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heory

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Boundary Conditions

2

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Boundary Conditions

Attempt to define and categorise BCs in financial PDEs

Mathematical and financial motivations

Unifying framework (Fichera function)

One-factor and n-factor examples

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Background

‘Fuzzy’ area in finance Boundary conditions motivated by

financial reasoning BCs may (or may not) be

mathematically correct A number of popular choices are in

use We justify them

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Challenges

Truncating a semi-infinite domain to a finite domain

Imposing BCs on near-field and far-field boundaries

Boundaries where no BC are needed (allowed)

Dirichlet, Neumann, linearity …

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Techniques

Using Fichera function to determine which boundaries need BCs

Determine the kinds of BCs to apply Discretising BCs (for use in FDM) Special cases and ‘nasties’

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The Fichera Method

Allows us to determine where to place BCs

Apply to both elliptic and parabolic PDEs

We concentrate on elliptic PDE Of direct relevance to computational

finance New development, not widely known

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Elliptic PDE (1/2)

Its quadratic form is non-negative (positive semi-definite)

This means that the second-order terms can degenerate at certain points

Use the Oleinik/Radkevic theory The application of the Fichera

function

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Domain of interest

Region and Boundary

Unit inward normal

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Elliptic PDE

Lu ´nX

i ;j =1

ai j@2u

@xi @xj+

nX

i=1

bi@u@xi

+ cu = f in

where

nX

i ;j =1

ai j »i »j ¸ 0in [P

8»= (»1; : : : ;»n) ² Rn

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Remarks

Called an equation with non-negative characteristic form

Distinguish between characteristic and non-characteristic boundaries

Applicable to elliptic, parabolic and 1st-order hyperbolic PDEs

Applicable when the quadratic form is positive-definite as well

Subsumes Friedrichs’ theory in hyperbolic case?

11

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Boundary Types

P3 =

nx²

P:P n

i ;j =1 ai j Ài Àj > 0o

OnP

¡P

3 examine Fichera function

b´nX

i=1

Ã

bi ¡nX

k=1

@aik

@xk

!

Ài

12

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Choices

OnP

¡P

3 de ne

P0 : b= 0

P1 : b> 0

P2 : b< 0

P´

P0 [

P1 [

P2 [

P3

13

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tion 

Example: Hyperbolic PDE (1/2)

¡ @u@x = f in = (0;L) £ (0;1)

b= ¡ À1

a@u@x + b@u

@y = f in (0;1) £ (0;1)

14

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Example: Hyperbolic PDE (2/2)y

x

1

L

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tion 

Example: Hyperbolic PDE

y

x

a;b> 0

Ficherab= aÀ1 + bÀ2

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Example: CIR Model

Discussed in FDM book, page 281 What happens on r = 0? We discuss the application of the

Fichera method Reproduce well-known results by

different means

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CIR PDE

@B@t + 1

2¾2r @2B@r 2 + (a¡ br)@B

@r ¡ rB = 0

Ficherab= (a¡ br) + 2¾

§ 2 : b< 0! ¾>p

2a

§ 0 : b= 0! ¾=p

2a

§ 1 : b> 0! ¾<p

2a (No BC needed)

18

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Convertible Bonds

Two-factor model (S, r)

Use Ito to find the PDE

19

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Two-factor PDE (1/2)

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Two-factor PDE (2/2)

S

V

21

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Asian Options

Two-factor model (S, A) Diffusion term missing in the A

direction Determine the well-posedness of

problem Write PDE in (x,y) form

22

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PDE for Asian

¡ @u@t + 1

2¾2x2 @2u@x2 + rx@u

@x + xT

@u@y ¡ ru = 0

y = 1T

Rt0 x(t)dt

Ficherab= (rx ¡ ¾2x)À1 + xT À2

= x(r ¡ ¾2)À1 + xT À2

23

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PDE Formulation I (1/2)

Lu = f in

u = g1 on§ 2

®u+ ¯ @u@l = g2 on§ 3 (®1¯ ¸ 0; l is a direction)

24

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PDE Formulation I (2/2)

x

y

25

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Special Case

Lu = f in

u = g3 on§ 2 [ § 3

26

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Example: Skew PDE

Pure diffusion degenerate PDE Used in conjunction with SABR model Critical value of beta (thanks to Alan Lewis)

27

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PDE

S

y

12S2¯ y2 @2u

@S2 + 12y2 @2u

@y2 = 0

28

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Fichera Function

Ficherab= ¡ § 2i=1

µbi ¡ § 2

k=1@aik

@xk

¶Ài

= ¡ § 2i=1§

2k=1

@aik

@xkÀi

= ¡ § 2i=1

@ai i

@xiÀi

= ¡¡¯S2¯ ¡ 1y2À1 + yÀ2

¢

29

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Boundaries¡¯ > 1

2

¢

¡ 1 : b= 0(§ 0)

¡ 2 : b= 0(§ 0)

¡ 3

¡ 4

9>>=

>>;belong to§ 3