The Annals of Applied Probability

Diffusion transformations, Black–Scholes equation and optimal stopping

Umut Çetin

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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 [Probab. Theory Related Fields 164 (2016) 1027–1069] 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.

Article information

Source
Ann. Appl. Probab., Volume 28, Number 5 (2018), 3102-3151.

Dates
Received: February 2017
Revised: January 2018
First available in Project Euclid: 28 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1535443244

Digital Object Identifier
doi:10.1214/18-AAP1385

Mathematical Reviews number (MathSciNet)
MR3847983

Zentralblatt MATH identifier
06974775

Subjects
Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 60J55: Local time and additive functionals 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)

Keywords
Linear diffusions local time potential operators optimal stopping strict local martingales financial bubbles nonuniqueness of the Cauchy problem h-transform

Citation

Çetin, Umut. Diffusion transformations, Black–Scholes equation and optimal stopping. Ann. Appl. Probab. 28 (2018), no. 5, 3102--3151. doi:10.1214/18-AAP1385. https://projecteuclid.org/euclid.aoap/1535443244


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