1 Analysis of Grid-based Bermudian – American Option Pricing Algorithms (presented in MCM2007)...
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Transcript of 1 Analysis of Grid-based Bermudian – American Option Pricing Algorithms (presented in MCM2007)...
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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: