Pricing European Options on Forwards with Heston Dynamics
Summary
The answer adapts the Heston stochastic volatility model to a forward by defining a zero-start log-return process whose exponential is a martingale under the pricing measure. It gives the characteristic function for that process, using the numerically stable Heston representation associated with Albrecher and colleagues. The construction models return variance with mean reversion and correlated Brownian shocks.
To obtain a forward's log-price characteristic function, the process characteristic function is shifted by the logarithm of the initial forward level. For a dividend-paying stock, the answer instead adds the risk-free rate less the dividend yield over time. The resulting characteristic function can be used with Fourier-based European vanilla option pricing methods, with the cosine expansion method cited as an example. The post provides formulas and references, but no numerical validation or discussion of calibration, parameter constraints, or implementation edge cases.
Key ideas
- The model uses Heston variance dynamics and a log-return process whose exponential is a pricing-measure martingale.
- A closed-form characteristic function is provided for the stochastic volatility process.
- A forward's log-price characteristic function follows by shifting for its initial forward level.
- Fourier-based vanilla option pricing can use the resulting characteristic function.
- The response gives formulas but no empirical tests or implementation guidance.
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Full text
# Heston with Forward Dynamics
# Heston with Forward Dynamics
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 by LocalVolatility (score 1, accepted)
https://quant.stackexchange.com/a/34745
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. 826-848Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.