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.
In this pose, we consider an Ornstein-Uhlenbeck stochastic clock whose instantaneous rate of activity process has an inverse Gaussian stationary distribution. We use the previously obtained relationship between the cumulant generation function of the stationary distribution and that of the background driving Lévy process and show that the latter can be represented as the sum of an inverse Gaussian processes and a compound Poisson process with chi-squared distributed increments.
In this post, we consider stochastic clocks whose instantaneous rate of activity follows a square root process. We present all intermediate steps in the derivation of the characteristic function of the corresponding time change. The corresponding steps closely resemble those in the derivation of the zero-coupon bond price in the Cox et al. (1985) model for the short-term interest rate.
In C++, we often use the substitution failure is not an error (SFINAE) rule in template overload resolution. A common problem is to write template specializations for class template arguments that expose member functions with a certain signature. The below macro allows to easily create member function detector type traits for user defined classes