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 $K\geq 0$.

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: $\left(\sum _{i}f_{i}(W)-K\right)_{+}=\left(\sum _{i}(f_{i}(W)-f_{i}(w))\right)_{+}=\sum _{i}(f_{i}(W)-f_{i}(w))1_{\{W\geq w\}}=\sum _{i}(f_{i}(W)-f_{i}(w))_{+}.$

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 $f_{i}(w)$.

References

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