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Time Series and Stochastic Differential Equations
Heston Model
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1.}, {0.4736458597280765, 1.6917786203203786`}}, PlotRangeClipping>True, PlotRangePadding>{{0.02, 0.02}, {0.024362655211846045`, 0.024362655211846045`}}]\), ImageSize > {144, 82}, AspectRatio > 76/144, LabelStyle > {FontSize > 6}, PlotLabel > Style["Volatility", FontFamily > "Helvetica", 10, GrayLevel[.2]], Ticks > False], FrameStyle > GrayLevel[203/255], FrameMargins > 2], "Image"]
Define an
ItoProcess
corresponding to the correlated 2D Wiener process.
In[1]:=
X
cW[\[Rho]_] := ItoProcess[{{0, 0}, IdentityMatrix[2]}, {{w1, w2}, {0, 0}}, t, {{1, \[Rho]}, {\[Rho], 1}}];
Define a Heston model by SDEs driven by the correlated 2D Wiener process.
In[2]:=
X
hm = ItoProcess[{ \[DifferentialD]s[t] == \[Mu] s[t] \[DifferentialD]t + Sqrt[r[t]] s[t] \[DifferentialD]Subscript[w, s][t], \[DifferentialD]r[ t] == \[Kappa] (\[Theta]  r[t]) \[DifferentialD]t + \[Xi] Sqrt[ r[t]] \[DifferentialD]Subscript[w, \[Nu]][t]}, {s[t], r[t]}, {{s, r}, {Subscript[s, 0], Subscript[r, 0]}}, t, {Subscript[w, s], Subscript[w, \[Nu]]} \[Distributed] cW[\[Rho]]];
Simulate the model using a stochastic RungeKutta scheme.
In[3]:=
X
td = BlockRandom[SeedRandom[1988]; RandomFunction[ hm /. {\[Mu] > 0, \[Kappa] > 2, \[Theta] > 1, \[Xi] > 1/2, \[Rho] > 1/3, Subscript[s, 0] > 25, Subscript[r, 0] > 1.25}, {0, 1, 0.005}, 6, Method > "StochasticRungeKutta"]];
Visualize the sample paths.
In[4]:=
X
Row[{ListLinePlot[td["PathComponent", 1], PlotLabel > "Price of the asset", ImageSize > 250], ListLinePlot[td["PathComponent", 2], PlotLabel > "Volatility of the asset", ImageSize > 250]}, Spacer[20]]
Out[4]=