Jamshidian's trick
Encyclopedia
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. The trick relies on the following simple, but very useful mathematical observation.
Consider a sequence of monotone (increasing) functions
,
of one real variable (which map onto 
), a random variable
, and a constant 
.
Since the function
is increasing and also maps onto 
, there is a unique solution 
to the equation 
Since the functions
are increasing,
Option
- Legal rights :*Option , an instrument that conveys the right, but not the obligation, to engage in a future transaction , or in a futures contract...
on a portfolio of assets
Portfolio (finance)
Portfolio is a financial term denoting a collection of investments held by an investment company, hedge fund, financial institution or individual.-Definition:The term portfolio refers to any collection of financial assets such as stocks, bonds and cash...
as a portfolio of options. The trick relies on the following simple, but very useful mathematical observation.
Consider a sequence of monotone (increasing) functions
Monotonic function
In mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....
,
of one real variable (which map onto ), a random variableRandom variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
, and a constant .Since the function
is increasing and also maps onto , there is a unique solution to the equation Since the functions
are increasing,-
In financial applications, each of the random variables
represents an asset value, the number 
is the strikeStrike priceIn options, the strike price is a key variable in a derivatives contract between two parties. Where the contract requires delivery of the underlying instrument, the trade will be at the strike price, regardless of the spot price of the underlying instrument at that time.Formally, 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 
.