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 $$f_i$$ of one real variable (which map onto $$[0,\infty)$$), a random variable $$W$$, and a constant $$K\ge0$$.

Since the function $$\sum_i f_i$$ is also increasing and maps onto $$[0,\infty)$$, there is a unique solution $$w\in\mathbb{R}$$ to the equation $$\sum_i f_i(w)=K.$$

Since the functions $$f_i$$ are increasing: $$\left(\sum_i f_i(W)-K\right)_+ = \left(\sum_i (f_i(W)-f_i(w))\right)_+ = \sum_i (f_i(W)-f_i(w))1_{\{W\ge w\}} = \sum_i(f_i(W)-f_i(w))_+.$$

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 $$f_i(w)$$.