Optimal Pair-Trading Decision Rules for a Class of Non-Linear Boundary Crossings by Ornstein-Uhlenbeck Processes Page: 90
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2.02
-o 2.00 -
a)
0
C/)
1.98-
1.96-Comparison between estimated and actual path
Key
/ZTrue path
- Long-term mean
Least Squares Estimated
- Maximum Likelihood Estimated9850
time
9900
FIGURE 5.31. Trend-stationary OU paths from parameter estimation by Max-
imum Likelihood and Method of Least Squares, using Monte Carlo averages
with 10,000 replications of paths of length 126. The true parameter values are
5.2.5. Artificial Stocks
We now generate a pair of artificial stock prices Qt and Pt, such that the spread
between their log-returns is trend-stationary and follows the trend-stationary OU process
5.2.1, and then perform parameter estimation for the pair.
5.2.5.1. Generating the Pair
For Qt we will use the same GBM we generated in subsection 5.1.5. We use relation 59
to generate a 10,000 steps trend-stationary OU process starting at x0 = 1.4, with parameter
values a = 0.0002, b = 0.8, A = 0.08 and a-= 0.005. We then combine these two processes
using equation 1.7, with r= = 0.3, to obtain the logarithm of the price time series Pt of a
second stock P. The graphs of these are shown in figures 5.32, 5.33 and 5.34 respectively.
We know from equation 58 that:90
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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/m1/109/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .