Minimal q-entropy martingale measures for exponential time-changed Lévy processes

In this paper, we consider structure preserving measure transforms for time-changed Lévy processes. Within this class of transforms preserving the time-changed Lévy structure, we derive equivalent martingale measures minimizing relative q-entropy. Structure preservation is found to be an inherent property of minimal q-entropy martingale measures under continuous time changes, whereas it imposes an additional restriction for discontinuous time changes. Our framework is demonstrated to allow for entropy minimization while remaining within a certain parametric class of (multidimensional) subordinated Brownian motion if the parametric class of subordinators is closed under Esscher transforms. Closedness under Esscher transforms is shown to hold for Gamma, Inverse Gaussian, Generalized Inverse Gaussian and Tempered Stable subordinators, while the Meixner class is generalized such that the corresponding class of subordinators is also closed. Within the class of time-changed Brownian motion, prices of options with convex payoff structure are decreasing in q . Moreover, option price orderings within the parametric classes under consideration are established.