Boundary Conditions
Boundary Conditions
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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
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‘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
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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
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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
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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)
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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
Unit inward normal
Region and Boundary
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Elliptic PDE
Xn
@2 u
Xn
@u
Lu ´
ai j
+
bi
+ cu = f in @
x
@
x
@
x
i
j
i
i ;j = 1
i= 1
where
Xn
ai j »i »j ¸ 0 in - [
P
i ;j = 1
8 » = (»1 ; : : : ; »n ) ² Rn
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Remarks
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Called an equation with non-negative
characteristic form
Distinguish between characteristic and noncharacteristic boundaries
Applicable to elliptic, parabolic and 1st-order
hyperbolic PDEs
Applicable when the quadratic form is positivedefinite as well
Subsumes Friedrichs’ theory in hyperbolic
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case?
Boundary Types
n
P
3
On
= x²
P
¡
P
:
Pn
i ;j = 1
o
ai j Ài Àj > 0
P
examine Fichera function
Ã
!
Xn
Xn @ai k
b´
bi ¡
Ài
@
x
k
i= 1
k= 1
3
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Choices
On
P
P
3
0
: b= 0
1
: b> 0
2
: b< 0
P
P
P
¡
P
´
P
0[
de¯ne
P
1[
P
2[
P
3
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Example: Hyperbolic PDE (1/2)
¡
@u
@x
= f in - = (0; L) £ (0; 1)
b= ¡ À1
@u
a @x
+
@u
b@y
= f in (0; 1) £ (0; 1)
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Example: Hyperbolic PDE (2/2)
y
1
x
L
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Example: Hyperbolic PDE
a; b> 0
y
Ficherab= aÀ1 + bÀ2
x
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Example: CIR Model
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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 2 @2 B
2 ¾ r @r 2
B
+ (a ¡ br) @
@r ¡ rB = 0
Fichera b = (a ¡ br ) + 2¾
§ 2 : b< 0 ! ¾>
§ 0 : b= 0 ! ¾=
§ 1 : b> 0 ! ¾<
p
p
p
2a
2a
2a (No BC needed)
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Convertible Bonds
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Two-factor model (S, r)
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Use Ito to find the PDE
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Two-factor PDE (1/2)
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Two-factor PDE (2/2)
V
S
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Asian Options
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Two-factor model (S, A)
Diffusion term missing in the A direction
Determine the well-posedness of problem
Write PDE in (x,y) form
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PDE for Asian
¡
@u
@t
y=
1 2 2 @2 u
2 ¾ x @x 2
+
1
T
Rt
0
x @u
T @y
¡ ru = 0
x(t)dt
Fichera b= (r x ¡
= x(r ¡
u
+ rx@
@x +
2
¾ )À1
+
x
2
¾ x)À1 + T À2
x
T À2
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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)
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PDE Formulation I (2/2)
y
x
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Special Case
Lu = f in u = g3 on § 2 [ § 3
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Example: Skew PDE
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Pure diffusion degenerate PDE
Used in conjunction with SABR model
Critical value of beta
(thanks to Alan Lewis)
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PDE
y
1 2¯ 2 @2 u
2 S y @S2
+
1 2 @2 u
2 y @y 2
=0
S
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Fichera Function
µ
Ficherab= ¡
§ 2i= 1
bi ¡
=¡
ai k
2
2 @
§ i = 1§ k= 1
Ài
@xk
=¡
ai i
2 @
§i= 1
@xi
=¡
¡
ai k
2 @
§ k= 1
@xk
¶
Ài
Ài
¯S2¯¡ 1y2À1 +
yÀ2
¢
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Boundaries
¡
¯>
1
2
¢
¡ 1 : b = 0(§ 0 )
¡ 2 : b = 0(§ 0 )
9
¡3 >
>
=
belong to § 3
¡4 >
>
;
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