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 ) .
