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
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)
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