Neha Patil (Editor)

Jamshidian's trick

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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 ) .

References

Jamshidian's trick Wikipedia