Optimal Pair-Trading Decision Rules for a Class of Non-Linear Boundary Crossings by Ornstein-Uhlenbeck Processes Page: 49
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We may similarly integrate over the time interval t - 1 < v <t, where At = t - (t - 1) is a
unit change in time, to obtain:
dY. = Ar c eA"dv + e "AvdBt
t-1 t-1 t-1
Y - Yt_1 = [vpe]'j +( e "dB.
eAtXt - eA(t-1)X-1 = et - ieA(t1) + uj eA"dBi
t-1
X = e-A(-(t-1))Xt-1 + - eA(t(t1)) + 0] e "A(t-)dBv
t-1
(32) = e- otXt-1 + pu(1 - e-t) + 07J e "A(t-)dBu
t-1
We consider the Ito integral f_1 e-A(t-)dB, in equation 32 above. Since e-A(t'-) is
a deterministic function of v which does not depend on By, it follows that the Ito integral,
ft _ e-M('-)dB, follows a normal distribution with zero mean and variance obtained as follow:
Itt 2~
VarA e-A(t-)dB1 = E [ -A(t-)dBt[)f2
t-1 t-1
(33) = E e-2A(t-v)dv
t-1
_ e-2A(t-v)
2A l t:
(34) = 1 (1 - e-2A~t),
2A
where we have used the isometry property of Ito integral in equation 33.
It then follows from equations 32 and 34 that:
(35) Xt Xt_1 ~ N (eAtXt-i + pu(1 - e ),2 (1 - e 2at)
A discretized form of the OU process Xt can be obtain from the distribution in 35 above.
We consider an example of this process with parameters ,u = 2.3, A = 0.08 and
a = 0.005. We also pick an initial value of Xo = 2.4, which is not too far from the long term49
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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/m1/68/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .