Analysis of Grid-based Bermudian

1
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 GAIKWAD
• INRIA Sophia-Antipolis
• France

2
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

Underlying price trajectory
Optimal exercise boundary
Exercise point
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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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Optimal Exercise Boundary Approach (3)Parallel
approach for high-dimensional option (I.Muni
Toke, 2006)

• 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

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Optimal Exercise Boundary Approach (4)Numerical
experimentations

• Benchmarks

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

• 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

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Optimal Exercise Boundary Approach (6)Some
others observations

• Number of iterations of the GLP points
  convergence

• Waiting period between each asset computation

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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.jacqu
  es/

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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(yx) 0
• Therefore the boundary is characterized by a
  function F(x) such that
• F(x) gt 0 whenever E(yx) gt 0 (hold/wait option)
• F(x) lt 0 whenever E(yx) lt 0 exercise

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Continuation Values Classification
(2)Description of the sequential algorithm

• 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

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

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

Training dataset
Objective function of the classification
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Continuation Values Classification (4)The
characterizations of the boundary during 12
opportunities
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Continuation Values Classification (5)Toward a
parallel classification

• 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

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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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Continuation Values Classification (7)

• The classification phase
• Support Vector Machine

• 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_at_sophia.inria.fr
  Abhijeet.Gaikwad_at_sophia.inria.fr