# Model Option Prices Using Merton Jump-Diffusion

Define Merton's jump-diffusion model for option pricing.

 In[1]:= XMerton\[ScriptCapitalP] = TransformedProcess[(v - \[Sigma]^2/2) t + \[Sigma] w[t] + j[t], {w \[Distributed] WienerProcess[r, \[Sigma]], j \[Distributed] CompoundPoissonProcess[\[Lambda], NormalDistribution[\[Mu], \[Delta]]]}, t];

Simulate the process.

 In[2]:= Xdata = RandomFunction[ Merton\[ScriptCapitalP] /. {v -> 1.2, \[Sigma] -> 0.7, \[Lambda] -> 1.3, \[Mu] -> 0.92, \[Delta] -> 0.425, r -> 1}, {0, 3, 0.02}, 3];
 In[3]:= XListLinePlot[data]
 Out[3]=

Compute slice properties for the process.

 In[4]:= X{Mean[Merton\[ScriptCapitalP][t]], Variance[Merton\[ScriptCapitalP][t]]}
 Out[4]=

Approximate slice distribution from sample.

 In[5]:= XsliceSample = RandomVariate[ Merton\[ScriptCapitalP][ 1] /. {v -> 1.2, \[Sigma] -> 0.7, \[Lambda] -> 1.3, \[Mu] -> 0.92, \[Delta] -> 0.425, r -> 1}, 10^6];
 In[6]:= XSmoothHistogram[sliceSample]
 Out[6]=

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