Non-Uniform Adaptive Meshing for One-Asset Problems in Finance

1. Non-Uniform Adaptive Meshing for One-Asset Problems in Finance Sammy Huen Supervisor: R. Bruce Simpson Scientific Computation Group University of Waterloo

2. Presentation Outline • Finance Background (Example) • Research Goals • Motivating Example • Non-Uniform Mesh Generation • Adaptive Meshing • Results – Digital Option • Conclusions

3. Call Option Example Today 1 year from today Maturity (T) Gas: $0.70 Exercise Buy: $0.60 Payoff: $0.10 Contract In 1 year, you have the right but not an obligation to buy gas at 0.60 cents per litre. ? Fair Market Value (V) of Contract Gas: $0.50 Let Expire Buy: $0.50 Strike Price (K)

4. Call Option Value V(S, t) r = 5%  = 20% K = $0.60 * At t < T, V satisfies the Black-Scholes PDE (1973)

5. Hedging • The issuers of the option can greatly reduce risk (hedging) by creating a portfolio that offsets the exposure to fluctuations in the asset price. • Portfolio composed of the option and a quantity of the asset. • For the Black-Scholes model, a possible hedging strategy is based on holding of the asset. • Delta Hedging

6. Our Research • The value V(S, t) of the option can be estimated by solving BS PDE numerically. • Solved using a static non-uniform mesh {Si}. • As time increases, V changes. • Mesh unchanged • Goal: We want a mesh generator that generates a mesh that adapts to the shape of V over time to efficiently control the error in V and in the portfolio.

7. Motivation t = 0.053

8. Motivation t = 0.43

9. Motivation t = 1.0

10. Goal – Dynamic Meshing t = 0.053 N = 35

11. Goal – Dynamic Meshing t = 0.43 N = 55

12. Goal – Dynamic Meshing t = 1.0 N = 66

13. Mesh Generator 1. Derefinement Density Function 2. Refinement 3. Equidistribution

14. Mesh Density Function w(S) > 0 for a  S  b S

15. Mesh (De)Refinement • Mesh size not known • Define a distributing weight tol • Insert and delete mesh points so that interval weight

16. Mesh Equidistribution • {Si} is an equidistributing mesh for w if • Mesh size fixed at N • Get a non-linear system of equations • Use frozen coefficient iteration to solve

17. Adaptive Meshing • Assume smooth profiles • Min. the Hm-seminorm* of error in piecewise linear interpolating fn of V • for m = 0, 1 MDF 1 & 2 *G.F. Carey and H.T. Dinh. Grading functions and mesh redistribution. SIAM Journal on Numerical Analysis, 22(5):1028-1040, 1985.

18. Adaptive Meshing • Taking the portfolio into account MDF 3

19. Other Issues • When to adapt mesh? • Every time step • Interpolation • Tensioned Spline* • Non-Smooth Profiles • Smooth (non-smooth) solution first • Apply previous methods * A.K. Cline. Scalar- and planar-valued curve fitting using splines under tension. Communications of the ACM, 17(4):218—220, 1974.

20. Results - Digital Option

21. Mesh Evolution

22. Mesh Evolution (cont.) * Designed by Forsyth and Windcliff for this problem

23. Global Option Price Error t = 0.0007 years t = 1 year * Exact values obtained in Matlab

24. Global Portfolio Error t = 0.0007 years t = 1 year

25. Conclusions • Adaptive meshing can be more efficient in controlling error • Option price profile • Portfolio profile • Each strategy worked well for digital call options • Similar results for vanilla and discrete barrier call options

26. Future Work • Consider different payoff functions • butterfly, straddle, bear spread • Consider early exercise • American style contracts • Consider other exotic options • Asian, Parisian • Consider other density functions