Jamshidian's trick

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

{\displaystyle f_{i}}

of one real variable (which map onto

{\displaystyle [0,\infty )}

), a random variable

, and a constant

Since the function

{\displaystyle \sum _{i}f_{i}}

is also increasing and maps onto

, there is a unique solution

to the equation

Since the functions

are increasing:

In financial applications, each of the random variables

represents an asset value, the number

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

with corresponding strikes