Heston with Forward Dynamics

Question

I'm just curious if it's possible to use forward dynamics and work out the pricing formulas. Does anyone know if there's a reference (paper/url) I can look at?

Answer 1

We define a process $X$ to start at $X_0 = 0$ have the Heston-like $\mathbb{Q}$-dynamics

\begin{eqnarray} \mathrm{d}X_t & = & -\frac{1}{2} V_t \mathrm{d}t + \sqrt{V_t} \mathrm{d}W_t^{(1)},\\ \mathrm{d}V_t & = & \kappa \left( \theta - V_t \right) \mathrm{d}t + \xi \sqrt{V_t} \mathrm{d}W_t^{(2)}. \end{eqnarray}

with $\mathrm{d} \langle W^{(1)}, W^{(2)} \rangle_t = \rho \mathrm{d}t$ and such that $e^{X_t}$ is a $\mathbb{Q}$-martingale. The characteristic function of $X_t$ under $\mathbb{Q}$ is given by

\begin{equation} \phi_{X_t}(\omega) = \exp \left\{ C(\omega, t) + D(\omega, t) V_0 \right\}, \end{equation}

where

\begin{eqnarray} C(\omega, t) & = & \frac{\kappa \theta}{\xi^2} \left[ \left( \kappa - \rho \xi \mathrm{i} \omega - d(\omega) \right) t - 2 \ln \left( \frac{1 - c(\omega) e^{-d(\omega) t}}{1 - c(\omega)} \right) \right],\\ D(\omega, t) & = & \frac{\kappa - \rho \xi \mathrm{i} \omega - d(\omega)}{\xi^2} \left( \frac{1 - e^{-d(\omega) t}}{1 - c(\omega) e^{-d(\omega) t}} \right),\\ c(\omega) & = & \frac{\kappa - \rho \xi \mathrm{i} \omega - d(\omega)}{\kappa - \rho \xi \mathrm{i} \omega + d(\omega)},\\ d(\omega) & = & \sqrt{(\rho\xi \mathrm{i} \omega - \kappa)^2 + \xi^2 \left( \mathrm{i} \omega + \omega^2 \right)}. \end{eqnarray}

This representation of the characteristic function is based on the numerically stable version provided in Albrecher et al. (2007).

Using $X$ you can now construct the underlying assets of interest and immediately obtain the relevant characteristic function. In case of a stock with dividend yield $q$, you set

\begin{equation} S_t = S_0 e^{X_t + (r - q) t} \qquad \Rightarrow \qquad \phi_{\ln \left( S_t \right)}(\omega) = \phi_{X_t}(\omega) e^{\mathrm{i} \omega \left( \ln \left( S_0 \right) + (r - q) t \right)}. \end{equation}

For a forward, you get

\begin{equation} F_t = F_0 e^{X_t} \qquad \Rightarrow \qquad \phi_{\ln \left( F_t \right)}(\omega) = \phi_{X_t}(\omega) e^{\mathrm{i} \omega \ln \left( F_0 \right)} \end{equation}

You can then directly employ your favourite characteristic function-based European vanilla option pricer - e.g. Fang and Oosterlee (2008).

References

Albrecher, Hansjoerg, Philipp Mayer, Wim Shoutens and Jurgen Tistaert (2007) "The Little Heston Trap," Wilmott Magazine, Mar.

Fang, Fang and Cornelis W. Oosterlee (2008) "A Novel Pricing Method for European Options Based on Fourier-Cosine Series Expansions," Siam Journal of Scientific Computing, Vol. 31, No. 2, pp. 826-848