Valuation models

Heston stochastic volatility

European calls and puts priced by Fourier inversion with QuantLib, the five parameters given or calibrated to an uploaded volatility surface, the implied smile, fit errors and Greeks.

Inputs

Results

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Methodology

Conventions

  • Variance follows a mean-reverting square-root process, correlated with the underlying.
  • Priced by Fourier inversion of the characteristic function (QuantLib's analytic Heston engine).
  • Parameters not given are fitted to the surface's vols at seven strikes per expiry.

References

  • Heston (1993)
  • Gatheral, The Volatility Surface, ch. 2

Formulas

Variance
dv = κ(θ − v) dt + σ √v dW₂, dW₁ dW₂ = ρ dt
Price
C = S e^(−qT) P₁ − K e^(−rT) P₂
Feller
2κθ > σ²