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 Gas: $0.70 Exercise
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 2. Refinement Function
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): , 1985.

18. Adaptive Meshing
Taking the portfolio into account
MDF 3

19. Other Issues When to adapt mesh? Interpolation Non-Smooth Profiles
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
Expiry (T)
1 year
Strike Price (K)
$100
Interest Rate (r)
10%
Volatility ()
20%

21. Mesh Evolution

22. Mesh Evolution (cont.) # of Mesh Points # Time Steps Type Min Max Avg
Static* 46 65
MDF1 45 55 47 73
MDF2 35 54 71
MDF3 59 72
* Designed by Forsyth and Windcliff for this problem

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

24. Global Portfolio Error
t = 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