Jamshidian's trick is a technique for one-factor asset price models, which re-expresses an option on a portfolio of assets as a portfolio of options. It was developed by Farshid Jamshidian in 1989.
The trick relies on the following simple, but very useful mathematical observation. Consider a sequence of monotone (increasing) functions
f
i
of one real variable (which map onto
[
0
,
∞
)
), a random variable
W
, and a constant
K
≥
0
.
Since the function
∑
i
f
i
is also increasing and maps onto
[
0
,
∞
)
, there is a unique solution
w
∈
R
to the equation
∑
i
f
i
(
w
)
=
K
.
Since the functions
f
i
are increasing:
(
∑
i
f
i
(
W
)
−
K
)
+
=
(
∑
i
(
f
i
(
W
)
−
f
i
(
w
)
)
)
+
=
∑
i
(
f
i
(
W
)
−
f
i
(
w
)
)
1
{
W
≥
w
}
=
∑
i
(
f
i
(
W
)
−
f
i
(
w
)
)
+
.
In financial applications, each of the random variables
f
i
(
W
)
represents an asset value, the number
K
is the strike of the option on the portfolio of assets. We can therefore express the payoff of an option on a portfolio of assets in terms of a portfolio of options on the individual assets
f
i
(
W
)
with corresponding strikes
f
i
(
w
)
.