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Heston Model

    
Define the Ito process of the correlated two‐dimensional Wiener process:
Wolfram Language code: cW[ρ_] := ItoProcess[{{0, 0}, IdentityMatrix[2]}, {{w1, w2}, {0, 0}}, t, {{1, ρ}, {ρ, 1}}];
Define the Heston model by SDEs driven by the correlated 2D Wiener process:
Wolfram Language code: hm = ItoProcess[{ ⅆs[t] == μ s[t]ⅆt + Sqrt[r[t]]s[t]ⅆSubscript[w, s][t], ⅆr[t] == κ (θ - r[t])ⅆt + ξ Sqrt[r[t]]ⅆSubscript[w, ν][t]}, {s[t], r[t]}, {{s, r}, {Subscript[s, 0], Subscript[r, 0]}}, t, {Subscript[w, s], Subscript[w, ν]}cW[ρ]];
Simulate the model using the stochastic Runge–Kutta scheme:
Wolfram Language code: td = BlockRandom[SeedRandom[1988];RandomFunction[hm /. {μ -> 0, κ -> 2, θ -> 1, ξ -> 1 / 2, ρ -> -1 / 3, Subscript[s, 0] -> 25, Subscript[r, 0] -> 1.25}, {0, 1, 0.005}, 6, Method -> "StochasticRungeKutta"]];
Visualize paths:
Wolfram Language code: Row[{ListLinePlot[td["PathComponent", 1], PlotLabel -> "Price of the asset"], ListLinePlot[td["PathComponent", 2], PlotLabel -> "Volatility of the asset"]}, Spacer[20]]