1. 1 Analysis of Grid-based Bermudian – American Option Pricing Algorithms (presented in MCM2007) Applications of Continuation Values Classification And Optimal Exercise Boundary Computation Viet Dung DOAN Mireille BOSSY Francoise BAUDE Ian STOKES-REES Abhijeet GAIKWAD INRIA Sophia-Antipolis France

2. 2 Outline  PicsouGrid current state  Building the optimal exercise boundary (Ibanez and Zapatero 2004)  Continuation exercise values classification (Picazo 2004)  Conclusion

3. 3 PicsouGrid  Current state :  Autonomy, scalability, and efficient distribution of tasks for complex option pricing algorithms  Master-Slave Architecture is incorporated

4. 4 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

5. 5 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 N 2 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

6. 6  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 nb 1 small independent packets  Breakdown in computational time Optimal Exercise Boundary Approach (3) Parallel approach for high-dimensional option (I.Muni Toke, 2006)

7. 7 Optimal Exercise Boundary Approach (4) Numerical experimentations K = 40, T = 1, r = 0.06, nbMC = 1000000, nb time step = 360  Benchmarks

8. 8  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. 9  Number of iterations of the GLP points convergence  Waiting period between each asset computation Optimal Exercise Boundary Approach (6) Some others observations

10. 10 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/

11. 11 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

12. 12  Standard American and basket American Asian put.  Step 1 : Compute the characterization of the boundary at each opportunity date  Simulate N 1 paths of the underlying, denote x i with i = (1,.., N 1 )  With each x i, simulate N 2 paths of simulations to compute the difference between the exercise and the continuation values, denote y i.  Classification with the training set (x i,y i )  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

13. 13  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

14. 14 Continuation Values Classification (4) The characterizations of the boundary during 12 opportunities

15. 15  Distributed approach  For step 1  Divide N 1 paths by nb small independents packets  Parallelize the classification process  Discuss more later  For step 2  Divide N paths by nb 1 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

16. 16 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  N 1 = 5000, N 2 = 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)

17. 17  The classification phase  Support Vector Machine Continuation Values Classification (7)  Parallelizing the classification phase  Application of Parallel Support Vector Machine

18. 18 Continuation Values Classification (7)  Preliminary simulation for the parallel classification using SVM

19. 19 Conclusion  PicsouGrid:  Parallel European option pricing algorithms (standard, barrier, basket)  Results published in  2 nd E-Science, Netherlands 12/2006  6 th 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

20. 20 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