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

Since the function $\sum _i{f}_i$ is also increasing and maps onto $[0,\mathrm{\infty })$, 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: ${(\sum _i{f}_i(W)-K)}_{+}={(\sum _i(f_i(W)-f_i(w)))}_{+}=\sum _i(f_i(W)-f_i(w))1_{\{W\ge w\}}=\sum _i(f_i(W)-f_i(w){)}_{+}.$

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 $f_i(w)$.

References

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