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Optimal Stopping Problems in Diffusion-Type Models with Running Maxima and Drawdowns

Part of: Functional-differential and differential-difference equations Markov processes Stochastic processes Mathematical economics Ordinary differential operators

Published online by Cambridge University Press:  30 January 2018

Pavel V. Gapeev  and
Neofytos Rodosthenous
Pavel V. Gapeev*
Affiliation:
London School of Economics
Neofytos Rodosthenous*
Affiliation:
London School of Economics
*
∗ Postal address: London School of Economics, Department of Mathematics, Houghton Street, London WC2A 2AE, UK. Email address: p.v.gapeev@lse.ac.uk
∗ Postal address: London School of Economics, Department of Mathematics, Houghton Street, London WC2A 2AE, UK. Email address: p.v.gapeev@lse.ac.uk
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Abstract

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We study optimal stopping problems related to the pricing of perpetual American options in an extension of the Black-Merton-Scholes model in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and maximum drawdown. The optimal stopping times of the exercise are shown to be the first times at which the price of the underlying asset exits some regions restricted by certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. We obtain closed-form solutions to the equivalent free-boundary problems for the value functions with smooth fit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting three-dimensional Markov process. We derive first-order nonlinear ordinary differential equations for the optimal exercise boundaries of the perpetual American standard options.

Keywords

Multidimensional optimal stopping problemBrownian motionrunning maximum and running maximum drawdown processfree-boundary probleminstantaneous stopping and smooth fitnormal reflectionchange-of-variable formula with local time on surfacesperpetual American option

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 34K10: Boundary value problems 91B70: Stochastic models
Secondary: 60J60: Diffusion processes 34L30: Nonlinear ordinary differential operators 91B25: Asset pricing models
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 3 , September 2014 , pp. 799 - 817
Copyright
© Applied Probability Trust 

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