On the Existence of Martingale Measures in Jump Diffusion Market Models

Abstract

In the context of jump-diffusion market models we construct examples that satisfy the weaker no-arbitrage condition of NA1 (NUPBR), but not NFLVR. We show that in these examples the only candidate for the density process of an equivalent local martingale measure is a supermartingale that is not a martingale, not even a local martingale. This candidate is given by the supermartingale deflator resulting from the inverse of the discounted growth optimal portfolio. In particular, we consider an example with constraints on the portfolio that go beyond the standard ones for admissibility.

No arbitrage and multiplicative special semimartingales

“…The latter is a process that acts multiplicatively and transforms nonnegative wealth processes into local martingales. The papers [16,19,26,27,37,41,46,50,56,60] and the textbooks [40,45] treat further developments and related topics concerning the FTAP. In certain situations, results about criteria for the absence of arbitrage have been derived, for example, in [13][14][15]51].…”

Weak and Strong No-Arbitrage Conditions for Continuous Financial Markets

“…A similar perspective is also adopted in the Stochastic Portfolio Theory (see [24]- [25]), where the NFLVR condition is not imposed as a normative assumption and it is shown that arbitrage opportunities may naturally arise in the market. Related works that explicitly consider situations where NFLVR may fail are [7], [8], [13], [27], [28], [35], [47], [49], [56], [73] and also, in the more specific case of diffusion models, [29], [53], [54] and [72] (see later in the text for more information). Somewhat surprisingly, these works have shown that the full strength of NFLVR is not necessarily needed in order to solve the fundamental problems of valuation, hedging and portfolio optimisation.…”