# zbMATH — the first resource for mathematics

Optimal prediction of resistance and support levels. (English) Zbl 1396.60041
Summary: Assuming that the asset price $$X$$ follows a geometric Brownian motion, we study the optimal prediction problem $\mathop{\operatorname{inf}}\limits_{0 \leq \tau \leq T}\mathsf{E}| X_\tau^x - \ell |$ where the infimum is taken over stopping times $$\tau$$ of $$X$$ and $$\ell$$ is a hidden aspiration level (having a potential of creating a resistance or support level for $$X$$). Adopting the ‘aspiration-level hypothesis’ and assuming that $$\ell$$ is independent from $$X$$, we show that a wide class of admissible (non-oscillatory) laws of $$\ell$$ lead to unique optimal trading boundaries that can be viewed as the ‘conditional median curves’ for the resistance and support levels (with respect to $$X$$ and $$T$$). We prove the existence of these boundaries and derive the (nonlinear) integral equations which characterize them uniquely. The results are illustrated through some specific examples of admissible laws and their conditional median curves.

##### MSC:
 60G35 Signal detection and filtering (aspects of stochastic processes) 60G40 Stopping times; optimal stopping problems; gambling theory
Full Text:
##### References:
 [1] Glover, K., H. Hulley, and G. Peskir. 2013. “Three-Dimensional Brownian Motion and the Golden Ratio Rule.” The Annals of Applied Probability 23: 895-922. doi:10.1214/12-AAP859. · Zbl 1408.60032 [2] Gomes, C., and H. Waelbroeck. 2010. “An Empirical Study of Liquidity Dynamics and Resistance and Support Levels.” Quantitative Finance 10: 1099-1107. doi:10.1080/14697680902814258. [3] Osler, C.2000. “Support for Resistance: Technical Analysis and Intraday Exchange Rates.” Federal Reserve Bank of New York Policy Review 6: 53-68. [4] Peskir, G.2005. “A Change-of-Variable Formula with Local Time on Curves.” Journal of Theoretical Probability 18: 499-535. doi:10.1007/s10959-005-3517-6. · Zbl 1085.60033 [5] Peskir, G.2012. “Optimal Detection of a Hidden Target: The Median Rule.” Stochastic Processes and Their Applications 122: 2249-2263. doi:10.1016/j.spa.2012.02.004. · Zbl 1253.60055 [6] Peskir, G.2014. “Quickest Detection of a Hidden Target and Extremal Surfaces.” The Annals of Applied Probability 24: 2340-2370. doi:10.1214/13-AAP979. · Zbl 1338.60115 [7] Peskir, G., and A. N. Shiryaev. 2006. Optimal Stopping and Free-Boundary Problems. Lectures in Mathematics. ETH Zürich, Birkhäuser: Basel. · Zbl 1115.60001 [8] Shefrin, H.2008. A Behavioral Approach to Asset Pricing. Elsevier Inc. [9] Simon, H. A.1955. “A Behavioral Model of Rational Choice.” The Quarterly Journal of Economics 69: 99-118. doi:10.2307/1884852. [10] Sonnemans, J.2006. “Price Clustering and Natural Resistance Points in the Dutch Stock Market: A Natural Experiment.” European Economic Review 83: 505-518.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.