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Analysis of Grid-based Bermudian – American Option Pricing Algorithms

(presented in MCM2007)

Applications of Continuation Values Classification AndOptimal Exercise Boundary Computation

Viet Dung DOAN

Mireille BOSSY

Francoise BAUDE

Ian STOKES-REES

Abhijeet GAIKWADINRIA Sophia-Antipolis

France

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Outline

PicsouGrid current state

Building the optimal exercise boundary (Ibanez and Zapatero 2004)

Continuation exercise values classification (Picazo 2004)

Conclusion

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PicsouGrid

Current state :

Autonomy, scalability, and efficient distribution of tasks

for complex option pricing algorithms

Master-Slave Architecture is incorporated

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Optimal Exercise Boundary Approach (1)Overview

Proposed by Ibanez and Zapatero in 2002 Time backward computing Base on the property that at each opportunity date:

There is always an exercise boundary i.e. exercise when the underlying price reaches the boundary

The boundary is a point (1 dimension) and a curve (high-dimension) where the exercise values match the continuation values

Estimate the optimal exercise boundary F(X) at each opportunity through a regression. F(X) is a quadratic or cubic polynomial

Advantages: Provides the optimal exercise rule Possible to compute the Greeks Possible to use straightforward Monte Carlo

simulation

Optimal exercise boundary

Exercise point

Underlying price

trajectory

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Optimal Exercise Boundary Approach (2)Description of the sequential algorithm

Maximum basket of d underlying American put Step 1 : compute the exercise boundary

At each opportunity, make a grid of J “good” lattice points

Compute the optimal boundary points Need N2 paths of simulations Need n iterations to converge

Regression Compute for all opportunity date

Step 2 : simulate a straightforward Monte Carlo simulation (easy to parallelize) N = nbMC

Complexity

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Distributed approach: For step 1

Divide the computation of J optimal boundary points by J independent tasks

Do the sequential regression on the master node

For step 2 Divide N paths by nb1 small

independent packets Breakdown in computational time

Optimal Exercise Boundary Approach (3)Parallel approach for high-dimensional option (I.Muni Toke, 2006)

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Optimal Exercise Boundary Approach (4)Numerical experimentations

S Sigma Premia Method LS OEB36 20% 4.483 4.472 +/- 0.029 4.477 +/- 0.01736 40% 7.106 7.114 +/- 0.020 7.091 +/- 0.03744 20% 1.118 1.101 +/- 0.007 1.114 +/- 0.01244 40% 3.954 3.945 +/- 0.017 3.944 +/- 0.032

K = 40, T = 1, r = 0.06, nbMC = 100000K = 40, T = 1, r = 0.06, nbMC = 1000000, nb time step = 360

Benchmarks

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Step 1: Estimate the optimal exercise boundary Use a grid of 256 points Simulate 5000 paths Use 360 time steps

Sequential regression : on the master node

Step 2: 1000000 Monte Carlo straightforward Use 100 packets

Optimal Exercise Boundary Approach (5)First benchmarks for the parallel approach

9 8 7 6 5 4 3 2 10

100

200

300

400

500

600

700

800

Speedup

Sequential 1 PC

Parallel 4 PCs

Opportunity

Tim

e (

s)

Speedup

0

20

40

60

80

100

120

140

1 P C 2 P Cs 4 P Cs

Number of PCT

ime

(s)

NbMC = 100000

NbMC = 1000000

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Number of iterations of the GLP points convergence

Waiting period between each asset computation

Optimal Exercise Boundary Approach (6)Some others observations

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Optimal Exercise Boundary Approach (7)Some others observations

Ssj package Piere L'Ecuyer

Normal Optimal Quantification http://perso-math.univ-mlv.fr/users/printems.jacques/

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Continuation Values Classification (1)Overview

Proposed by Picazo in 2004 Time backward computing Base on the property that at each opportunity date:

Classify the continuation values to have the characterization of the waiting zone and the exercise zone

At a fixed time t, define the value of continuation y at the current underlying assets x as : y = Avg. discounted payoff – value of exercise(of the

sampling paths starting from x) The exercise boundary is given by the set of points x such

that E(y|x) = 0 Therefore the boundary is characterized by a function

F(x) such that: F(x) > 0 whenever E(y|x) > 0 (hold/wait option) F(x) < 0 whenever E(y|x) < 0 exercise

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Standard American and basket American Asian put. Step 1 : Compute the characterization of the

boundary at each opportunity date Simulate N1 paths of the underlying, denote xi

with i = (1,.., N1 ) With each xi, simulate N2 paths of simulations to

compute the difference between the exercise and the continuation values, denote yi.

Classification with the training set (xi,yi) Need n iterations to converge

Step 2 : simulate a straightforward Monte Carlo simulation (easy to parallelize) N = nbMC

Complexity

Continuation Values Classification (2)Description of the sequential algorithm

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Characterization of the boundary for an American sample option at a given opportunity.

Objective function of the classification

Training dataset

Continuation Values Classification (2)An illustration of the classification phase

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Continuation Values Classification (4)The characterizations of the boundary during 12 opportunities

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Distributed approach For step 1

Divide N1 paths by nb small independents packets

Parallelize the classification process

Discuss more later For step 2

Divide N paths by nb1 small independents packets

Breakdown computational time

Computational overhead for Sequential Classification: about 40% of the total time

Continuation Values Classification (5)Toward a parallel classification

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Continuation Values Classification (6)First benchmarks

Current state Implementation of the proposed scheme Investigate techniques for parallelizing the classification phase

e.g. transition from boosting algorithm to Support Vector Machine based approach

Preliminary results Sequential standard American put option

N1 = 5000, N2 = 500 Time to generate the training set : 13 (s) Time for the sequential classification : 1200 (s)

Need to improve the implementation and the benchmarks Time for the final 1000000 Monte Carlo straightforward simulations :

40 (s)

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The classification phase

Support Vector Machine

Continuation Values Classification (7)

Parallelizing the classification phase

Application of Parallel Support Vector Machine

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Continuation Values Classification (7)

Preliminary simulation for the parallel classification using SVM

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Conclusion

PicsouGrid: Parallel European option pricing algorithms (standard,

barrier, basket) Results published in

2nd E-Science, Netherlands 12/2006 6th ISGC, Taiwan 3/2007

Parallel American option pricing algorithms Sequential implementation Parallel approaches and benchmarks

Further results to be published in Mathematics and Computers in Simulation journal

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Thank you

Questions?

Project links :

Sub-project PicsouGrid (in English) (secure-email for access)

https://gforge.inria.fr/project/picsougrid/

Contact us:

viet_dung.doan@sophia.inria.fr Abhijeet.Gaikwad@sophia.inria.fr