Optimal Pair-Trading Decision Rules for a Class of Non-Linear Boundary Crossings by Ornstein-Uhlenbeck Processes Page: 29
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4.5. The Optimization Problem
1.0
Key
--Long-term mean
0.5-Spread
--Threshold
C)
a
C0.0- - ---------- -------------------------------------- T
-0.5-
-1.0
0 100 200
Time
FIGURE 4.1. Spread and threshold.
Given the spread between the logarithmic returns of the pair of stocks, our objective
is to find thresholds that maximize the expected return of a complete trade cycle. The trade
cycle involves two stages. The first stage is the movement from the long-term mean or long-
term trend to crossing either the upper or lower threshold, while the second stage involves
the movement back to the long-term mean or long-term trend. Since the thresholds we are
considering are symmetric 11, we will focus on a one-sided threshold for our optimization
problem. The return on the trade cycle corresponds to the height of the spread at the time
of crossing the threshold. We demonstrated this in figure 4.1 for the constant threshold for
simplicity. Since we are considering a time-dependent threshold this height will vary over
time.
In section 4.1, we showed that the OU process 1 and the trend-stationary OU process
3 can be converted to the standardized (dimensionless)OU process. Thus we will undertake
the optimization in the dimensionless system, and then the optimal thresholds in the original
process can be obtained by translation and scaling, as shown in section 4.3, equations 14, 1529
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Tamakloe, Emmanuel Edem Kwaku. Optimal Pair-Trading Decision Rules for a Class of Non-Linear Boundary Crossings by Ornstein-Uhlenbeck Processes, dissertation, December 2021; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc1873709/m1/48/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .