Optimal Pair-Trading Decision Rules for a Class of Non-Linear Boundary Crossings by Ornstein-Uhlenbeck Processes Page: 72
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sidering how the one realization estimates perform, particularly in estimating the long-term
mean pu when all four parameters are estimated together, and the standard error obtained
for p, we recommend estimating rq separately by simple linear regression of Pt on Qt and
then estimate the remaining three parameters using maximum likelihood estimates as shown
in subsections 5.1.3 and 5.1.4. This is in line with the methods used by [40] and [17].
5.1.6.1. Long Path
Comparison between estimated and actual path
Key
- Actual
- - Long-term mean
- Least Squares Estimated
1.6 - Maxmum Likelihood Estimated
0~
C/)
1.5 -- - --- -- - - - - - - - --14
1.4-
9000 9500 10000
time
FIGURE 5.19. Paths generated from one realization of parameter estimates
by maximum likelihood and least squares methods, for the OU representation
of the spread of length 1260 between ln(Qt) and ln(Pt)
5.1.6.2. Case 1: 1 = 0
The thresholds for this case are given by
(50) g_(t) = /1 + 17 3e -.
2a
Substituting the time horizon T = 5 into the corresponding h(f3) in chapter 4 and solving
gives the optimizer as /3 0.9715, approximated to four decimal places. Thus the optimal
thresholds for this case are:72
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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/m1/91/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .