Optimal Pair-Trading Decision Rules for a Class of Non-Linear Boundary Crossings by Ornstein-Uhlenbeck Processes Page: 82
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2.0-
1.5-
1.0-
CD)
0.5-
0.0-0
2500
5000
time7500
FIGURE 5.25. Discretized trend-stationary Ornstein-Uhlenbeck Process of
length 10000, with parameters a = 0.0002, b = 0.02, A = 0.08 and a-= 0.005,
starting from x0 = 0.4
We then obtain the log-likelihood as:N
l(a, b, A,u o-Xt) = log fJ1 e
t=1 27T (1-e-2at)(Xt-(e-AA Xt-1+b(1-e-Aat)+a(1-e-AAt)t+a/te-A\t))2
2(2 1-e-2^^t))N N _(.2 2Aot))
= log (2 r) - log ( ( - e-2a )
2 2 2A
N
(60) - 2(1 -2At) (Xt - e-AAtXt-i - b(1 - e-AAt) - a(1 - e-AAt)t - aAte-AAt))2
t~i
Thus the maximum likelihood estimates for a, b, A and - will be the values a, b, A and
6 respectively, that maximize the above expression. We will solve this numerically from
equation 60.82
10000
-0
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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/m1/101/?rotate=90: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .