Diffusion transformations, Black-Scholes equation and optimal stopping

Umut Cetin

LSE Research Online Documents on Economics from London School of Economics and Political Science, LSE Library

Abstract: We develop a new class of path transformations for one-dimensional diffusions that are tailored to alter their long-run behaviour from transient to recurrent or vice versa. This immediately leads to a formula for the distribution of the first exit times of diffusions, which is recently characterised by Karatzas and Ruf [26] as the minimal solution of an appropriate Cauchy problem under more stringent conditions. A particular limit of these transformations also turn out to be instrumental in characterising the stochastic solutions of Cauchy problems defined by the generators of strict local martingales, which are well-known for not having unique solutions even when one restricts solutions to have linear growth. Using an appropriate diffusion transformation we show that the aforementioned stochastic solution can be written in terms of the unique classical solution of an alternative Cauchy problem with suitable boundary conditions. This in particular resolves the long-standing issue of non-uniqueness with the Black-Scholes equations in derivative pricing in the presence of bubbles. Finally, we use these path transformations to propose a unified framework for solving explicitly the optimal stopping problem for one-dimensional diffusions with discounting, which in particular is relevant for the pricing and the computation of optimal exercise boundaries of perpetual American options.

Keywords: linear diffusions; local time; potential operators; optimal stopping; strict local martingales; financial bubbles; nonuniqueness of the Cauchy problem; h-transform (search for similar items in EconPapers)
JEL-codes: C1 (search for similar items in EconPapers)
Date: 2018-10-01
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Citations: View citations in EconPapers (1)

Published in Annals of Applied Probability, 1, October, 2018, 28(5), pp. 3102-3151. ISSN: 1050-5164

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