The MertonJumpDiffusion command creates a new jump diffusion process that is governed by the stochastic differential equation (SDE)

where

$\mathrm{\mu }\left(t\right)$ is the drift parameter

$\mathrm{\sigma }\left(t\right)$ is the volatility parameter

$W\left(t\right)$ is the standard Wiener process

and

$J\left(t\right)$ is a compound Poisson process of the form

such that $\log \left(Y_i\right)$ is independent and lognormally distributed with mean $a$ and standard deviation $b$.

Both the drift parameter mu and the volatility parameter sigma can be either constant or time-dependent. In the second case they can be specified either as an algebraic expression containing one indeterminate, or as a procedure that accepts one parameter (the time) and returns the corresponding value of the drift (volatility).

Similar to the drift and the volatility parameters, the intensity parameter lambda can be either constant or time-dependent. In the second case it can be specified either as an algebraic expression containing one indeterminate or as a procedure that accepts one parameter (the time).

Both the scale parameter a and the shape parameter b of the underlying lognormal Poisson process must be real constants.

$\mathrm{with}\left(\mathrm{Finance}\right)\:$

$\mathrm{S0}≔100$

$\mathrm{S0}≔100$

$r≔0.05$

$r≔0.05$

$d≔0.01$

$d≔0.01$

$\mathrm{\sigma }\left[1\right]≔0.01$

${\mathrm{\sigma }}_1≔0.01$

$a≔0.$

$a≔0.$

$b≔0.5$

$b≔0.5$

$\mathrm{\lambda }\left[1\right]≔2.0$

${\mathrm{\lambda }}_1≔2.0$

$\mathrm{\lambda }\left[2\right]≔0.2$

${\mathrm{\lambda }}_2≔0.2$

$X\left[1\right]≔\mathrm{MertonJumpDiffusion}\left(\mathrm{S0}\,\mathrm{\sigma }\left[1\right]\,r\,d\,\mathrm{\lambda }\left[1\right]\,a\,b\right)\:$

$\mathrm{PathPlot}\left(X\left[1\right]\left(t\right)\,t=0..1\,\mathrm{timesteps}=100\,\mathrm{replications}=5\,\mathrm{color}=\mathrm{red}..\mathrm{blue}\,\mathrm{thickness}=3\,\mathrm{axes}=\mathrm{BOXED}\,\mathrm{gridlines}=\mathrm{true}\right)$

$X\left[2\right]≔\mathrm{MertonJumpDiffusion}\left(\mathrm{S0}\,\mathrm{\sigma }\left[1\right]\,r\,d\,\mathrm{\lambda }\left[2\right]\,a\,b\right)\:$

$\mathrm{PathPlot}\left(X\left[2\right]\left(t\right)\,t=0..1\,\mathrm{timesteps}=100\,\mathrm{replications}=5\,\mathrm{color}=\mathrm{red}..\mathrm{blue}\,\mathrm{thickness}=3\,\mathrm{axes}=\mathrm{BOXED}\,\mathrm{gridlines}=\mathrm{true}\right)$

$\mathrm{\sigma }\left[2\right]≔0.5$

${\mathrm{\sigma }}_2≔0.5$

$X\left[3\right]≔\mathrm{MertonJumpDiffusion}\left(\mathrm{S0}\,\mathrm{\sigma }\left[2\right]\,r\,d\,\mathrm{\lambda }\left[2\right]\,a\,b\right)\:$

$\mathrm{PathPlot}\left(X\left[3\right]\left(t\right)\,t=0..1\,\mathrm{timesteps}=100\,\mathrm{replications}=5\,\mathrm{color}=\mathrm{red}..\mathrm{blue}\,\mathrm{thickness}=3\,\mathrm{axes}=\mathrm{BOXED}\,\mathrm{gridlines}=\mathrm{true}\right)$

$Y≔\mathrm{BlackScholesProcess}\left(\mathrm{S0}\,\mathrm{\sigma }\left[2\right]\,r\,d\right)\:$

$X\left[4\right]≔\mathrm{MertonJumpDiffusion}\left(Y\,\mathrm{\lambda }\left[1\right]\,0\,b\right)\:$

$\mathrm{PathPlot}\left(X\left[4\right]\left(t\right)\,t=0..1\,\mathrm{timesteps}=100\,\mathrm{replications}=5\,\mathrm{color}=\mathrm{red}..\mathrm{blue}\,\mathrm{thickness}=3\,\mathrm{axes}=\mathrm{BOXED}\,\mathrm{gridlines}=\mathrm{true}\right)$

$\mathrm{ExpectedValue}\left(\max \left(X\left[4\right]\left(1\right)-90\,0\right)\,\mathrm{timesteps}=100\,\mathrm{replications}={10}^5\right)$

$\left[\mathrm{value}=65.32590075\,\mathrm{standarderror}=0.5535149183\right]$

$S\left[1\right]≔\mathrm{SampleValues}\left(Y\left(1\right)\,\mathrm{timesteps}=100\,\mathrm{replications}={10}^5\right)$

$S\left[2\right]≔\mathrm{SampleValues}\left(X\left[4\right]\left(1\right)\,\mathrm{timesteps}=100\,\mathrm{replications}={10}^5\right)$

$\mathrm{P1}≔\mathrm{Statistics}\left[\mathrm{FrequencyPlot}\right]\left(S\left[1\right]\,\mathrm{range}=0..300\,\mathrm{thickness}=3\,\mathrm{color}=\mathrm{red}\,\mathrm{bincount}=50\right)\:$

$\mathrm{P2}≔\mathrm{Statistics}\left[\mathrm{FrequencyPlot}\right]\left(S\left[2\right]\,\mathrm{range}=0..300\,\mathrm{thickness}=3\,\mathrm{color}=\mathrm{blue}\,\mathrm{bincount}=50\right)\:$

$\mathrm{plots}\left[\mathrm{display}\right]\left(\mathrm{P1}\,\mathrm{P2}\,\mathrm{axes}=\mathrm{BOXED}\,\mathrm{gridlines}=\mathrm{true}\right)\:$

$J≔\mathrm{PoissonProcess}\left(\mathrm{\lambda }\left[2\right]\,\mathrm{Normal}\left(a\,b\right)\right)$

$J≔\mathrm{\_P}$

$Z≔t\mapsto Y\left(t\right)\cdot \exp \left(J\left(t\right)\right)$

$Z≔t\mapsto Y\left(t\right)\cdot {\ⅇ}^{J\left(t\right)}$

$\mathrm{ExpectedValue}\left(\max \left(Z\left(1\right)-90\,0\right)\,\mathrm{timesteps}=100\,\mathrm{replications}={10}^5\right)$

$\left[\mathrm{value}=30.41454912\,\mathrm{standarderror}=0.1764124093\right]$

Glasserman, P., Monte Carlo Methods in Financial Engineering. New York: Springer-Verlag, 2004.

Hull, J., Options, Futures, and Other Derivatives, 5th. edition. Upper Saddle River, New Jersey: Prentice Hall, 2003.

Merton, R.C., On the pricing when underlying stock returns are discontinuous, Journal of Financial Economics, (3) 1976, pp. 125-144.

The Finance[MertonJumpDiffusion] command was introduced in Maple 15.

For more information on Maple 15 changes, see Updates in Maple 15.