Optimal Pair-Trading Decision Rules for a Class of Non-Linear Boundary Crossings by Ornstein-Uhlenbeck Processes Page: 30
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and 16.
In our study, it came to light that although theoretically we can always find values
for our parameters 0 and y that maximize the expected return, practically, this can only be
done for a reasonable time horizon, which usually should not be more than 11 years. As a
result, our objective function will entail two pieces, the first piece focuses on time horizon
T < 1.25 where we optimize the expected return, while the second piece focuses on T > 1.25
where we optimize a scaled form of the probability that the first passage time is greater
than 1.25 but less than the time horizon. This is to ensure that 0 does not blow out, nor
y become too small such that we do not obtain any trades. In fact this second piece of the
objective function is a nice behaved function that maximizes our return.
4.5.1. Case 1: - = 0
The dimensionless system threshold for this case is
9(t) = e-t.
The objective function is
E [3e-TI(o<T<T)] , 0 < T < 1.25
e-3P(1.25 <T< T), 1.25 < T < oo
THEOREM 4.5. For any 0 < T < oc, there exists some / E (0, oc) which maximizes h(/3)
and it is unique.
LEMMA 4.6. There exists some /3 E (0, oc) which maximizes E [3e-TI(o<T<T)] and it is
unique.
PROOF. Let IHJ(0) = E [Oe-TI(o<T;T)] , where / > 0.
By theorem 4.1, the pdf of T is given by
f2/() = - exp, t > 0
F /(e2t- P13 (2(e2 _1)30
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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/m1/49/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .