In this post, I augment a constant coefficient geometric Brownian motion process by a single jump whose time of occurrence is known. The random variable representing the jump size follows a normal mixture distribution. To get a feeling for the impact of predictable jumps on option prices, we inspect the shape of a few model implied probability densities and volatility smiles.
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.