On the minimal entropy martingale measure

Abstract

Let X be a locally bounded semimartingale. Using the theory of BMOmartingales we give a sufficient criterion for a martingale measure for X to minimize relative entropy among all martingale measures. This is applied to prove convergence of the q-optimal martingale measure to the minimal entropy martingale measure in entropy for q ↓ 1 under the assumption that X is continuous and that the density process of some equivalent martingale measure satisfies a reverse LLogL-inequality.

Indifference valuation in incomplete binomial models

“…We refer the reader to [7] (see, also, [4], [8], [13] and [23]) for its properties and the role of this measure in stochastic optimization problems of exponential utility.…”

Options Prices in Incomplete Markets

“…For example, giving a typical paper in each rubric, there is the idea of minimizing quadratic hedging risk error [8], the idea of choosing a risk neutral measure via indifference pricing [2], or the idea of minimizing the entropy between the objective measure and the risk neutral measure [10,11]. More recently there is the idea that the market can choose a risk neutral measure for pricing, as proposed in [13,19] and implemented in the form of calibration methods in the industry.…”

Robust Exponential Hedging and Indifference Valuation

“…(a) When P is a singleton, a measure transformation argument (see [7]) reduces the problem (2.2) to the minimization of relative entropy alone, which goes back to Csizár [5], and is studied by Miyahara [26], Frittelli [15], Grandits and Rheinländer [17] among others in the context of mathematical finance. However, we can not remove the penalty term −αE Q [H] in the general case, although this measure transformation is still useful.…”