Expected Profit from an Option 

Assume to be the price of a financial asset at time and assume it follows a WienerProcess. The value of a simple option is the amount it exceeds 12 at time . Find the expected value of the option.

In[1]:=

X priceProcess = WienerProcess[\[Mu], \[Sigma]];

In[2]:=

X optionValue = Expectation[Max[x[t] - 12, 0], x \[Distributed] priceProcess]

Out[2]=

Plot the value for and different assumptions on maturity time and volatility .

In[3]:=

X Plot3D[Evaluate[optionValue /. \[Mu] -> 1], {t, 0, 25}, {\[Sigma], 1, 5}, AxesLabel -> Automatic, ImageSize -> Medium]

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