec 3.00E+09 3.46E+07 4.77E+07 9.35E+07 9.32E+07

MRG32k3a: GPU benchmarking – single precision Note: for double precision all sequences were identical For single precision GPU and CPU identical GPU and VSL differ max abs err 5.96E-008 Which output preferred? use statistical tests of randomness GPU CPU-ICC CPU-VC++ VSL-ICC VSL-VC++ Samples/ sec 3.49E+09 3.58E+07 5.24E+07 1.02E+08 9.75E+07

LIBOR Market Model on GPU Equally weighted portfolio of 15 swaptions each with same maturity, but different lengths and different strikes

Numerical Libraries for GPUs The problem The time-consuming work of writing basic numerical components should not be repeated The general user should not need to spend many days writing each application The solution Standard numerical components should be available as libraries for GPUs

Example program: generate random numbers on GPU ... // Allocate memory on Host host_array = (double *)calloc(N,sizeof(double)); // Allocate memory on GPU cudaMalloc((void **)&device_array, sizeof(double)*N); // Call GPU functions // Initialise random number generator nag_gpu_mrg32k3a_init(V1, V2, offset); // Generate random numbers nag_gpu_mrg32k3a_uniform(nb, nt, np, device_array); // Read back GPU results to host cudaMemcpy(host_array,gpu_array,sizeof(double)*N,cudaMemcpyDeviceToHost); ...

Example program – kernel function __global__ void mrg32k3a_kernel(int np, FP *d_P){ unsigned int v1[3], v2[3]; int n, i0; FP x, x2 = nanf(""); // initialisation for first point nag_gpu_mrg32k3a_stream_init(v1, v2, np); // now do points i0 = threadIdx.x + np*blockDim.x*blockIdx.x; for (n=0; n<np; n++) { nag_gpu_mrg32k3a_next_uniform(v1, v2, x); } d_P[i0] = x; i0 += blockDim.x; }

Library issues: Auto-tuning Performance affected by mapping of algorithm to GPU via threads, blocks and warps Implement a code generator to produce variants using the relevant parameters Determine optimal performance Li, Dongarra & Tomov (2009)

Working with Fixed Income Research & Strategies Team (FIRST) NAG mrg32k3a works well in BNP Paribas CUDA “Local Vol Monte-Carlo” Passes rigorous statistical tests for randomness properties (Diehard, Dieharder,TestU01) Performance good Being able to match the GPU random numbers with the CPU version of mrg32k3a has been very valuable for establishing validity of output Early Success with BNP Paribas

BNP Paribas Results – local vol example

“ The NAG GPU libraries are helping us enormously by providing us with fast, good quality algorithms. This has let us concentrate on our models and deliver GPGPU based pricing much more quickly.” And with Bank of America Merrill Lynch

“ Thank you for the GPU code, we have achieved speed ups of x120” In a simple uncorrelated loss simulation: Number of simulations 50,000 Time taken in seconds 2.373606 Simulations per second 21065 Simulated default rate 311.8472 Theoretical default rate 311.9125 24 trillion numbers in 6 hours “ A N Other Tier 1” Risk Group

NAG routines for GPUs – 1 Currently available Random Number Generator (L’Ecuyer mrg32k3a) Uniform distribution Normal distribution Exponential distribution Gamma distribution Sobol sequence for Quasi-Monte Carlo (to 19,000 dimensions) Brownian Bridge

NAG routines for GPUs – 2 Future plans Random Number Generator – Mersenne Twister Linear algebra components for PDE option pricing methods Time series analysis – wavelets ...

Summary GPUs offer high performance computing for specific massively parallel algorithms such as Monte Carlo simulations GPUs are lower cost and require less power than corresponding CPU configurations Numerical libraries for GPUs will make these an important computing resource Higher order methods for GPUs being considered

Acknowledgments Mike Giles (Mathematical Institute, University of Oxford) – algorithmic input Technology Strategy Board through Knowledge Transfer Partnership with Smith Institute NVIDIA for technical support and supply of Tesla C1060 and Quadro FX 5800 See www.nag.co.uk/numeric/GPUs/ Contact: [email_address]

Monte Carlo on GPUs

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