May 29, 2003 74.757 Final Presentation Sajib Barua1 Development of a Parallel Fast Fourier Transform...

May 29, 2003 74.757 Final Presentation Sajib Barua

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Development of a Parallel Fast Fourier Transform Algorithm for Derivative Pricing Using MPI

Sajib BaruaCS. Dept. University of Manitoba

Course Professor: Dr. Ruppa K. Thulasiram


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Outline

Introduction Background and related work Problem Statement Solution Strategy and Implementation Experimental Platform Results Conclusions and Future Work


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Introduction

Derivatives are knows as options Two basic types of option

– Call Option

– Put Option

Two types of option depending on the exercise time:– American Option

– European Option Determining the optimum exercise policy is a key issue in pricing American option to achieve maximum profit.


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Background and Related Work


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Methods used for option pricing

Binomial Lattice method Monte Carlo approach Finite-Difference method Fast Fourier transform Finite-element method

FFT is used recently to study multifactor model for option pricing problems and

FFT is highly suitable for parallel computing.


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Parallel FFT Algorithm

Most Common parallel FFT Algorithm:– Cooley-Tukey Algorithm– Gentleman-Sande Algorithm

Two Schemes for FFT Algorithm– Recursive Scheme– Iterative Scheme


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Problem Statement

There are two distinct features in this project: – mathematical treatment of the option pricing problem – and its computation

Design, development and implementation of FFT algorithm with improved data locality.

Use of this algorithm in option pricing problem with appropriate mapping


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Challenges

Computation:– In parallel system two types of latencies are incurred

• Communication latency• Synchronization latency

– FFT is inherently a synchronous algorithm.– Communication latency is circumvented by providing good

data locality. Mathematical treatment:

– Applying appropriate risk-neutral probability for finishing in-the-money

– obtaining the Fourier Transform of the call price function with a known risk-neutral density function

– etc


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Solution Strategy and Implementation

Parallel FFT AlgorithmFFT in option Pricing

A New Parallel FFT Algorithm


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How FFT Equation can be parallized?

A sequence of X of length n– X = <X[0], X[1] …. X[n-1]>

The DFT of this sequence is – Y = <Y[0], Y[1] …. Y[n-1]>


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Contd.

Overall sequential complexity is Θ(n log n) FFT calculation can be parallelized A parallel FFT calculation requires log N number of stages A parallel FFT algorithm with block data distribution of N data

points with P processors requires – (log N – log P) number of iterations for local calculation called local

algorithm– log P number of stages for communication among processors called

remote algorithm.


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Fourier Analysis in Option Pricing

Call value is a function of the log of its strike price. This function is not square integrable. Its Fourier transform can not be

calculated. To get the analytically solvable Fourier transform, this function is

multiplied by an exponential to make it square integrable. The call price function is

ΨT (v) is the Fourier transform of the call price


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Contd. If M = eαk/2Π and ω = ei then

The discrete form of this equation will be

This DFT equation can be parallelized. Call price can be calculated accurately and efficiently

from a good parallel FFT algorithm with an efficient data distribution.


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A New Parallel FFT Algorithm

Improving Data Locality Approach


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Cooley-Tukey Algorithm

x0 w80

w80

Iteration 1Iteration 0 Iteration 2

w

a

b

a+bw

a-bw

w

Butterfly Operation

x4

x2

x6

x1

x5

x3

x7

w80

w80

w80

w82

w80

w82

w80

w81

w82

w83

y0

y1

y2

y3

y4

y5

y6

y7

P0

P1


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Parallel FFT Algorithm With Improved Data Locality

0

1

2

3

4

5

6

7

0

2

1

3

4

6

5

7

0

4

1

5

2

6

3

7

Iteration 0

Iteration 1

Iteration 2

P0

P1

Iteration No = i1st node index = j2nd node index = 2i + j


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Parallel FFT Algorithm With 16 number of nodes

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0

2

1

3

4

6

5

7

8

10

9

11

12

14

13

15

0

4

1

5

2

6

3

7

8

12

9

13

10

14

11

15

0

8

1

9

2

10

3

11

4

12

5

13

6

14

7

15

p0

p1

p2

p3

Iteration 0 Iteration 1 Iteration 2 Iteration 3


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Experimental Platform…


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Programming Language– Unix C– Message Passing Interface (MPI)

Single Processor Bewoulf Cluster (Distributed Architecture)

– 2 no of processors– 4 no of processors– 8 no of processors

Experimental Testbed or Environment


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Experimental Results…


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Performance result with varying processor size

0

2

4

6

8

10

12

14

No of Processor

Exe

cutio

n tim

e(in

sec

)

2^12

2^13

2^15

2^16

2^17

2^18

2^19

2^20

2^12 0.02535093 0.02060401 0.01717699 0.01517296

2^13 0.05539596 0.04329801 0.02915406 0.02441597

2^15 0.2773631 0.183461 0.1334109 0.10645

2^16 0.615428 0.392648 0.265748 0.2157681

2^17 1.332534 0.915802 0.5701129 0.4130991

2^18 2.87687 1.789844 1.19493 0.8592221

2^19 6.18839 3.80007 2.502013 1.786862

2^20 11.83851 7.009985 3.72315

1 2 4 8


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Performance result with varying Data size

0

2

4

6

8

10

12

14

2^12

2^13

2^15

2^16

2^17

2^18

2^19

2^20

Data Size

Ex

ec

uti

on

tim

e (

in S

ec

)

1 Processor2 Processors4 Processors8 Processors


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Comparison of Cooley-Tukey and Parallel FFT Algorithm

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

2^12 2^13 2^15 2^16 2^17 2^18 2^19 2^20

Data Size

Ex

ec

uti

on

Tim

e (

in S

ec

)

Parallel FFT

Cooley-Tukey

With 8 no of processors


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Conclusions & Future Work


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1. A New Parallel FFT algorithm(with improved data locality)Reduce communication overheadPerformance is more than 15% over Cooley-Tukey

Algorithm with larger data sets.

2. Mathematical Contribution– Applying appropriate risk neutral density function– Specification of dampening co-efficient – Mathematical derivation of the dampened call price

function and its Fourier Transform

Contribution:


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Rapid solution and quick information processing

Fast algorithm with high performance computing capability

Better performance and accuracy Market Forecasting FFT is a data- and communication-intensive

Problem. Implementation of FFT is a serious challenge in scientific-computing.

Contd.


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Future Work

Better risk neutral density function For now the results are presented only in terms of the performance improvement of the FFT algorithm.

The results need to be presented in terms of the option prices Implementation of our algorithm on shared-memory architecture (OpenMP) More efficient node distribution technique and tuning parallel FFT algorithm.


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Questions…?

May 29, 2003 74.757 Final Presentation Sajib Barua1 Development of a Parallel Fast Fourier Transform...

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