Approximations

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 $b$. The starting point for the approximation is the (Black-Scholes) stochastic differential equation valid for the value of any derivative with price $V$.

$$\frac{1}{2}\sigma^2 S^2 V_SS + bSV_S -rV +V_t = 0$$ (12.1)

Here $V$ is the (unknown) formula that determines the price of the contingent claim. For an European option the value of $V$ 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 exercise premium

$$C(S,T) = c(S,T) + \varepsilon_C(S,T)$$

Here $\varepsilon_C$ is the early exercise premium. The insight used by BAW is that $\varepsilon_C$ 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 $V_t$ 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 $S^*$. This is the classical problem of finding a root of the equation

$$g(S^*)= S^*-X-c(S^*)-\frac{S^*}{q_2}\left(1-e^{(b-r)(T-t)}N\left(d_1(S^*)\right)\right)=0$$

This is solved using Newton's algorithm for finding the root. We start by finding a first ``seed'' value $S_0$. The next estimate of $S_i$ is found by:

$$S_{i+1}= S_i-\frac{g()}{g^\prime}$$

At each step we need to evaluate $g()$ and its derivative $g^\prime()$.

$$g(S) = S-X-c(S)-\frac{1}{q_2}S\left(1-e^{(b-r)(T-t)}N(d_1)\right)$$

$$g^\prime\left(S\right) = (1-\frac{1}{q_2})\left(1-e^{(b-r)(T-... ... \frac{1}{q_2}(e^{(b-r)(T-t)}n(d_1))\frac{1}{\sigma\sqrt{T-t}}$$

where $c(S)$ is the Black Scholes value for commodities. Code 12.1 shows the implementation of this formula for the price of a call option.