Radial Basis Functions generated FiniteDifferences (RBF-FD) for Solving

High-Dimensional PDEs in Finance

S. Milovanovic, L. von Sydow

Uppsala UniversityDepartment of Information Technology

Division of Scientific Computing

SIAMStudent Computational Finance DayTU Delft, Netherlands, 2016-May-23

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Overview

IntroductionOption Pricing

MethodsRadial Basis Functions generated Finite Difference methods

ResultsNumerical Experiments

ConclusionFurther Research

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Option PricingI Black-Scholes-Merton model in higher dimensions

dB(t) = rB(t)dt,

dS1(t) = µ1S1(t)dt+ σ1S1(t)dW1(t),

dS2(t) = µ2S2(t)dt+ σ2S2(t)dW2(t),

...dSD(t) = µDSD(t)dt+ σDSD(t)dWD(t).

(1)

I Option price

u(S1(t), . . . ,SD(t), t) = e−r(T−t)EQt [g(S1(T ), . . . ,SD(T ))].

I The Black-Scholes-Merton equation∂u

∂t+ r

D∑i

si∂u

∂si+

1

2

D∑i,j

ρijσiσjsisj∂2u

∂usisj− ru = 0,

u(s1, s2, . . . , sD,T ) = g(s1, s2, . . . , sD).

(2)

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BENCHOP – The BENCHmarking project in OptionPricing∗

I Monte Carlo methods converge slowly but are increasinglycompetitive in higher dimensions.

I Fourier-methods are fast, especially the COS-method.I Finite difference methods are reasonably fast in lower

dimensions but suffer from the curse of dimensionality.I Methods based on approximations with Radial Basis

Functions have the potential to be competitive in higherdimensions.

∗ L. von Sydow, L.J. Hook, E. Larsson, E. Lindstrom, S. Milovanovic,J. Persson, V. Shcherbakov, Y. Shpolyanskiy, S. Siren, J. Toivanen, J.Walden, M. Wiktorsson, J. Levesley, J. Li, C.W. Oosterlee, M.J.Ruijter, A. Toropov, and Y. Zhao. In International Journal of ComputerMathematics, volume 92, pp 2361-2379, 2015.

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Radial Basis Functions MethodI Discretize space using N nodes.I Approximate solution

u(s, t) ≈N∑k=1

λk(t)φ(ε‖s− sk‖), k = 1, 2, . . . ,N , (3)

where φ is a radial basis function and ε is a shapeparameter.

I The linear combination coefficients λk are found byenforcing interpolation conditions.

Figure 1: Gaussian RBFs.5 / 27

Local Radial Basis Functions Methods

I The global approximation leads to a dense linear system ofequations which tends to be ill-conditioned when ε is small⇒ Local RBF approximations might be a better approach!

I Ideas

I Come up with a localization which leads to a sparse linearsystem of equations.

I Radial Basis Functions Partition of Unity (RBF-PU)I Radial Basis Functions generated Finite Differences

(RBF-FD)

I Come up with basis transformation which will heal theill-conditioning.

I RBF-QRI RBF-GA

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Local Radial Basis Functions Methods

I The global approximation leads to a dense linear system ofequations which tends to be ill-conditioned when ε is small⇒ Local RBF approximations might be a better approach!

I Ideas

I Come up with a localization which leads to a sparse linearsystem of equations.

I Radial Basis Functions Partition of Unity (RBF-PU)I Radial Basis Functions generated Finite Differences

(RBF-FD)

I Come up with basis transformation which will heal theill-conditioning.

I RBF-QRI RBF-GA

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Radial Basis Functionsgenerated Finite Differences

I Try to exploit the best properties from both FD and RBFwith minimal loss.

I For each point si in space, define its neighborhood ofM − 1 points and observe it as a stencil.

I Approximate the differential operator at every point

[Lu(s)]i ≈M∑k=1

w(i)k u

(i)k . (4)

I Compute the RBF-FD weights and place them in matrix Wφ(‖s(i)1 − s

(i)1 ‖) . . . φ(‖s(i)1 − s

(i)M ‖)

.... . .

...φ(‖s(i)M − s

(i)1 ‖) . . . φ(‖s(i)M − s

(i)M ‖)

w1

...wM

=

[Lφ(‖s− s(i)1 ‖)]s=si

...[Lφ(‖s− s(i)M ‖)]s=si

.

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Radial Basis Functionsgenerated Finite Differences

I Try to exploit the best properties from both FD and RBFwith minimal loss.

I For each point si in space, define its neighborhood ofM − 1 points and observe it as a stencil.

I Approximate the differential operator at every point

[Lu(s)]i ≈M∑k=1

w(i)k u

(i)k . (4)

I Compute the RBF-FD weights and place them in matrix Wφ(‖s(i)1 − s

(i)1 ‖) . . . φ(‖s(i)1 − s

(i)M ‖)

.... . .

...φ(‖s(i)M − s

(i)1 ‖) . . . φ(‖s(i)M − s

(i)M ‖)

w1

...wM

=

[Lφ(‖s− s(i)1 ‖)]s=si

...[Lφ(‖s− s(i)M ‖)]s=si

.

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Adding polynomial termsφ(‖s(i)1 − s(i)1 ‖) . . . φ(‖s(i)1 − s(i)

M‖) 1

.

.

.. . .

.

.

....

φ(‖s(i)M− s(i)1 ‖) . . . φ(‖s(i)

M− s(i)

M‖) 1

1 . . . 1 0

w1

.

.

.wMwM+1

=

[Lφ(‖s− s(i)1 ‖)]s=si

.

.

.[Lφ(‖s− s(i)

M‖)]s=si

L1

.

40 60 80 100 120 140 160

s

0.5

1

1.5

2

2.5

3

∆u

max

×10 -4 Absolute Error for European Call Option

regular grid, standard stencil, standard matrix

regular grid, standard stencil, polynomial matrix

regular grid, adapted stencil, polynomial matrix

K=100

T=1

r=0.03

σ=0.15

n=5

Figure 2: Error in the solution.8 / 27

Disctretization

I Discretize the Black-Scholes-Merton equation operator inspace using RBF-FD

ut = −

r D∑i

siusi +1

2

D∑i,j

ρijσiσjsisjusisj − ru

≈Wu.

I Integrate in time using the standard implicit schemesI BDF-1,I BDF-2.

⇒

Aun+1 = bn

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Solution of linear systems

0 100 200 300 400 500 600 700 800

0

100

200

300

400

500

600

700

800

n=25

Figure 3: Structure of A.

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Grids

Figure 4: The nearest-neighbor based stencils used for approximatingthe differential operator.

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Error distribution

s1

s2

RBF−FD absolute error

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

4

0

0.002

0.004

0.006

0.008

0.01

0.012

Figure 5: The absolute error distribution computed using 41 point ineach dimension and a 5-point stencil on a full regular grid.

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Grids

0 K 2K 8K0

K

2K

8K

s1

s 2

Figure 6: Square domain.13 / 27

Grids

0 K 2K 8K0

K

2K

8K

s1

s 2

Figure 7: Triangular grid.14 / 27

Grids

0 K 2K 8K0

K

2K

8K

s1

s 2

Figure 8: Adapted grid.15 / 27

Grids

0 K 2K 8K0

K

2K

8K

s1

s 2

Figure 9: Adapted grid with some possible stencils.16 / 27

Experiment

I European-style basket option withT = 1,K = 1,r = 0.03,σ1 = σ2 = 0.15,ρ = 0.5.

I Payoff function

u(s1, s2,T ) =

[s1(T ) + s2(T )

2−K

]+. (5)

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Details

I Boundary conditions

u(0, 0, t) = 0, (6)

u(s∗1, s∗2, t) =

1

2[s∗1(t) + s∗2(t)]− e−r(T−t)K. (7)

I RBF choice: Gaussian

φ(r) = e−(ε·r)2 . (8)

I Error measured on a subdomain: (s1, s2) ∈ [13K, 53K].

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Solution

8

6

4

S1

2

00

2S2

4

6

0

0.5

3

2.5

2

1.5

1

8

V(S

1,S2)

Figure 10: The computed solution on a regular triangular grid.

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Choice of ε, n = 3.

10-1

100

101

102

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

Δumax

n=3

ε

h=0.01h=0.003h=0.001h=0.0003h=0.0001

ε(h)=0.0020228(1/h)+0.3451057

Figure 11: Error as a function of ε.

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Choice of ε, n = 7.

100

101

102

103

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102 n=7

Δumax

ε

h=0.01h=0.003h=0.001h=0.0003h=0.0001

ε(h)=0.05256(1/h)+7.60214

Figure 12: Error as a function of ε.

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Results

10−3 10−2 10−1

10−5

10−4

10−3

10−2

10−1

h

∆u

max

equidistant grid, n=3equidistant grid, n=5equidistant grid, n=7adapted grid, n=3adapted grid, n=5adapted grid, n=7

Figure 13: Spatial Convergence.22 / 27

Results

10−3 10−2 10−1 100 10110−7

10−4

10−1

102

t [s]

∆u

max

FD, equidistant gridRBF-FD, equidistant grid, εconst, n = 5RBF-FD, equidistant grid, εopt, n = 5RBF-FD, adapted grid, εavg, n = 5

Figure 14: Computational performance.23 / 27

Conclusions

Proposed method:

I Works equally well with American options using operatorsplitting.

I Allows for local grid refinement which may deliver highaccuracy in desired regions of the computational domain.

I Performs well due to sparsity and reduction in thecomputational domain allowed by the mesh-freediscretization.

I Promises extendability to higher dimensional problems aswell as for other models in the field.

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Future Research

I Finalize study of shape parameter behavior.I Developing adaptivity both in node placement and stencil

size (hp-adaptivity).I Optimizing stencils close to the boundaries.I Smoothing of the terminal condition.I Least squares instead of interpolation.I Solving advanced financial models.

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Future ResearchI Higher dimensional problems.

0

0

2

0 5

4

s1

6s3

8

s2

10

5

12

1010

0

0

2

4

2

6

s3

4

8

10

s1

610

12

8

s2

510

12 0

00

20

2

4

4

European Call Basket Option

6

s1

s3

6

8

5

s2

10

8

12

101012

0

0.5

1

1.5

2

2.5

3

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Thank you for your attention.

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