Finance:Jamshidian's trick

Short description: Price model technique

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 [math]\displaystyle{ f_i }[/math] of one real variable (which map onto [math]\displaystyle{ [0,\infty) }[/math]), a random variable [math]\displaystyle{ W }[/math], and a constant [math]\displaystyle{ K\ge0 }[/math].

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

Since the functions [math]\displaystyle{ f_i }[/math] are increasing: [math]\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\ge w\}} = \sum_i(f_i(W)-f_i(w))_+. }[/math]

In financial applications, each of the random variables [math]\displaystyle{ f_i(W) }[/math] represents an asset value, the number [math]\displaystyle{ K }[/math] 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 [math]\displaystyle{ f_i(W) }[/math] with corresponding strikes [math]\displaystyle{ f_i(w) }[/math].

References

• Jamshidian, F. (1989). "An exact bond option pricing formula," Journal of Finance, Vol 44, pp 205-209