Jamshidian's trick explained
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.
of one real variable (which map onto
), a
random variable
, and a constant
.
Since the function
is also increasing and maps onto
, there is a unique solution
to the equation
Since the functions
are increasing:
\left(\sumifi(W)-K\right)+=\left(\sumi(fi(W)-fi(w))\right)+=\sumi(fi(W)-fi(w))1\{W\ge
} = \sum_i(f_i(W)-f_i(w))_+.
In financial applications, each of the random variables
represents an asset value, the number
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
with corresponding strikes
.
References
- Jamshidian, F. (1989). "An exact bond option pricing formula," Journal of Finance, Vol 44, pp 205-209