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In-Depth Information
Chapter 8
Multi-asset Options
In Chap. 6, we considered exotic options written on a single underlying. Further
examples of exotic options are given by the so-called
multi-asset options
. These
are options derived from
d
≥
2 underlying risky assets, whose price movement can
be described by a system of SDEs. The pricing functions of multi-asset options
are multivariate functions satisfying a parabolic partial differential equation in
d
dimensions, together with an appropriate terminal value depending on the type of
the option. We distinguish between different types of European multi-asset options.
A
basket option
is an option whose payoff is linked to a portfolio or basket of un-
derlier values. The basket can be any weighted sum of underlier values as long
as the weights are all positive. A typical example of a basket option is the arith-
metic mean put option with payoff
g(s)
0
,
i
=
1
α
i
s
i
−
=
max
{
K
}
. Examples of
rainbow option
are better-of-options, where
g(s)
=
max
1
≤
i
≤
d
{
α
i
s
i
}
, or maximum
call options, with
g(s)
.A
quanto option
(also called
cross-currency option
) is a vanilla option on a foreign underlying, but with a payout
in domestic currency. The payout of the option is converted to the domestic cur-
rency at expiration, at a predefined exchange rate
s
2
. The payoff function of a call is
g(s
1
,s
2
)
=
max
1
≤
i
≤
d
{
max
{
0
,s
i
−
K
i
}}
=
s
2
max
{
0
,s
1
−
K
}
.
8.1 Pricing Equation
As in the case of a single underlying, the price of a multi-asset option on
d
assets is
given as the conditional expectation
= E
e
−
t
r(X
s
)
d
s
g(X
T
)
x
,
V(t,x)
|
X
t
=
(X
t
,...,X
t
)
d
-valued stochastic process modeling the dy-
where
X
t
=
R
is an
C
0
(
d
namics of the
d
assets,
r
∈
R
; R
≥
0
)
is the deterministic interest rate,
g
:
d
R
→ R
≥
0
denotes the payoff of the option and
R
≥
0
the non-negative real num-
bers.
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