The minimal k-entropy martingale measure

We introduce the notion of κ-entropy (κ ∈ R, |κ| ≤ 1), starting from Kaniadakis’ (2001, 2002, 2005) one-parameter deformation of the ordinary exponential function. The κ- entropy is in duality with a new class of utility functions which are close to the exponential utility functions, for small values of wealth, and to the power law utility functions, for large values of wealth. We give conditions on the existence and on the equivalence to the basic measure of the minimal κ-entropy martingale measure. Moreover, we provide characterizations of its density as a κ-exponential function. We show that the minimal κ-entropy martingale measure is closely related to both the standard entropy martingale measure and the well known q-optimal martingale measures. We finally establish the convergence of the minimal κ-entropy martingale measure to the minimal entropy martingale measure as κ tends to 0.