We consider plain vanilla European options written on an underlying asset that follows a continuous time semi-Markov multiplicative process. derive formula and renewal type equation for the martingale option price. In case in which intertrade times follow Mittag-Leffler distribution, under appropriate scaling, we prove these prices converge to price of geometric Brownian motion time-changed wit...

We investigate the optimal strategy over a finite time horizon for a portfolio of stock and bond and a derivative in an multiplicative Markovian market model with transaction costs (friction). The optimization problem is solved by a Hamilton-Bellman-Jacobi equation, which by the verification theorem has well-behaved solutions if certain conditions on a potential are satisfied. In the case at ha...

In the Black-Scholes option price model Brownian motion and the underlying Normal distribution play a fundamental role. Empirical evidence however shows that the normal distribution is a very poor model to fit real-life data. In order to achieve a better fit we replace the Brownian motion by a special Lévy process: the Meixner process. We show that the underlying Meixner distribution allows an ...