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An approximation to the American Put due to Geske and Johnson

In Geske and Johnson (1984) a very nice approximation is developed for the American put problem. The solution technique is to view the American put as an sequence of Bermudan options, with the number of exercise dates increasing. The correct value is the limit of this sequence.

Define $P_i$ to be the price of the put option with $i$ dates of exercise left. $P_1$ is then the price of an european option, with the one exercise date the expiry date. $P_2$ is the price of an option that can be exercised twice, once halfway between now and expiry, the other at expiry. Geske-Johnson shows how these options may be priced, and then develops a sequence of approximate prices that converges to the correct price. An approximation involving 3 option evaluations is

\begin{displaymath}\hat{P}=P_3 + \frac{7}{2}(P_3-P_2) - \frac{1}{2} (P_2-P_1) \end{displaymath}

$P_1$ is the ordinary (european) Black Scholes value

\begin{displaymath}P_2 = Xe^{-r\frac{(T-t)}{2}}{\cal N}\left(-d_2\left(\overline
S_{\frac{T-t}{2}}\right)\right)
....
\end{displaymath}


\begin{displaymath}P_3 =
....
\end{displaymath}

where


\begin{displaymath}d_1(q,\tau) = \end{displaymath}


\begin{displaymath}d_2(q,\tau) = \end{displaymath}

The evaluations of $P_1$ and $P_2$ are easy, the problem is the evalutation of $P_3$, since it involves the evaluation of a trivariate normal cumulative distribution. For this see the notes on the normal distribution, but it is a nontrivial problem, involving some numerical integration.


next up previous contents index
Next: Futures algoritms. Up: Approximations Previous: A quadratic approximation to   Contents   Index
Bernt Arne Odegaard
1999-09-09