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Chapter 4
4.1 Lookback options and Neumann boundary condition
Lattice Methods for American Opt4.2 ions
Applications of Lattice Methods
4.3 Options with two underlying assets
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4.1 Lookback options and Neumann Boundary Condition
22 2
2
1( ) 0, 0 >
=S t T
V S S T f S SS
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Finite difference schemes for options on two
assets
1, 2, 1 2, , ) ( , , ) : 0,1, , ;
:
0,1, , ; 0,1, , }
{( i j nS t ih jh n t n N
i I j J
S = =
= =
Mesh
, 1 2( , , ).Denoten
i jV ih jh nV t=
Two questions:
the discretization of the cross derivative;boundary conditions.
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Discretization of the cross derivative
21, 1 1, 1 1, 1 1, 1
1 2 1 2
21, 1 1, 1 , 1, 1, , 1 , 1
1 2 1 2
21, 1 1, 1 , 1, 1,
1 2
1.
22.
23.
4
2
n n n n
i j i j i j i j
n n n n n n n
i j i j i j i j i j i j i j
n n n n n
i j i j i j i j i j
V V
V V V
V VV
S S h h
V VV
S S h hV VV
S
V V
V V V
S
+ + + +
+ + + +
+ + +
+
+
+
+
+ , 1 , 1
1 22
n n
i j i j
h
V V
h
+
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Boundary conditions
1 2Consider the spread option with payoff ( ) .S SX+
Let ( , be the price function of a European vanilla put.
at time for a given asset with current price , v
, , )
and
continuous
olatili
dividend yi
ty
eld .
EP S
t
q t
S
q
1
2
1 2
1
2 2 20
1 10
Boundary conditions for the spread option can be imposed
as follows:
( , , , ),( , ,
0
, ),
.
E
E
IS J
S
S
S
S q tS q
V PtV P
V V
=
=
= =
=
=
= =
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Implicit difference scheme
Proceed with similar choice of FD formulae as for the 1-asset
case and with FD formula (1) for the cross derivative yields
the FDE:
1
, , 1, , 1,2 2 2
1 1 2
1
1, 1 1, 1 1, 1 1, 1
1 2 1 2
1 2
, 1 , , 12 2 2
1
2
1
2 2 2
+
21
2
4
21
(
2
)
+ +
+ + + +
+
++
+
+
++
n n n n
i j i j i j i j i j
n n n n
i j i j i j i j
n n
i j i j i j
V V V V i h
t
V Vij
V
h
V V
Vj
h
Vr
h hh h
q
V Vh
ih1, 1, ,
2
1
1
2
, 1
2
,
1, , 1, 1, 2, , 1.2,
( 0,2 2
)+ +
= =
+
=n n n n
i j i j i j i j n
i j
I Ji j
Vr q rV
h
V Vjh
h