A Visual Classification of Local Martingales

Hardy Hulley and Eckhard Platen ()

No 238, Research Paper Series from Quantitative Finance Research Centre, University of Technology, Sydney

Abstract: This paper considers the problem of when a local martingale is a martingale or a universally integrable martingale, for the case of time-homogeneous scalar diffusions. Necessary and suffcient conditions of a geometric nature are obtained for answering this question. These results are widely applicable to problems in stochastic finance. For example, in order to apply risk-neutral pricing, one must first check that the chosen density process for an equivalent change of probability measure is in fact a martingale. If not, risk-neutral pricing is infeasible. Furthermore, even if the density process is a martingale, the possibility remains that the discounted price of some security could be a strict local martingale under the equivalent risk-neutral probability measure. In this case, well-known identities for option prices, such as put-call parity, may fail. Using our results, we examine a number of basic asset price models, and identify those that suffer from the above-mentioned difficulties.

Keywords: diffusions; first-passage times; Laplace transforms; local martingales; ordinary differential equations (search for similar items in EconPapers)
Date: 2008-12-01

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