forward - Characteristic functions for options on futures

Using simple delta-probability decomposition, the price European call options a non- dividend paying asset can be computed as

\begin{equation} C(T,K) = {S_0}{\rm{ }}{\Pi _1} - {e^{ - rT}}K{\rm{ }}{\Pi _2}, \end{equation}

with

\begin{equation} {\Pi _1} = \frac{1}{2} + \frac{1}{\pi }\int_0^\infty {{\mathop{\rm Re}\nolimits} \left[ {\frac{{{e^{ - iw\ln (K)}}{\psi _{\ln {S_T}}}(w - i)}}{{iw{\psi _{\ln {S_T}}}( - i)}}} \right]} \;dw, \end{equation}

\begin{equation}\label{pi2} {\Pi _2} = \frac{1}{2} + \frac{1}{\pi }\int_0^\infty {{\mathop{\rm Re}\nolimits} \left[ {\frac{{{e^{ - iw\ln (K)}}{\psi _{\ln {S_T}}}(w)}}{{iw}}} \right]} \;dw, \end{equation}

and where ${\psi _{\ln {S_T}}}$ is the characteristic function of the log-asset price. For instance, for the Heston and Variance Gamma models, the corresponding $\psi^{H}_{\ln {S_T}}$ and $\psi^{VG}_{\ln {S_T}}$ are given by:

• Variance Gamma

\begin{equation} \psi _{\ln ({S_t})}^{VG}(w) = {\left( {\frac{1}{{1 - i\theta vw + ({\sigma ^2}v/2){w^2}}}} \right)^{t/v}} \end{equation}

\begin{equation} \psi_{\ln(S_t)}^{H} (w) = e^{ C(t,w) \overline{V}+ D(t,w) V_0 +iw \ln(S_0 e^{rt})}, \end{equation} where \begin{eqnarray*} C(t,w) &=& a \left[ r_{-} \cdot t - \frac{2}{\eta^2} \ln \left( \frac{1-g e^{-ht}}{1-g} \right) \right], \\ D(t,w) &=& r_{-} \frac{1-e^{-ht}}{1-g e^{-ht}}, \\ \alpha &=& - \frac{w^2}{2}- \frac{iw}{2}, \quad \beta = a - \rho \eta i w , \quad \gamma = \frac{\eta^2}{^2}, \\ r_{\pm} &=& \frac{ \beta \pm h}{\eta^2}, \quad h= \sqrt{ \beta^2- 4 \alpha \gamma}, \quad g= \frac{r_{-}}{r_{+}}. \end{eqnarray*}

These equations, however, are intended for options on spot prices. If I were interested in options on futures, which modifications should be made to the formulas above?(because it is not as straightforward as simply using $F_0$ instead of $S_0$ and removing the effects of $r$, right?)

Answer

I generally agree with Quantuple's comment.

I explicitly discuss the case of the variance gamma model, though most of this also applies to Heston. First note that in case of the variance gamma model, the characteristic function that you presented is not the one of the logarithmic stock price under the risk-neutral probability measure. Let $X$ be a variance gamma process as in Madan et al. (1998). The characteristic function of $X_t$ is

\begin{equation} \phi_{X_t}(\omega) = \left( 1 - \mathrm{i} \theta \nu \omega + \frac{1}{2} \sigma^2 \nu \omega^2 \right)^{-t / v}; \end{equation}

see Equation (7) in the original paper (note that there was originally a typo in your question). We now look for a drift term $\gamma$, such that the process

\begin{equation} Y_t = \exp \left\{ \gamma t + X_t \right\} \end{equation}

is a martingale. You find that

\begin{equation} \gamma = -\ln \left( \phi_{X_t}(-\mathrm{i}) \right). \end{equation}

Your model for the stock and forward prices respectively is then

\begin{eqnarray} S_t & = & S_0 \exp \left\{ (r + \gamma) t + X_t \right\}\\ F_t(T) & = & F_0(T) \exp \left\{ \gamma t + X_t \right\} \end{eqnarray}

with characteristic functions

\begin{eqnarray} \phi_{\ln \left( S_t \right)}(\omega) & = & \exp \left\{ \mathrm{i} \left( \ln \left( S_0 \right) + r + \gamma \right) \omega t \right\} \phi_{X_t}(\omega)\\ \phi_{\ln \left( F_t(T) \right)}(\omega) & = & \exp \left\{ \mathrm{i} \left( \ln \left( F_0(T) \right) + \gamma \right) \omega t \right\} \phi_{X_t}(\omega) \end{eqnarray}

You can now re-use your general expressions for $\Pi_1$ and $\Pi_2$ in terms terms of the appropriate characteristic function as these are just the corresponding exercise probabilities under the respective measure. Your pricing formula then becomes

\begin{equation} C_0 = e^{-r T} \left( F_0(T) \Pi_1 - K \Pi_2 \right). \end{equation}

If you want to convince yourself that this is correct, then you could work through the detailed steps given e.g. in Schmelzle (2010).

References

Madan, Dilip B, Peter P. Carr and Eric C. Chang (1998) "The Variance Gamma Process and Option Pricing", European Finance Review, Vol. 2, pp. 79-105

Schmelzle, Martin (2010) "Option Pricing Formulae Using Fourier Transform: Theory and Application", Technical Report