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 ${\displaystyle f_{i}}$ of one real variable (which map onto ${\displaystyle [0,\infty )}$), a random variable ${\displaystyle W}$, and a constant ${\displaystyle K\geq 0}$.

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

Since the functions ${\displaystyle f_{i}}$ are increasing: ${\displaystyle \left(\sum _{i}f_{i}(W)-K\right)_{+}=\left(\sum _{i}(f_{i}(W)-f_{i}(w))\right)_{+}=\sum _{i}(f_{i}(W)-f_{i}(w))1_{\{W\geq w\}}=\sum _{i}(f_{i}(W)-f_{i}(w))_{+}.}$

In financial applications, each of the random variables ${\displaystyle f_{i}(W)}$ represents an asset value, the number ${\displaystyle K}$ 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 ${\displaystyle f_{i}(W)}$ with corresponding strikes ${\displaystyle f_{i}(w)}$.

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