Optimal Pair-Trading Decision Rules for a Class of Non-Linear Boundary Crossings by Ornstein-Uhlenbeck Processes Page: 81
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= e-A^tXt-1+ b(1 - e-AAt) + a(1 - e-AAt)t + aAte-at
+Q- e ")dBv
t-1
(58)
As discussed earlier, the Ito integral jt_1 e-A(t-)dB, in equation 58 above follows a normal
distribution with mean zero and variance 22(1 - e-2a~t).
So, it follows that:
(59)
Xt X_1 ~ N e-a^tXt-1 + b(1 - e-AAt) + a(1 - e-AAt)t + aAte-AAt, (1 - e-2at)
As before, a discretized form of the trending OU process X can be obtained from
the distribution in 59 above.
Let us consider an example of this process with parameter values a = 0.0002, b = 0.02,
A = 0.08 and a-= 0.005, which starts at xo = 0.4. We generate a path of length 10,000 based
on he model. This is shown in figure 5.25. We see here that the process oscillates around
the line Xt = 0.0002t + 0.02, and always reverts to it whenever there is a deviation.
5.2.2. Parameter Estimation
We show in this subsection parameter estimation for the trend-stationary OU process
both with maximum likelihood and least squares methods.
5.2.2.1. Maximum Likelihood Estimator
From relation 59, the likelihood function for this model is:
N (Xt-(e- 1tXt1+b(1-e-Z)+a(1-e-A i)t+apte-At))2
L(a, b, A, -Xt2) = ]71 e ) -e-2"*)
t=1 2 r2(1 - e- )81
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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/m1/100/?rotate=90: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .