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Perpetual American Double Lookback Options on Drawdowns and Drawups with Floating Strikes

Pavel V. Gapeev ()
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Pavel V. Gapeev: London School of Economics

Methodology and Computing in Applied Probability, 2022, vol. 24, issue 2, 749-788

Abstract: Abstract We present closed-form solutions to the problems of pricing of the perpetual American double lookback put and call options on the maximum drawdown and the maximum drawup with floating strikes in the Black-Merton-Scholes model. It is shown that the optimal exercise times are the first times at which the underlying risky asset price process reaches some lower or upper stochastic boundaries depending on the current values of its running maximum or minimum as well as the maximum drawdown or maximum drawup. The proof is based on the reduction of the original double optimal stopping problems to the appropriate sequences of single optimal stopping problems for the three-dimensional continuous Markov processes. The latter problems are solved as the equivalent free-boundary problems by means of the smooth-fit and normal-reflection conditions for the value functions at the optimal stopping boundaries and the edges of the three-dimensional state spaces. We show that the optimal exercise boundaries are determined as either the unique solutions of the associated systems of arithmetic equations or the minimal and maximal solutions of the appropriate first-order nonlinear ordinary differential equations.

Keywords: Perpetual American double lookback options; Optimal double stopping problem; Geometric Brownian motion; Running maximum and minimum processes; Maximum drawdown and maximum drawup; First passage time; Free-boundary problem; Instantaneous stopping and smooth fit; Normal reflection; A change-of-variable formula with local time on surfaces; Primary 60G40; 34B40; 91G20; Secondary 60J60; 60J65; 91B70 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s11009-021-09917-y

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