Optimal Pair-Trading Decision Rules for a Class of Non-Linear Boundary Crossings by Ornstein-Uhlenbeck Processes Page: 32
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Thus,
(23) log[1 - Ib(C/32)] = logo(C3) + log(/3) + log(C) - log(1 + 02), for i = 1, 2.
Since both 01 and X32 are maximizers of JT (0), then UT(01) = UT(032) with # /32.
Then from equation 22, we have
(24) log[1 - <(C/31)] - log[1 - b(C2)] = log() + j(/ - ).
Equations 23 and 24 together implies:
log (~i) + log + -([l - 02) + 2log (1)= 0
#(C02) 1 + N? 2 02
This is equivalent to
/2 2(c2-1)1 _ e2 - (c2-1)32
1+N 3 1+/2
Consider the function:
xe2(2)
(x)- 1 for x > 0
The derivative of 1(x) is given by:
-_[1 - 2(C2 - 1)x(x + 1)] e1(c2-1)x
(x + 1)2
We note that the equation 1 - 1(C2 - 1)x(x + 1) = 0 has two roots: One is negative and the
other is positive, which is given by
-1+ + 8
2
When x > x, l'(x) < 0, which implies 1(x) is strictly decreasing on ( , oc). When 0 <
x < , l'(x) > 0, which implies 1(x) is strictly increasing on (0, x). Since l(131) = l(/3) and
#1 /02, by Rolle's theorem, there exists at least one r between #3and/32 such that l'(r) = 0.
W.L.O.G., we assume 01 </32. Then r E (+,132). Using the fact that l'(x) = 0 has only one
positive root x, we conclude that r = . Since /32 is a maximizer of ]JT(/), we obtain
2C
HT (02) = l(/?).
2wr32
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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/m1/51/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .