Boundary Conditions
29
T h e F i n i t e D i f f e r e n c e M e t h o d i n C o m p u t a t i o n a l F i n a n c e T h e o r y , A p p l i c a t i o n s a n d C o m p u t a t i o n Boundary Conditions
description
Boundary Conditions. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A. Boundary Conditions. Attempt to define and categorise BCs in financial PDEs Mathematical and financial motivations Unifying framework (Fichera function) - PowerPoint PPT Presentation
Transcript of Boundary Conditions
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
Boundary Conditions
2
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
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
3
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
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
4
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
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 …
5
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
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’
6
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
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
7
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
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
8
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
Domain of interest
Region and Boundary
Unit inward normal
9
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
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
10
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
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
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
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
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
Choices
OnP
¡P
3 de ne
P0 : b= 0
P1 : b> 0
P2 : b< 0
P´
P0 [
P1 [
P2 [
P3
13
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
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
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
Example: Hyperbolic PDE (2/2)y
x
1
L
15
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
Example: Hyperbolic PDE
y
x
a;b> 0
Ficherab= aÀ1 + bÀ2
16
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
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
17
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
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
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
Convertible Bonds
Two-factor model (S, r)
Use Ito to find the PDE
19
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
Two-factor PDE (1/2)
20
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
Two-factor PDE (2/2)
S
V
21
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
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
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
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
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
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
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
PDE Formulation I (2/2)
x
y
25
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
Special Case
Lu = f in
u = g3 on§ 2 [ § 3
26
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
Example: Skew PDE
Pure diffusion degenerate PDE Used in conjunction with SABR model Critical value of beta (thanks to Alan Lewis)
27
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
PDE
S
y
12S2¯ y2 @2u
@S2 + 12y2 @2u
@y2 = 0
28
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
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
Th
e F
ini t
e D
iffere
nce
Meth
od
in
Com
pu
tati
on
al
Fin
an
ce T
heory
, A
pp
l i cati
ons
and C
om
puta
tion
Boundaries¡¯ > 1
2
¢
¡ 1 : b= 0(§ 0)
¡ 2 : b= 0(§ 0)
¡ 3
¡ 4
9>>=
>>;belong to§ 3
