A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Merton model

\begin{eqnarray*} dS_t&=& rS_t dt+\sigma S_t dW_t+S_{t-}(e^{J_t}-1) dN_t-S_t\lambda(e^{\mu_j+\sigma_J^2/2}-1)\text{d}t \end{eqnarray*} where $J_t\in \text{Norm}(\mu_J,\sigma^2_J)$, $N$ is a Poisson process with intensity $\lambda$. \begin{multline*} E[e^{iy\ln(S(T))}|S(t)]=\exp(iy\ln(S(t))+iyr(T-t)+A(t,T,iy))\\ A(t,T,iy)=(T-t)((-\sigma^2/2)iy-y^2\sigma^2/2+\lambda\left((e^{iy\mu_j-y^2\sigma^2_J/2}-1)\right.\\\left.-iy(e^{\mu_j+\sigma^2/2}-1)\right) \end{multline*} Note that $S_{t-}=\lim_{s\uparrow t} S_s$.
A simulation of the log stock price can be seen below.
(C) Magnus Wiktorsson (2011) Error

References

Merton, R. C., 1976. Option Pricing When the Underlying Stock Returns are Discontinuous. Journal of Financial Economics 5, 125-144.