Boundary Conditions

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Boundary Conditions

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

2

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

3

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 …

4

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’

5

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

6

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

7

Domain of interest

Unit inward normal

Region and Boundary

8

Elliptic PDE

X n

Lu ´ where i ;j = 1 a i j

@ u

@x i

@x j

+

X n i = 1

@u b i

@x i

+ cu = f in -

X n i ;j = 1 a i j

» i

» j

¸ 0 in - [

P

8 » = (»

1

; : : : ; » n

) ² R n

9

Remarks

Called an equation with non-negative characteristic form

Distinguish between characteristic and noncharacteristic boundaries

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

Applicable when the quadratic form is positivedefinite as well

Subsumes Friedrichs’ theory in hyperbolic case?

10

Boundary Types

P

3

= n x²

P

:

P n i ;j = 1 a i j

À i

À j

> 0 o

On

P b ´

X n i = 1

¡

Ã

P

3 examine Fichera function b i

¡

X n

@a i k k= 1

@x k

!

À i

11

On

P

¡

Choices

P

3 de¯ne

P

0

: b = 0

P

1

: b > 0

P

2

: b < 0

P

´

P

0

[

P

1

[

P

2

[

P

3

12

Example: Hyperbolic PDE (1/2)

¡ @u

@x

= f in - = (0; L) £ (0; 1) b= ¡ À

1 a @u

@x

+ b @u

@y

= f in (0; 1) £ (0; 1)

13

Example: Hyperbolic PDE (2/2) y

1

L x

14

Example: Hyperbolic PDE a; b> 0 y

Fichera b= aÀ

1

+ bÀ

2 x

15

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

16

CIR PDE

@B

@t

+ 1

2

¾ 2 r @ B

@r 2

+ (a ¡ br ) @B

@r

¡ r B = 0

Fichera b = (a ¡ br ) + 2¾

§

2

: b < 0 ! ¾> p

2a

§

0

: b = 0 ! ¾= p

2a

§

1

: b > 0 ! ¾< p

2a (No BC needed) 17

Convertible Bonds

 Two-factor model (S, r)

 Use Ito to find the PDE

18

Two-factor PDE (1/2)

19

V

Two-factor PDE (2/2)

S

20

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

21

PDE for Asian

¡

@u

@t

+

1

2

¾ 2 x 2 @

2 u

@x 2

+ r x

@u

@x

+ x

T

@u

@y

¡ r u = 0 y = 1

T

R t

0 x(t)dt

Fichera b= (r x ¡ ¾ 2 x)À

1

+ x

T

À

2

= x(r ¡ ¾ 2 )À

1

+ x

T

À

2

22

PDE Formulation I (1/2)

Lu = f in u = g

1 on §

2

®u + ¯ @u

@l

= g

2 on §

3

1

¯ ¸ 0; l is a direction)

23

PDE Formulation I (2/2) y x

24

Special Case

Lu = f in u = g

3 on §

2

[ §

3

25

Example: Skew PDE

Pure diffusion degenerate PDE

Used in conjunction with SABR model

Critical value of beta

(thanks to Alan Lewis)

26

PDE y

1

2

S 2¯ y 2 @ u

@S 2

+ 1

2 y 2 @ u

@y 2

= 0

S

27

Fichera Function

Fichera b= ¡ § 2 i = 1

µ b i

¡ § 2 k= 1

@a i k

@x k

À i

= ¡ § 2 i = 1

§ 2 k= 1

@a i k

@x k

À i

= ¡ § 2 i = 1

@a i i

@x i

= ¡

À i

¡

¯S 2¯¡ 1 y 2 À

1

+ yÀ

2

¢

28

¡

¯ >

Boundaries

1

2

¢

¡

1

: b = 0(§

0

)

¡

2

: b = 0(§

0

)

¡

3

9

>

¡

4

;

> belong to §

3

29

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