Subsections

Alternatives to the Black Scholes type option formula

A large number of alternative formulations to the Black Scholes analysis has been proposed. Very few of them have seen any widespread use, but we will look at some of these alternatives.


Merton's Jump diffusion model.

Merton has proposed a model where in addition to a Brownian Motion term, the price process of the underlying is allowed to have jumps. The risk of these jumps is assumed to not be priced.

In the following we look at an implementation of a special case of Merton's model, described in (Hull, 1993, pg 454), where the size of the jump has a normal distribution. $ \lambda$ and $\kappa$ are parameters of the jump distribution. The price of an European call option is

\begin{displaymath}c = \sum_{n=0}^\infty
\frac{e^{\lambda^\prime\tau}(\lambda^\prime\tau)^n}{n!}
C_{BS}(S,X,r_n,\sigma_n^2,T-t)
\end{displaymath}

where

\begin{displaymath}\tau=T-t \end{displaymath}


\begin{displaymath}\lambda^\prime = \lambda(1+\kappa) \end{displaymath}

$C_{BS}(\cdot)$ is the Black Scholes formula, and

\begin{displaymath}\sigma_n^2 = \sigma^2+\frac{n\delta^2}{\tau} \end{displaymath}


\begin{displaymath}r_n = r-\lambda\kappa + \frac{n\ln(1+\kappa)}{\tau} \end{displaymath}

In implementing this formula, we need to terminate the infinite sum at some point. But since the factorial function is growing at a much higher rate than any other, that is no problem, terminating at about $n=50$ should be on the conservative side. To avoid numerical difficulties, use the following method for calculation of

\begin{displaymath}\frac{e^{\lambda^\prime\tau}(\lambda^\prime\tau)^n}{n!}
= exp...
...e \tau+ n\ln (\lambda^\prime\tau) - \sum_{i=1}^n \ln i
\right)
\end{displaymath}


\begin{program}
% latex2html id marker 4241\caption{Mertons jump diffusion for...
...{/home/bernt/2003_algor/all_cc_tex_files/merton_jump_diff_call.cc}
\end{program}

2003-10-22