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Timing in the Presence of Directional Predictability: Optimal Stopping of Skew Brownian Motion

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  • Luis H. R. Alvarez E.
  • Paavo Salminen

Abstract

We investigate a class of optimal stopping problems arising in, for example, studies considering the timing of an irreversible investment when the underlying follows a skew Brownian motion. Our results indicate that the local directional predictability modeled by the presence of a skew point for the underlying has a nontrivial and somewhat surprising impact on the timing incentives of the decision maker. We prove that waiting is always optimal at the skew point for a large class of exercise payoffs. An interesting consequence of this finding, which is in sharp contrast with studies relying on ordinary Brownian motion, is that the exercise region for the problem can become unconnected even when the payoff is linear. We also establish that higher skewness increases the incentives to wait and postpones the optimal timing of an investment opportunity. Our general results are explicitly illustrated for a piecewise linear payoff.

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  • Luis H. R. Alvarez E. & Paavo Salminen, 2016. "Timing in the Presence of Directional Predictability: Optimal Stopping of Skew Brownian Motion," Papers 1608.04537, arXiv.org.
    • Handle: RePEc:arx:papers:1608.04537
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    Cited by:

    1. Lempa, Jukka & Mordecki, Ernesto & Salminen, Paavo, 2024. "Diffusion spiders: Green kernel, excessive functions and optimal stopping," Stochastic Processes and their Applications, Elsevier, vol. 167(C).
    2. Antoine Lejay & Paolo Pigato, 2019. "A Threshold Model For Local Volatility: Evidence Of Leverage And Mean Reversion Effects On Historical Data," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(04), pages 1-24, June.
    3. Haoyan Zhang & Yingxu Tian, 2022. "Hitting Time Problems of Sticky Brownian Motion and Their Applications in Optimal Stopping and Bond Pricing," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 1237-1251, June.
    4. Antoine Lejay, 2018. "Estimation of the bias parameter of the skew random walk and application to the skew Brownian motion," Statistical Inference for Stochastic Processes, Springer, vol. 21(3), pages 539-551, October.
    5. Antoine Lejay & Paolo Pigato, 2017. "Data and methods for A threshold model for local volatility: evidence of leverage and mean reversion effects on historical data [Données et méthodes pour "A threshold model for local volatilit," Working Papers hal-01668975, HAL.

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