A Scalable Dataflow Implementation of Curran's Approximation Algorithm
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This document describes a dataflow implementation of Curran's approximation algorithm for pricing Asian options. The implementation computes the value at risk of a portfolio containing Asian options. It supports an arbitrary number of averaging points and achieves high precision using fixed-point arithmetic. Experimental results show that a single Maxeler dataflow engine can price portfolios over 10 times faster than a 48-core CPU. Further optimization including multi-DFE processing and porting to newer FPGA hardware could improve energy efficiency and performance.
![16
Speedups and Energy Efficiencies
s CPU 48 Cores Single DFE
DataSet30
DataSet780
Tabella 1-1
DataSet30 DataSet780
PU 1 Core 21 400
PU 24 Cores 7 36
PU 48 Cores 8 27
ingle DFE 5 6
s CPU 48 Cores Single DFE
DataSet30
DataSet780
Tabella 1-1-1
DataSet30 DataSet780
PU 1 Core 11 30
PU 24 Cores 7 36
PU 48 Cores 8 27
ingle DFE 11 12
RunTime[s]
1
100
10000
CPU 1 Core CPU 24 Cores CPU 48 Cores Single DFE 8 DFEs
2,181
11,99
238,564240,017
3789,277
1,461,25
10,3310,49
158,81
DataSet30
DataSet780
SocketEnergy[Wh]
1
10
100
CPU 1 Core CPU 24 Cores CPU 48 Cores Single DFE 8 DFEs
8
6
27
36
400
55
87
21
DataSet30
DataSet780](https://image.slidesharecdn.com/asianoption5minpdf-170713144233/85/A-Scalable-Dataflow-Implementation-of-Curran-s-Approximation-Algorithm-16-320.jpg)
A Scalable Dataflow Implementation of Curran's Approximation Algorithm
- 2. 2 Contributions Thanks to the Maxeler Tools productivity features, we aimed to create an efficient parametric design which: 1. Computes Value at Risk (VaR) of a portfolio of Asian Options based on Curran’s approximation method 2. Supports arbitrary number of averaging points
- 3. 3 • Black-Scholes model: option payoff variable no closed-form representation for its probability distribution • Curran's Approximation: expected option payoff conditional on the geometric mean of the prices at averaging points • Curran’s algorithm characterised by: 1. High degree of precision 2. Computational intensive • High number of invocations to the Normal Cumulative Distribution Function (NCDF), exponentials and logarithms • Highly parallel computation, completely independent variables are calculated • Evaluation of one portfolio takes from one to many hours Curran’s Approximation for Asian Option Pricing
- 5. 5 • DFE input: N x N x #optionFields • Initialisation K1, intermediate K3 and finalisation K5 kernels do not require multi-cycling • Summation kernels K2 and K4 unroll k summand computations • DFE output: N S Data Flow Architecture Single DFE O S Infiniband link Infiniband link
- 6. 6 • DFE input: N x (N / #DFEs) x #optionFields • Initialisation K1, intermediate K3 and finalisation K5 kernels do not require multi-cycling • Summation kernels K2 and K4 unroll k summand computations • DFE output: N / #DFEsS Data Flow Architecture Multi-DFEs O S DFEs Infiniband link Infiniband link
- 7. 7 • Two test data sets: DataSet30 and DataSet780 • Precision analysis performed exploiting fixed-point and floating-point data types, one per build, for the entire design • DFE resource usage analysis for the same data types • Dynamic ranges analysis Experiments
- 9. 9 • 54 and 64 bits fixed-point data representation leads to less resources than in case of a floating point (through 48, 54 and 64 bits) DFE Resource Analysis Results
- 10. 10 • Assuming worst case (linear) scalability resource utilisation with parameter k • With Fix54(16,38) maximum value of the unrolling factor k=3 • Dynamic range analysis aiming to increase the unrolling factor Dynamic Ranges Analysis Results
- 11. 11 • Assuming worst case (linear) scalability resource utilisation with parameter k • With Fix54(16,38) maximum value of the unrolling factor k=3 • Dynamic range analysis aiming to increase the unrolling factor k=3 Dynamic Ranges Analysis Results
- 12. 12 • Assuming worst case (linear) scalability resource utilisation with parameter k • With Fix54(16,38) maximum value of the unrolling factor k=3 • Dynamic range analysis aiming to increase the unrolling factor K1 K2 K2 K2 K2 K3 K4 K4 K4 K4 K5 … … k=3 Dynamic Ranges Analysis Results
- 13. 13 • Assuming worst case (linear) scalability resource utilisation with parameter k • With Fix54(16,38) maximum value of the unrolling factor k=3 • Dynamic range analysis aiming to increase the unrolling factor K1 K2 K2 K2 K2 K3 K4 K4 K4 K4 K5 … … Float(11,32) Float(11,32) Float(11,32) k=3 Dynamic Ranges Analysis Results
- 14. 14 • Assuming worst case (linear) scalability resource utilisation with parameter k • With Fix54(16,38) maximum value of the unrolling factor k=3 • Dynamic range analysis aiming to increase the unrolling factor K1 K2 K2 K2 K2 K3 K4 K4 K4 K4 K5 … … Float(11,32) Float(11,32) Float(11,32) Fix48(14,34), Fix48(8,40), Fix64(20,40), Fix54(21,33) and Fix32(32,0) Fix48(14,34), Fix48(8,40), Fix64(20,40), Fix54(21,33) and Fix32(32,0) k=3 Dynamic Ranges Analysis Results
- 15. 15 • Assuming worst case (linear) scalability resource utilisation with parameter k • With Fix54(16,38) maximum value of the unrolling factor k=3 • Dynamic range analysis aiming to increase the unrolling factor K1 K2 K2 K2 K2 K3 K4 K4 K4 K4 K5 … … Float(11,32) Float(11,32) Float(11,32) Fix48(14,34), Fix48(8,40), Fix64(20,40), Fix54(21,33) and Fix32(32,0) Fix48(14,34), Fix48(8,40), Fix64(20,40), Fix54(21,33) and Fix32(32,0) k=3 k=15 Dynamic Ranges Analysis Results
- 16. 16 Speedups and Energy Efficiencies s CPU 48 Cores Single DFE DataSet30 DataSet780 Tabella 1-1 DataSet30 DataSet780 PU 1 Core 21 400 PU 24 Cores 7 36 PU 48 Cores 8 27 ingle DFE 5 6 s CPU 48 Cores Single DFE DataSet30 DataSet780 Tabella 1-1-1 DataSet30 DataSet780 PU 1 Core 11 30 PU 24 Cores 7 36 PU 48 Cores 8 27 ingle DFE 11 12 RunTime[s] 1 100 10000 CPU 1 Core CPU 24 Cores CPU 48 Cores Single DFE 8 DFEs 2,181 11,99 238,564240,017 3789,277 1,461,25 10,3310,49 158,81 DataSet30 DataSet780 SocketEnergy[Wh] 1 10 100 CPU 1 Core CPU 24 Cores CPU 48 Cores Single DFE 8 DFEs 8 6 27 36 400 55 87 21 DataSet30 DataSet780
- 17. 17 • An example of large class of HPC application with numerical solvers used as case study in EXTRA European Project • Improvements in runtime and energy utilisation offer a compelling advantage to financial institutions that want to reduce both option pricing time and energy usage • DFE: 1. Multi-DFE energy efficiency in progress 2. Porting to the new Maxeler MAX5 based on Xilinx Virtex UltraScale+ • CPU: 1. More improvements to be done Conclusions and Future Works