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

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

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)

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

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

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.

Motivation t = 0.053

Motivation t = 0.43

Motivation t = 1.0

Goal – Dynamic Meshing t = 0.053 N = 35

Goal – Dynamic Meshing t = 0.43 N = 55

Goal – Dynamic Meshing t = 1.0 N = 66

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

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

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

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

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.

Adaptive Meshing Taking the portfolio into account MDF 3

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.

Results - Digital Option Expiry (T) 1 year Strike Price (K) $100 Interest Rate (r) 10% Volatility () 20%

Mesh Evolution

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

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

Global Portfolio Error t = 0.0007 years t = 1 year

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

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