Andersen-Piterbarg Integration Limits for the Time Dependent Heston Model

The normalized characteristic function \phi_t(z) of a piecewise constant time dependent Heston model

\begin{array}{rcl} d \ln S_t&=& \left(r_t - q_t - \frac{\nu_t}{2}\right)dt + \sqrt\nu_t dW^{S}_t \nonumber \\ d\nu_t&=& \kappa_t\left(\theta_t-\nu_t \right ) dt + \sigma_t\sqrt\nu_t dW^{\nu}_t \nonumber \\ \rho_t dt &=& dW^{S}_tdW^{\nu}_t \end{array}

with n time intervals [t_0=0, t_1], ... ,[t_{n-1}, t_n] and constant parameters within these intervals

\kappa_t = \kappa_j \wedge \theta_t = \theta_j \wedge \sigma_t=\sigma_j \wedge \rho_t=\rho_j \forall j\in [1, n] \wedge t\in [t_{j-1}, t_j]

is given by the recurrence relation

\begin{array}{rcl}  k_j &=& \kappa_j - iz\rho_j\sigma_j \nonumber \\ d_j &=& \sqrt{k_j^2 +\sigma_j^2(z^2+iz)} \nonumber \\  g_j &=&\displaystyle \frac{k_j- d_j}{k_j + d_j} \nonumber \\ \tilde{g_j} &=& \displaystyle \frac{k_j- d_j - D_{j+1}\sigma_j^2}{k_j + d_j - D_{j+1}\sigma_j^2} \nonumber \\ D_j &=&  \displaystyle \frac{k_j + d_j}{\sigma_j^2}\frac{g_j-\tilde{g_j} e^{-d_j \tau_j}}{1-\tilde{g_j} e^{-d_j\tau_j}} \nonumber \\ C_j &=& \displaystyle \frac{\kappa_j\theta_j}{\sigma_j^2} \left( (k_j- d_j )\tau_j - 2\ln\left(\frac{1-\tilde{g_j}e^{-d_j\tau_j}}{1-\tilde{g_j}}\right)\right) + C_{j+1} \nonumber \\ \tau_j &=& t_j - t_{j-1} \nonumber \\ \phi_{t_n}(z) &=& \exp{\left(C_1(z)+D_1(z)\nu_0\right)}\end{array}

and the initial condition

C_{n+1} = D_{n+1} = 0

The should be noted that the complex logarithm in this formulation can be restricted to the principal branch without introducing any discontinuities [1]. It is important to know the asymptotic behaviour of C_1(z)+D_1(z)\nu_0 in order to calculate the truncation point for the integral over the characteristic function when using the Andersen-Piterbarg approach with control variate [2]. A Mathematica script gives

\begin{array}{rcl} \displaystyle \lim_{u\to\infty}\frac{D_1(u)}{u} &=& \displaystyle-\frac{i\rho_1 + \sqrt{1-\rho_1^2} }{\sigma_1} \nonumber \\ \displaystyle\lim_{u\to\infty}\frac{C_1(u)}{u} &=& \displaystyle -\sum_{j=1}^n \frac{\kappa_j\theta_j}{\sigma_j}\left(\sqrt{1-\rho_j^2} + i\rho_j\right)\tau_j\end{array}

The implementation of the Andersen-Piterbarg method for the piecewise constant time dependent Heston model is part of the pull request #251.

[1] Afshani, S. (2010) Complex logarithms and the piecewise constant extension
of the Heston model.

[2] Andersen, L. and Piterbarg, V. (2010) Interest Rate Modeling, Volume I: Foundations and Vanilla Models, (Atlantic Financial Press London).

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