Option Pricing When a Single Jump is Anticipated I

We consider a simple and tractable model for an asset price process that is subject to a large jump with a known time of occurrence. These situations often arise around scheduled news releases such as quarterly earnings announcements, monetary policy decisions or political elections. This post is the first in a series on this topic. I provide the general pricing setup and derive the characteristic function of the corresponding logarithmic return process including drift adjustment. Future posts then make specific choices for the jump size distribution and provide examples for the corresponding implied volatility smiles.

General Setup

We assume that the logarithmic asset price process X_t = \ln \left( S_t / S_0 \right) is driven by a Brownian motion W = \left\{ W_t : t \in \mathbb{R}_{+, 0} \right\} plus a single jump. The assumption of the a Brownian motion is not essential and could be replaced by another process whose characteristic function is available. The jump time t_{\text{J}} is known and has a random jump size Y, independent of the process W. Define

    \begin{equation*} X_t = \int_0^t \gamma(u) \mathrm{d}u + \sigma W_t + Y \mathrm{1} \left\{ t \geq t_{\text{J}} \right\}, \end{equation*}

where \gamma: \mathbb{R}_{+, 0} \rightarrow \mathbb{R} is the deterministic instantaneous drift which we specify later and \sigma \in \mathbb{R}_{+, 0} is the diffusion coefficient. X has the characteristic function

    \begin{equation*} \phi_{X_t}(\omega) = \exp \left\{ \mathrm{i} \omega \int_0^t \gamma(u)  \mathrm{d}u - \frac{1}{2} \sigma^2 \omega^2 t \right\} \left( \mathrm{1} \left\{ t < t_{\text{J}} \right\} + \phi_Y(\omega) \mathrm{1} \left\{ t \geq t_{\text{J}} \right\} \right), \end{equation*}

where \phi_Y(\omega) is the characteristic function of Y. In order for the discounted asset price process to be a martingale, we require that

    \begin{equation*} \mathbb{E} \left[ e^{X_t} \right] = e^{r t} \qquad \Leftrightarrow \qquad \ln \left( \phi_{X_t}(-\mathrm{i}) \right) = r t \end{equation*}

and get

    \begin{equation*} \gamma(t) = r - \frac{1}{2} \sigma^2 - \ln \left( \phi_Y(-\mathrm{i}) \right) \delta \left( t - t_{\text{J}} \right), \end{equation*}

where \delta(x) is the Dirac delta function. The last term corresponds to a point jump at t_{\text{J}} that ensures that the asset price on expectation stays unchanged as we cross the jump time. This amounts to assuming that all information that affects the average future asset price is already incorporated in the current spot price.

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