Simulation of Conditional Value-at-Risk via Parallelism on Graphic Processing Units
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1 Simulation of Conditional Value-at-Risk via Parallelism on Graphic Processing Units Hai Lan Dept. of Management Science Shanghai Jiao Tong University Shanghai, 252, China July 22, 22 H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic JulyProcessing 22, 22 Units / 23
2 Outline of Topics Motivation Sequential procedure of CVaR via nested simulation Paralleled nested simulation with screening Experimental performance Conclusion & Question H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic JulyProcessing 22, 22 Units 2 / 23
3 Motivations Some simulation procedures take days, weeks or even months to run. Traditional supercomputers are inaccessible for most researchers and practitioners. New development of high performance computing catches out attention: cloud computing: heterogeneous, massive storage, massive computing parallel computing on GPUs: homogeneous, data light, moderate computing Characteristics of simulation in financial engineering: data security,moderate size computing. H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic JulyProcessing 22, 22 Units 3 / 23
4 GPU Computing Characteristics of GPU Computing: same program on different data, which makes it attractive for embarrassingly paralleled simulation work. Reported work shows 2% 8% faster for sorting (key only) 3 5 times faster for random number generating almost faster for Monte Carlo simulation of Black Shorel model H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic JulyProcessing 22, 22 Units 4 / 23
5 GPU Computing Characteristics of GPU Computing: same program on different data, which makes it attractive for embarrassingly paralleled simulation work. Reported work shows 2% 8% faster for sorting (key only) 3 5 times faster for random number generating almost faster for Monte Carlo simulation of Black Shorel model How about its effectiveness on complex simulation such as nested simulation? H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic JulyProcessing 22, 22 Units 4 / 23
6 Risk Measurement V is the dollar change of a portfolio over T time. Risk Measure: from distribution F V (especially the loss part) one number. H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic JulyProcessing 22, 22 Units 5 / 23
7 Risk Measurement V is the dollar change of a portfolio over T time. Risk Measure: from distribution F V (especially the loss part) one number. F V the right continuous inverse of F V. H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic JulyProcessing 22, 22 Units 5 / 23
8 Risk Measurement V is the dollar change of a portfolio over T time. Risk Measure: from distribution F V (especially the loss part) one number. F V the right continuous inverse of F V. VaR p = F V (p) is the p-quantile of F V. H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic JulyProcessing 22, 22 Units 5 / 23
9 Risk Measurement V is the dollar change of a portfolio over T time. Risk Measure: from distribution F V (especially the loss part) one number. F V the right continuous inverse of F V. VaR p = F V (p) is the p-quantile of F V. CVaR p = p F V (s)ds. H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic JulyProcessing 22, 22 Units 5 / 23
10 Nested Simulation Goal: provide a framework to estimate the market risk of an investment portfolio H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic JulyProcessing 22, 22 Units 6 / 23
11 Nested Simulation Goal: provide a framework to estimate the market risk of an investment portfolio outer level: generating Z(T) from market scenario model in the real world probability measure. underlying stock price S i (T),i =,,k. H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic JulyProcessing 22, 22 Units 6 / 23
12 Nested Simulation Goal: provide a framework to estimate the market risk of an investment portfolio outer level: generating Z(T) from market scenario model in the real world probability measure. underlying stock price S i (T),i =,,k. inner level: estimating V conditionally on Z(T) by Monte Carlo Simulation in risk neutral probability measure. generate the underlying stock prices S(T t U) S i (T), i =,,k H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic JulyProcessing 22, 22 Units 6 / 23
13 Plain Nested Simulation Simulate scenarios Z,,Z k. Unknown true value V i = E[X Z = Z i ]. Simulate payoffs X i,,x in given Z i. Estimate V i by (X i + +X in )/n. H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic JulyProcessing 22, 22 Units 7 / 23
14 Nested Simulation with Screening H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic JulyProcessing 22, 22 Units 8 / 23
15 Screening and Restarting Motivation: Only the loss part is interesting. Identified as subset selection with uncommon variance H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic JulyProcessing 22, 22 Units 9 / 23
16 Screening and Restarting Motivation: Only the loss part is interesting. Identified as subset selection with uncommon variance The comparison of Scenario i and j is taken as a hypothesis test: H o : V i > V j v.s.h : V i V j we say, j is beaten by i if H o is true. Screening can be taken as multiple comparisons of each pairs of market scenarios. A market scenario will survive from screening only when it is beaten less than kp times of such comparisons. H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic JulyProcessing 22, 22 Units 9 / 23
17 Screening and Restarting Motivation: Only the loss part is interesting. Identified as subset selection with uncommon variance The comparison of Scenario i and j is taken as a hypothesis test: H o : V i > V j v.s.h : V i V j we say, j is beaten by i if H o is true. Screening can be taken as multiple comparisons of each pairs of market scenarios. A market scenario will survive from screening only when it is beaten less than kp times of such comparisons. Restarting is used to avoid the selection bias inherited from screening. H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic JulyProcessing 22, 22 Units 9 / 23
18 Embarrassingly Paralleled: Generating Scenarios and st Stage Conditional Payoffs Random Number Generator on GPU: long period, multiple streams and sub-streams. 32 bits computation only combined multiple recursive generator H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic July Processing 22, 22 Units / 23
19 Embarrassingly Paralleled: Generating Scenarios and st Stage Conditional Payoffs Random Number Generator on GPU: long period, multiple streams and sub-streams. 32 bits computation only combined multiple recursive generator Gaussian Random Variable: Muller-Box method to generate normal variables H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic July Processing 22, 22 Units / 23
20 Embarrassingly Paralleled: Generating Scenarios and st Stage Conditional Payoffs Random Number Generator on GPU: long period, multiple streams and sub-streams. 32 bits computation only combined multiple recursive generator Gaussian Random Variable: Muller-Box method to generate normal variables Each thread equipped with one random number generator starting with the same/different seeds when common/independent random numbers applied. H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic July Processing 22, 22 Units / 23
21 Embarrassingly Paralleled: Generating Scenarios and st Stage Conditional Payoffs Random Number Generator on GPU: long period, multiple streams and sub-streams. 32 bits computation only combined multiple recursive generator Gaussian Random Variable: Muller-Box method to generate normal variables Each thread equipped with one random number generator starting with the same/different seeds when common/independent random numbers applied. Experiments show 5-6 times faster than the sequential procedure. H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic July Processing 22, 22 Units / 23
22 Parallel Screening : Strategy Sequential procedure: for any scenario i =,...,k, compute at least kp variances of (X il X jl ) 2 n n l= (X il X jl ) n l= (X il X jl ) difference Sij 2 = n l= n (n ) scenario i is screened out if it is beaten no less than kp times. thus, a lot of data communication has to be done. To save data communication, we adopt a divide-and-conquer strategy. H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic July Processing 22, 22 Units / 23
23 Parallel Screening : Strategy Sequential procedure: for any scenario i =,...,k, compute at least kp variances of (X il X jl ) 2 n n l= (X il X jl ) n l= (X il X jl ) difference Sij 2 = n l= n (n ) scenario i is screened out if it is beaten no less than kp times. thus, a lot of data communication has to be done. To save data communication, we adopt a divide-and-conquer strategy. randomly group the total k market scenario into G groups, each GPU processes one group. apply screening procedure within each group to eliminate some unimportant market scenarios. collect remaining scenarios from each GPU and perform screening again. H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic July Processing 22, 22 Units / 23
24 Parallel Screening 2: Flow Chart Thead invoke GPU Thead 3 invoke GPU 3 Simulate k/g scenarios Simulate conditional payoffs for each scenario estimated conditional portfolio values CPU part GPU part Sorting Yes More comparisons need? Group Comparison Comparison Results Scenarios retained after the first stage on GPUs and their conditional payoffs together with comparison results Partition all retained scenario after the first stage screeninginto small chunks Create G threads to process the chunks according to the sequential screening procedure Dynamic Schedule of OpenMP applied Screen resut: scenario set I H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic July Processing 22, 22 Units 2 / 23
25 Parallel Screening 3: Implementation on GPU scenarios payoffs scenarios payoffs Thread i Thread j 6X6 6X6 Block i H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic Processing Units July 22, 22 3 / 23
26 Parallel Screening 3: Implementation on GPU scenarios payoffs scenarios payoffs Thread i Thread j 6X6 6X6 Block i Scheme is about 2 times faster than CPU only, scheme 2 is 3-5 times faster. H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic Processing Units July 22, 22 3 / 23
27 Second Stage Sampling Scenaros 2 3 Sample size Threads GPU GPU GPU 2 GPU 3 H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic July Processing 22, 22 Units 4 / 23
28 Second Stage Sampling Scenaros 2 3 Sample size Threads GPU GPU GPU 2 GPU 3 balance the uneven load among different GPUs and threads experiments show -4 times faster H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic July Processing 22, 22 Units 4 / 23
29 Experimental Setting hardware: one 3.G Hz Core 2 Duo CPU, four GeForce 98 GPUs, 6 GB memory. software: gcc on Ubuntu 64 bits version. testing portfolios: shorting one put option and a portfolio with 8 options. number of replications: and more. H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic July Processing 22, 22 Units 5 / 23
30 Speedup of Sampling serial procedure parallel procedure Average Running Time (s) Number of Simulated Payoffs at the 2nd Stage C (millions of simulated payoffs) H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic July Processing 22, 22 Units 6 / 23
31 Speedup of Computing the CI serial procedure parallel procedure Average Running Time (s)... Number of Scenarios Retained after Screening I H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic July Processing 22, 22 Units 7 / 23
32 Speedup in Running Time serial procedure parallel procedure Average Running Time (s) Computational Budget C (millions of simulated payoffs) H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic July Processing 22, 22 Units 8 / 23
33 Conclusion H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic July Processing 22, 22 Units 9 / 23
34 Conclusion For complex simulation procedure, like nested simulation with screening, GPU computing can still help to improve the overall performance. H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic July Processing 22, 22 Units 9 / 23
35 Conclusion For complex simulation procedure, like nested simulation with screening, GPU computing can still help to improve the overall performance. In a parallel computing environment, simulation procedure, like ranking and selection (screening here) need to be trimmed or even redesigned to achieve really high performance. H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic July Processing 22, 22 Units 9 / 23
36 Conclusion For complex simulation procedure, like nested simulation with screening, GPU computing can still help to improve the overall performance. In a parallel computing environment, simulation procedure, like ranking and selection (screening here) need to be trimmed or even redesigned to achieve really high performance. Parallelism on GPU is very similar as that on CPUs and distributed computing, data communication and synchronization are the bottlenecks of scalability. H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic July Processing 22, 22 Units 9 / 23
37 Conclusion For complex simulation procedure, like nested simulation with screening, GPU computing can still help to improve the overall performance. In a parallel computing environment, simulation procedure, like ranking and selection (screening here) need to be trimmed or even redesigned to achieve really high performance. Parallelism on GPU is very similar as that on CPUs and distributed computing, data communication and synchronization are the bottlenecks of scalability. Research opportunities exist in providing standard scientific computing package like BLAST on GPU computing. H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic July Processing 22, 22 Units 9 / 23
38 H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic July Processing 22, 22 Units 2 / 23
39 Multi-GPU and OpenMP OpenMP shared memory(fast cached) high performance yet limited scalability branching adaptability CUDA on GPU many cores shared memory (cached and un-cached) fast on-board memory (register, local, share, global, const,texture) hardware thread management (zero overhead) 4 GPUs allowed on one computer. Affordable personal supercomputer. H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic July Processing 22, 22 Units 2 / 23
40 CUDA Program Model 6 successive threads is called half-wrap, processed simultaneously on one SP. no diversity access successive memory in one command H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic July Processing 22, 22 Units 22 / 23
41 Memory Model H. Lan (SJTU) Simulation of Conditional Value-at-Risk via Parallelism on Graphic July Processing 22, 22 Units 23 / 23
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