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 {\displaystyle f_{i}} of one real variable (which map onto [ 0 , ∞ ) {\displaystyle [0,\infty )} ), a random variable W {\displaystyle W} , and a constant K ≥ 0 {\displaystyle K\geq 0} . Since the function ∑ i f i {\displaystyle \sum _{i}f_{i}} is also increasing and maps onto [ 0 , ∞ ) {\displaystyle [0,\infty )} , there is a unique solution w ∈ R {\displaystyle w\in \mathbb {R} } to the equation ∑ i f i ( w ) = K . {\displaystyle \sum _{i}f_{i}(w)=K.}
Since the functions f i {\displaystyle 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 ) ) + . {\displaystyle \left(\sum _{i}f_{i}(W)-K\right)_{+}=\left(\sum _{i}(f_{i}(W)-f_{i}(w))\right)_{+}=\sum _{i}(f_{i}(W)-f_{i}(w))1_{\{W\geq w\}}=\sum _{i}(f_{i}(W)-f_{i}(w))_{+}.}
In financial applications, each of the random variables f i ( W ) {\displaystyle f_{i}(W)} represents an asset value, the number K {\displaystyle 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 ) {\displaystyle f_{i}(W)} with corresponding strikes f i ( w ) {\displaystyle f_{i}(w)} .
References Jamshidian, F. (1989). "An exact bond option pricing formula," Journal of Finance, Vol 44, pp 205-209
