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

Sammy HuenSupervisor: R. Bruce Simpson

Scientific Computation GroupUniversity 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

ContractIn 1 year, you havethe right but not anobligation to buygas at 0.60 cents per litre.

Today 1 year from today

Gas: $0.70ExerciseBuy: $0.60Payoff: $0.10

Gas: $0.50Let ExpireBuy: $0.50

Maturity (T)

Strike Price (K)

? Fair MarketValue (V) ofContract

4

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.80

0.05

0.1

0.15

0.2

0.25

Asset Price (S)

Opt

ion

Val

ue (

V)

At maturity1 month6 months1 year (Today)

Call Option Value V(S, t)

r = 5% = 20%K = $0.60

)0,max(),( KSTtSV

* 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

S

V

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

50 100 150-2

0

2

4

6

8

10

12x 10

-3

Asset Price ($)

Err

or

($)

50 100 150-5

0

5

10

15

20

25

30

35

40

45

50

Asset Price ($)

Op

tio

n P

ric

e (

$)

Motivationt = 0.053

8

50 100 150-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Asset Price ($)

Err

or

($)

50 100 150-5

0

5

10

15

20

25

30

35

40

45

50

Asset Price ($)

Op

tio

n P

ric

e (

$)

Motivationt = 0.43

9

50 100 150-5

0

5

10

15

20

25

30

35

40

45

50

Asset Price ($)

Op

tio

n P

ric

e (

$)

50 100 150-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Asset Price ($)

Err

or

($)

Motivationt = 1.0

10

50 100 150-5

0

5

10

15

20

25

30

35

40

45

50

Asset Price ($)

Op

tio

n P

ric

e (

$)

Goal – Dynamic Meshingt = 0.053

N = 35

11

50 100 150-5

0

5

10

15

20

25

30

35

40

45

50

Asset Price ($)

Op

tio

n P

ric

e (

$)

Goal – Dynamic Meshingt = 0.43

N = 55

12

50 100 150-5

0

5

10

15

20

25

30

35

40

45

50

Asset Price ($)

Op

tio

n P

ric

e (

$)

Goal – Dynamic Meshingt = 1.0

N = 66

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Mesh Generator

1. Derefinement

2. Refinement

3. Equidistribution

DensityFunction

14

Mesh Density Function

0 50 100 150 2000

0.1

0.2

0.3

0.4

0.5

x

w

w(S) > 0for a S b

S

15

Mesh (De)Refinement

Mesh size not known Define a distributing weight tol Insert and delete mesh points so that

1

5.1)(5.0i

i

S

Stoldzzwtol

interval weight

16

Mesh Equidistribution {Si} is an equidistributing mesh for w if

1

constant)(i

i

S

Sdzzw

Mesh size fixed at N Get a non-linear system of equations Use frozen coefficient iteration to solve

bSSA kk )1()( )(

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Adaptive Meshing Assume smooth profiles Min. the Hm-seminorm* of error in

piecewise linear interpolating fn of V

*G.F. Carey and H.T. Dinh. Grading functions and mesh redistribution.SIAM Journal on Numerical Analysis, 22(5):1028-1040, 1985.

)1)2(2/(2

2

2

)(

m

S

VSw

for m = 0, 1 MDF 1 & 2

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Adaptive Meshing Taking the portfolio into account

SS

VVP

5/2

3

3

2

2

)(

S

VS

S

VSw

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.

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Results - Digital Option

0 50 100 150 200

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Asset Price ($)

Op

tio

n P

ric

e (

$)

1 month4 months1 year (today)at maturity

otherwise0

if1)(

KSSVpayoff

Expiry (T) 1 year

Strike Price (K) $100

Interest Rate (r)

10%

Volatility () 20%

21

Mesh Evolution

0 50 100 150 200

10

20

30

40

50

60

70

S

tim

e s

tep

nu

mb

er

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Mesh Evolution (cont.)

10 20 30 40 50 60 70

35

40

45

50

55

60

time interval

# o

f m

es

h p

oin

ts

mdf1mdf2mdf3 # of Mesh

Points#

Time Step

sType Mi

nMax

Avg

Static* 46 46 46 65

MDF1 45 55 47 73

MDF2 35 54 46 71

MDF3 35 59 47 72

* Designed by Forsyth and Windcliff for this problem

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Global Option Price Error

t = 0.0007 years t = 1 year

* Exact values obtained in Matlab

0 50 100 150 200-10

-8

-6

-4

-2

0

2

4x 10

-3

S ($)

V e

rro

r ($

)staticmdf1mdf2mdf3

95 100 105-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

S ($)

V e

rro

r ($

)

staticmdf1mdf2mdf3

24

Global Portfolio Error

t = 0.0007 years t = 1 year

SS

VVP

95 100 105-30

-20

-10

0

10

20

30

S ($)

Po

rtfo

lio

err

or

($)

staticmdf1mdf2mdf3

0 50 100 150 200-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

S ($)

Po

rtfo

lio

err

or

($)

staticmdf1mdf2mdf3

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

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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