The characteristic function of the Heston model, as modified according to Gatheral (2006), is given by:
φ_H(t; w) = exp[C(t,w)̅V + D(t,w)V₀ + iwln(S₀e^(rt))]
whereV ̅ is the long-term variance, V₀ is the initial variance, S₀ is the initial stock price, r is the risk-free rate, and w is the frequency variable. The coefficients C(t,w) and D(t,w) are given by:
C(t,w) = ((r i w - λ θ)/σ^2) * (1 - e^(-λ t)) + (θ/(2λ)) * ((λ - r i w) t - 2 ln((1 - g e^(-λ t))/(1 - g)))
and
D(t,w) = (1/σ^2) * (1 - e^(-λ t)) * ((r i w - λ θ)/λ - 1/2) + (1/λ^2) * {(λ - r i w)(1 - e^(-λ t)) - 2(ln(1 - g e^(-λ t)) - g e^(-λ t))}
where λ is the mean reversion rate, θ is the long-term variance, σ is the volatility of volatility, and g = (λ - r i w)/(λ + r i w) is the damping factor.
φ_H(t; w) = exp[C(t,w)̅V + D(t,w)V₀ + iwln(S₀e^(rt))]
whereV ̅ is the long-term variance, V₀ is the initial variance, S₀ is the initial stock price, r is the risk-free rate, and w is the frequency variable. The coefficients C(t,w) and D(t,w) are given by:
C(t,w) = ((r i w - λ θ)/σ^2) * (1 - e^(-λ t)) + (θ/(2λ)) * ((λ - r i w) t - 2 ln((1 - g e^(-λ t))/(1 - g)))
and
D(t,w) = (1/σ^2) * (1 - e^(-λ t)) * ((r i w - λ θ)/λ - 1/2) + (1/λ^2) * {(λ - r i w)(1 - e^(-λ t)) - 2(ln(1 - g e^(-λ t)) - g e^(-λ t))}
where λ is the mean reversion rate, θ is the long-term variance, σ is the volatility of volatility, and g = (λ - r i w)/(λ + r i w) is the damping factor.
