STOPPING TIME FOR ORNSTEIN-UHLENBECK
PROCESS VS. STOPPING TIME FOR
DISCRETE AR(1) PROCESS
By Rafal Lochowski
Warsaw School of Economics
In [4] there was considered trading strategy based on hypothesis
of existence of cointegrating relationship between series of prices of
contracts for two commodities. More precisely, it was assumed that
the cointegrating series has AR(1) structure and from this assumption
long-run gain for (approximately) optimal trading strategy consisting
of simultaneous selling and buying contracts for these commodities
was determined (for some range of parameters characterizing AR(1)
series). The formula for the approximate optimal gain was derived
from reasoning based on heuristic assumption that the stopping time
for AR(1) process may be approximated by stopping time for its continuous counterpart - Ornstein-Uhlenbeck process. This paper deals
with more exact results concerning this approximation.
1. Introduction. Let P, Q, ..., R be some commodities and Pt , Qt , ..., Rt
denote the prices of the contracts for these commodities at time t.
Henceforth we will assume that there exists cointegrating relationship
between processes (Pt ) , (Qt ) , ... and (Rt ) , i.e. there exist such constants
A, B, ..., C that the process
Xt = [A, B, ..., C] ◦ [Pt , Qt , ..., Rt ]
= A · Pt + B · Qt + ... + C · Rt
is a stationary one (in a strong sense).
Let a be such a positive constant that
P ∀T ≥ 0
∃ta (T ) , t−a (T ) ≥ T | Xta (T ) ≥ a, Xt−a (T ) ≤ −a = 1.
We consider the following trading strategy based on the assumption of stationarity of the process (Xt ) .
If Xt exceeds threshold a then we take short positions in the contracts
for all commodities, prices of which correspond to the positive coefficients
A, B, ..., C, and simultaneously take long positions in the contracts for all
commodities, prices of which correspond to negative coefficients A, B, ..., C.
AMS 2000 subject classifications: Primary 91B70
Keywords and phrases: Cointegration, AR(1) process, Ornstein-Uhlenbeck process
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If Xt decreases below −a, then we do opposite, after closing old positions.
The amounts of contracts for commodities P, Q, ..., R, which we buy shall
be equal (or proportional to) |A| , |B| , ..., |C| respectively.
The pair of this tradings gives the profit which is equal (or proportional
to) 2a. But the nominal rate of return of this strategy depends on the time
elapsed between two tradings: Ta,−a (0) = T−a (Ta (0)) − Ta (0) , where
Ta (t) = inf {T ≥ t|XT ≥ a} ,
T−a (t) = inf {T ≥ t|XT ≤ −a} .
We will consider the following models of the market:
1) continuous model, where parameter t attains all positive real values;
2) discrete model, where parameter t attains only values from the set
δZ+ = {0, δ, 2δ, 3δ, ...} for some δ > 0.
Moreover, we will assume exact structure of the process (Xt ) for both
models. Namely, we will assume that (Xt ) is AR(1) process, when discrete
model is considered and (Xt ) is Ornstein-Uhlenbeck process, (which is continuous cunterpart for AR(1) process in some sense), when we consider continuous model.
In the next section we recall known or easily derivable quantitative facts
about the distribution of Ta (0) and Ta,−a (0) for continous model.
The last section deals with the discrete time model and we prove there
that the distribution of Ta (0) and Ta,−a (0) for discrete model converges to
its continuous counterpart when δ → 0+ .
2. Continuous model. Let us assume that (Xt )t≥0 is an OrsteinUhlenbeck process. Xt may be introduced as unique strong solution of the
following stochastic differential equation
p
dXt = −βXt dt + α 2βdBt ,
(1)
where α, β > 0 and (Bt )t≥0 is a standard Wiener process (Brownian motion).
(Xt )t≥0 has continuous trajectories.
√
In order to ease the notation we will assume that α 2β = 1 (which is
equivalent with some space scaling of the process Xt ).
For a ∈ R let us define
Ta = inf {t ≥ 0 : Xt = a} .
(2)
It is known (see [2]) that for t > 0
−tTa
E e
√ 2
eβx /2 D−t/β εx 2β
√ ,
|X0 = x = βa2 /2
e
D−t/β εa 2β
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3
STOPPING TIME
where ε = sign (x − a) and Dν stands for the parabolic cylinder function
(see [1]) which admits for ν < 0 the following integral representation
2
e−z /4
Dν (z) =
Γ (−ν)
!
∞
Z
u
u2
exp − − zu du.
2
−ν−1
0
Now, since (Xt )t≥0 has continuous trajectories we have
(
Ta if a ≥ X0 ,
0 if a < X0
is independent from (Bt )t≥0 then
Ta (0) =
1
and if X0 ∼ N 0, α2 = N 0, 2β
−tTa (0)
E e
a
=1−Φ
α
s
+
β
π
Z
√ 2
eβx /2 D−t/β εx 2β −βx2
√ e
dx,
eβa2 /2 D−t/β εa 2β
a
−∞
where Φ is a distribuant of standard normal distribution.
Remark. If we assume that X0 ∼ N 0, α2 is independent from (Bt )t≥0
then (Xt )t≥0 defined by (1) admits the following Doob’s representation
Xt = αe−βt W e2βt ,
(3)
where W (.) is a standard Wiener process.
From Laplace transform we may obtain ETa (0) and V arTa (0) but it is
rather laborious task. More interesting are formulae for E (Ta,−a (0)) and
V ar (Ta,−a (0)) . We have (compare [2] and [5])
√ Z a/√2β
π
2
E (Ta,−a (0)) =
et /2 dt
β 0
!k
√ ∞
1
πX
a2
=
β k=0 4k · (k + 1) · k! β
and a bit more complicated formula for variance
r
V ar (Ta,−a (0)) = 8
π
β
Z
a
−a
Z
t
Z
−∞
a
e−β (u
2 −v 2 −t2
) dt du dv
u
−E (Ta,−a (0))2 .
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4
3. Discrete model. Henceforth, in order to distinguish continuous and
discrete models, instead of writing Xt in the discrete case we will denote
cointegrating series by X̃t .
In the second model we will assume that X̃t has discrete AR(1) structure
X̃t+δ = γ X̃t + ε̃t ,
where
•
•
•
•
X̃0 ∼ N 0, α2
ε̃t , t ∈ δZ+ , are i.i.d. normal variables,
for every t ∈ δZ+ variables ε̃t and X̃t are independent,
γ > 0.
From this assumptions we have that X̃t
is a stationary Markov
t∈δZ+ 0, α2 1 − γ 2 .
chain if and only if γ < 1 and ε̃t ∼ N
Let γ = e−βδ for some β > 0, then ε̃t ∼ N 0, α2 1 − e−2βδ
and it is
easy to check that the same relationships hold for process defined by (3):
Xt+δ = αe−β(t+δ) W e2β(t+δ)
h
= αe−β(t+δ) W e2βt + αe−β(t+δ) W e2β(t+δ) − W e2βt
i
= e−βδ Xt + εt ,
where εt ∼ N 0, α2 1 − e−2βδ
since
h
Eε2t = α2 e−2β(t+δ) e2β(t+δ) − e2βt
h
i
= α2 e−2β(t+δ) e2β(t+δ) 1 − e−2βδ
i
= α2 1 − e−2βδ .
Hence
X̃t
may be treated as the process (Xt ) sampled at the
t∈δZ+
points t ∈ δZ+ .
Let us now denote
n
o
Ta(δ) = inf t ∈ δZ+ : X̃t ≥ a .
(δ)
Of course Ta ≤ Ta , where Ta is defined by (2). We will show
Theorem 1.
(δ)
Ta
→ Ta in distribution, when δ → 0+ .
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5
STOPPING TIME
In order to prove the above theorem, first we will prove the following
lemmas.
Lemma 1. If x, y ∈ R, c > 0 then from inequalities |x − y| ≥ 2 |y| and
x ≤ c it follows that x − y ≤ 2c.
Proof. We have three following possibilities
• x < y then x − y < 0 ≤ 2c.
• y > 0 then x − y < x ≤ c < 2c.
• x ≥ y and y ≤ 0 then x − y = |x − y| ≥ 2 |y| = −2y, hence x ≥ −y
and x − y ≤ 2x ≤ 2c.
Lemma 2. Let W (.) be a standard Wiener process,
α > 0 and 0 ≤ t1 ≤
t2 ≤ ... ≤ tn be such a sequence that tk+1 ≥ n6 + 1 tk and tk+1 ≥ tk + α
for k = 1, 2, ..., n − 1. Then for every c > 0
c
P (max {W (t1 ) , W (t2 ) , ..., W (tn )} ≤ c) ≤ εn ( √ ),
α
where
n2
c
2
2c
εn ( √ ) = + 2ne− 8 + Φ √
α
n
α
√
n−1
√
.
tk+1 −tk
n2
Proof. For k = 1, 2, ..., n − 1 let ck ∈ n tk ,
(since tk+1 ≥
√
√
√
tk+1 −tk
n6 t
≥ n2 k = n tk ) and let G be a random
n6 + 1 tk we have
n2
variable with standard normal distribution. We have
P (|W (tk+1 ) − W (tk )| < 2 |W (tk )|)
≤ P (|W (tk+1 ) − W (tk )| ≤ ck ) + P (2 |W (tk )| > ck )
ck
ck
= P |G| < √
+ P |G| > √
tk+1 − tk
2 tk
2
≤
c
n2
2ck
2
− k
√
+ 2e 8tk ≤ 2 + 2e− 8 .
tk+1 − tk
n
√1 2u ≤ 2u and P (|G| > u)
2π
2
2
2
2e−u eu /2 = 2e−u /2 which hold for u > 0.
We used the estimates P (|G| < u) ≤
2P (G > u) ≤
2
2e−u EeuG
≤
=
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6
From the previous lemma
P (max {W (t1 ) , W (t2 ) , ..., W (tn )} ≤ c)
2
2
− n8
≤ (n − 1)
+
2e
n2
+ P (W (tk+1 ) ≤ c & |W (tk+1 ) − W (tk )| ≥ 2 |W (tk )| for k = 1, 2, ..., n − 1)
n2
2
+ 2ne− 8 + P (W (tk+1 ) − W (tk ) ≤ 2c for k = 1, 2, ..., n − 1)
≤
n
n−1
Y n2
2
2c
+ 2ne− 8 +
=
P G≤ √
n
tk+1 − tk
k=1
n2
2c
2
+ 2ne− 8 + Φ √
n
α
≤
n−1
.
Now we will estimate P Taδ ≥ Ta +
√ δ . In order to ease notation we
will assume that Xt = e−t/2 W et , since it is equivalent with space and
time scaling ot the process (Xt ). We have
P Taδ ≥ t +
√
δ|Ta = t
√ i
δ Xs < a|Ta = t
h
√ i
∀s ∈ δZ+ ∩ t, t + δ Xs < a|Xt = a
h √ i
∀s ∈ (δZ+ − t) ∩ 0, δ Xs < a|X0 = a
h √ i
∀s ∈ (δZ+ − t) ∩ 0, δ e−s/2 W (es ) < a|W (1) = a
√ i
h
√
∀u ∈ eδZ+ −t ∩ 1, e δ W (u) < a u|W (1) = a
h
= P ∀s ∈ δZ+ ∩ t, t +
= P
= P
= P
= P
h
h
≤ P ∀u ∈ eδZ+ −t ∩ 1, e
= P ∀u ∈ eδZ+ −t ∩ 1, e
√
√
√
δ
i
W (u) < ae
δ
i
W (u − 1) < a e
h
= P ∀u ∈ eδZ+ −t − 1 ∩ 0, e
δ/2
√
δ
|W (1) = a
√
δ/2
i
−1
√
− 1 W (u) < a e
δ/2
−1
.
Let us now take the smallest integer z0 such that δz0 − t ≥ δ. Then we have
δz0 −t δ and s = eδz0 −t+δk −1
δ (z0 − 1)−t < δ andjδz0 −t
k
k < 2δ. Defining α = e
1
√
for k = 1, 2, ..., N = 2 δ we get
1
sn = eδz0 −t+δn − 1 ≤ e2δ e 2
δ √1
δ
1
− 1 ≤ e2
√
δ+2δ
√
−1≤e
δ
− 1 for δ ≤
1
,
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STOPPING TIME
sk+1 − sk = eδz0 −t+δ(k+1) − eδz0 −t+δk
= eδz0 −t eδk eδ − 1 ≥ eδz0 −t δ = α.
√
Since 0 ≤ δz0 − t + δk ≤ δ ≤ ln 2 for δ ≤ (ln 2)2 and for 0 ≤ x ≤ ln 2 we
ln 2
have x ≤ ex − 1 ≤ e ln 2−1 x = ln12 x, hence
1
(k + 2) δ.
ln 2
(k + 1) δ ≤ δz0 − t + δk ≤ sk = eδz0 −t+δk − 1 ≤
(4)
Let now n = n (N ) be the greatest integer such that
l
2
and define ki =
ln 2
i = 1, 2, ..., n we get
m
n6 + 1
i
l
2
ln 2
m
n6 + 1
n
≤N
for i = 1, 2, ..., n. Defining ti = ski for
l
ti+1
(ki+1 + 1) δ
ki+1
≥ 1
≥ 2
=
ti
ln 2 (ki + 2) δ
ln 2 ki
2
ln 2
2
ln 2
m
n6 + 1 ≥ n6 + 1.
Hence for the sequence t1 , ..., tn the assumptions of the previous lemma hold
and we have
√
P Taδ ≥ t + δ|Ta = t
h
≤ P ∀u ∈ eδZ+ −t − 1 ∩ 0, e
√
δ
i
√
≤ P max {W (t1 ) , W (t2 ) , ..., W (tn )} < a e
 √
a e
≤ εn  √
√
and since

≤
2a ln12 2δ
√
eδ/2 δ
Z
√ δ =
+∞
0
Z
2
ln 2 a
≤
εn
√
δ|Ta = t P (Ta ∈ [t, t + dt))
2
a P (Ta ∈ [t, t + dt))
ln 2
2
a .
ln 2
Since for δ → 0+ we have N =
2
ln 2 a
we get
P Taδ ≥ t +
+∞
= εn
εn
−1

0
−1
√
≤
δ/2
δ/2
−1
eδz0 −t δ
2a e δ/2 −1
√
eδz0 −t δ
P Taδ ≥ Ta +
δ/2
√
− 1 W (u) < a e
j
1
√
2 δ
k
→ ∞, n = n (N ) → ∞ and
→ 0, we get
lim P Ta ≤ Taδ ≤ Ta +
δ→0+
√ δ =1
which is even stronger result than convergence in distribution.
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8
Acknowledgements. I would like to thank prof. Krzysztof Oleszkiewicz
for his comments and ideas concerning the convergence of stopping times.
Without them this paper would be not completed.
References.
[1] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental
Functions, based on notes left by H. Bateman, McGraw-Hill, 1953
[2] S. Finch, Ornstein-Uhlenbeck Process, preprint, 2005, available at the
http://algo.inria.fr/csolve/ou.pdf
[3] S. R. Finch, Euler-Mascheroni constant, Mathematical Constants, Cambridge Univ.
Press, 2003, pp. 2840
[4] R.
Lochowski,
On
approximate
upper
gain
bound
for
trading
strategy
based
on
cointegration,
preprint,
2007,
available
at
the
http://akson.sgh.waw.pl/~rlocho/AR 1.pdf
[5] M. U. Thomas, Some mean .rst-passage time approximations for the OrnsteinUhlenbeck process, J. Appl. Probab. 12 (1975) pp. 600-604
Address
Department of Mathematical Economics
Warsaw School of Economics
Al. Niepodleglości 164
02-554 Warszawa, Poland
E-mail: [email protected]
URL: http://akson.sgh.waw.pl/:rlocho
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