There has been developed some useful approximations to various specific options. It is of course
American options that are approximated. The particular example we will look at, is a general quadratic
approximation to American call and put prices.
A quadratic approximation to American prices due to Barone-Adesi and Whaley.
We now discuss an approximation to the option price of an American option on a commodity, described in
Barone-Adesi and Whaley (1987) (BAW).12.1 The
commodity is assumed to have a continuous payout . The starting point for the approximation is the
(Black-Scholes) stochastic differential equation valid for the value of any derivative with price .
Here is the (unknown) formula that determines the price of the contingent claim. For an European
option the value of has a known solution, the adjusted Black Scholes formula. For American options,
which may be exercised early, there is no known analytical solution.
To do their approximation, BAW decomposes the American price into the European price and the early
Here is the early exercise premium. The insight used by BAW is that must
also satisfy the same partial differential equation. To come up with an approximation BAW
transformed equation (12.1) into one where the terms involving are neglible, removed
these, and ended up with a standard linear homeogenous second order equation, which has a known solution.
The functional form of the approximation is shown in formula 12.1.
In implementing this formula, the only problem is finding the critical value .
This is the classical problem of finding a root of the equation
This is solved using Newton's algorithm for finding the root.
We start by finding a first ``seed'' value .
The next estimate of is found by:
At each step we need to evaluate and its derivative .
where is the Black Scholes value for commodities.
Code 12.1 shows the implementation of this formula for the price of a call option.