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# Heston model

The Heston model (Heston, 1993) is a generalisation of the standard Black-Scholes model to incorporate stochastic volatility.
The Heston model satisfies the following system of SDE:s

dS(t)=μS(t)dt+(V(t))^{1/2}dW_{1}(t), S(0)=s_{0},

dV(t)=κ(θ-V(t))dt+σ(V(t))^{1/2}dW_{2}(t),V(0)=v_{0},

where μ>0,κ>0,θ>0,σ>0. Usually one also wants to have κθ>σ^{2}/2 (Feller, 1951) to guarantee that V stays strictly positive.
## References

Heston, Steven L. (1993). A Closed-Form Solution for Options with Stochastic Volatility with Application to Bond and Currency Options, *The Review of Financial Studies* **6**, 327-343. Journal article.

Feller, W. (1951): Two Singular Diffusion Problems, Annals of Mathematics,** 54 **, 173-182. Journal article

dS(t)=μS(t)dt+(V(t))

dV(t)=κ(θ-V(t))dt+σ(V(t))

where μ>0,κ>0,θ>0,σ>0. Usually one also wants to have κθ>σ

Feller, W. (1951): Two Singular Diffusion Problems, Annals of Mathematics,

Questions: Magnus Wiktorsson

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