Optimal Pair-Trading Decision Rules for a Class of Non-Linear Boundary Crossings by Ornstein-Uhlenbeck Processes Page: 65
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least squares methods overestimate the p, the level of the long term mean, but the maximum
likelihood method performed better. This is not very surprising, given that we only picked
one instance of the process. To better understand the performance of the model, we perform
Monte Carlo simulations with 10,000 sample paths and obtain the average estimates for the
parameters, using both maximum likelihood and least squares methods. Table 5.7 shows the
result. We again reproduce the paths based on these averages, and they are plotted in figure
5.14. We observe here that the estimates perform very well and it is almost impossible to
distinguish among the true path, the path from the maximum likelihood estimates and the
path from the least squares estimates.
5.1.5.4. Short Path
We now turn our attention to the last 126 days of the processes discussed in 5.1.5.1.
Similar to subsection 5.1.5.2, we show the OU process and the logarithm of the price time
series of the two artificial stocks in figures 5.15 and 5.16 respectively.
True Value Method Estimate Bias Standard Error RMSE
MLE 0.28454530 -0.01545470 0.01557660 0.00048148
r7 0.30000000
LS 0.29773916 -0.00226084
MLE 1.61252270 0.11252270 0.11299430 0.02542907
p 1.50000000
LS 1.82803207 0.32803207
MLE 0.05121930 0.00121930 0.00966090 0.00009482
A 0.05000000
LS 0.05324758 0.00324758
MLE 0.01318950 0.00018950 0.00027020 0.00000011
- 0.01300000
LS 0.01319774 0.00019774
TABLE 5.6. Parameter estimation for the OU process representing the spread
between lnP and lnQt, by maximum likelihood and method of least squares.65
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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/m1/84/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .