Optimal switching for pairs trading rule: A viscosity solutions approach

Abstract

This paper studies the problem of determining the optimal cut-off for pairs
trading rules. We consider two correlated assets whose spread is modelled by a
mean-reverting process with stochastic volatility, and the optimal pair trading
rule is formulated as an optimal switching problem between three regimes: flat
position (no holding stocks), long one short the other and short one long the
other. A fixed commission cost is charged with each transaction. We use a
viscosity solutions approach to prove the existence and the explicit
characterization of cut-off points via the resolution of quasi-algebraic
equations. We illustrate our results by numerical simulations.

Full text

Optimal switching for pairs trading rule:
a viscosity solutions approach
Minh-Man NGO
John von Neumann (JVN) Institute
Vietnam National University
Ho-Chi-Minh City,
man.ngo at jvn.edu.vn
Huyˆen PHAM
Laboratoire de Probabilit´es et
Mod`eles Al´eatoires, CNRS UMR 7599
Universit´e Paris 7 Diderot,
CREST-ENSAE,
and JVN Institute
pham at math.univ-paris-diderot.fr
March 23, 2016
Abstract
This paper studies the problem of determining the optimal cut-off for pairs trading
rules. We consider two cointegrated assets whose spread is modelled by a general
mean-reverting process, and the optimal pair trading rule is formulated as an optimal
switching problem between three regimes: flat position (no holding stocks), long one
short the other and short one long the other. A fixed commission cost is charged with
each transaction. We use a viscosity solutions approach to prove the existence and the
explicit characterization of cut-off points via the resolution of quasi-algebraic equations.
We illustrate our results by numerical simulations.
Keywords: pairs trading, optimal switching, mean-reverting process, viscosity solutions.
MSC Classification: 60G40, 49L25.
JEL Classification: C61, G11.
1

1 Introduction
Pairs trading consists of taking simultaneously a long position in one of the assets Aand
B, and a short position in the other, in order to eliminate the market beta risk, and be
exposed only to relative market movements determined by the spread. A brief history and
discussion of pairs trading can be found in Ehrman [5], Vidyamurthy [15], Elliott et al. [6],
or Gatev et al. [7]. The main aim of this paper is to rationale mathematically these rules
and find optimal cutoffs, by means of a stochastic control approach.
Pairs trading problem has been studied by stochastic control approach in the recent
years. Mudchanatongsuk et al. [11] consider self-financing portfolio strategy for pairs
trading, model the log-relationship between a pair of stock prices by an Ornstein-Uhlenbeck
process and use this to formulate a portfolio optimization and obtain the optimal solution to
this control problem in closed form via the corresponding Hamilton-Jacobi-Bellman (HJB)
equation. They only allow positions that are short one stock and long the other, in equal
dollar amounts. Tourin and Yan [14] study the same problem, but allow strategies with
arbitrary amounts in each stock. Chiu and Wong [3], [4] investigated optimal strategies for
conintegrated assets using mean-variance criterion and CRRA utility function. We mention
also the recent paper by Liu and Timmermann [10] who studied optimal trading strategies
for cointegrated assets with both fixed and random Poisson horizons. On the other hand,
instead of using self-financing strategies, one can focus on determining the optimal cut-offs,
i.e. the boundaries of the trading regions in which one should trade when the spread lies
in. Such problem is closely related to optimal buy-sell rule in trading mean reverting asset.
Zhang and Zhang [16] studied optimal buy-sell rule, where they model the underlying asset
price by an Ornstein-Uhlenbeck process and consider an optimal trading rule determined
by two regimes: buy and sell. These regimes are defined by two threshold levels, and
a fixed commission cost is charged with each transaction. They use classical verification
approach to find the value function as solution to the associated HJB equations (quasi-
variational inequalities), and the optimal thresholds are obtained by smooth-fit technique.
The same problem is studied in Kong’s PhD thesis [8], but he considers trading rules with
three aspects: buying, selling and shorting. Song and Zhang [13] use the same approach
for determining optimal pairs trading thresholds, where they model the difference of the
stock prices Aand Bby an Ornstein-Uhlenbeck process and consider an optimal pairs
trading rule determined by two regimes: long Ashort Band flat position (no holding
stocks). Leung and Li [9] studied the optimal timing to open or close the position subject
to fixed transaction costs (entry and exit), and the effect of Stop-loss level under the
Ornstein-Uhlenbeck (OU) model. They directly construct the value functions instead of
using variational inequalities approach, by characterizing the value functions as the smallest
concave majorant of reward function. Lei and Xu [18] studied the optimal pairs trading
rule in finite horizon with proportional transaction cost by applying numerical method for
solving the system of variational inequalities.
In this paper, we consider a pairs trading problem as in Song and Zhang [13], but differ
in our model setting and resolution method. We consider two cointegrated assets whose
spread is modelled by a more general mean-reverting process, and the optimal pairs trading
2

rule is based on optimal switching between three regimes: flat position (no holding stocks),
long one short the other and vice-versa. A fixed commission cost is charged with each
transaction. We use a viscosity solutions approach to solve our optimal switching problem.
Actually, by combining viscosity solutions approach, smooth fit properties and uniqueness
result for viscosity solutions proved in Pham [12], we are able to derive directly the structure
of the switching regions, and thus the form of our value functions. This contrasts with the
classical verification approach where the structure of the solution should be guessed ad-hoc,
and one has to check that it satisfies indeed the corresponding HJB equation, which is not
trivial in this context of optimal switching with more than two regimes.
The paper is organized as follows. We formulate in Section 2 the pairs trading as an
optimal switching problem with three regimes. In Section 3, we state the system of varia-
tional inequalities satisfied by the value functions in the viscosity sense and the definition
of pairs trading regimes. In Section 4, we state some useful properties on the switching
regions, derive the form of value functions, and obtain optimal cutoff points by relying
on the smooth-fit properties of value functions. In Section 5, we illustrate our results by
numerical examples.
2 Pair trading problem
Let us consider the spread Xbetween two cointegrated assets, say Aand Bmodelled by a
mean-reverting process with boundaries `−∈ {−∞,0}, and `+=∞:
dXt=µ(L−Xt)dt +σ(Xt)dWt,(2.1)
where Wis a standard Brownian motion on (Ω,F,F= (Ft)t≥0,P), µ > 0 and L≥0
are positive constants, σis a Lipschitz function on (`−, `+), satisfying the nondegeneracy
condition σ > 0. The SDE (2.1) admits then a unique strong solution, given an initial
condition X0=x∈(`−, `+), denoted Xx. We assume that `+=∞is a natural boundary,
`−=−∞ is a natural boundary, and `−= 0 is non attainable. The main examples are
the Ornstein-Uhlenbeck (OU in short) process or the inhomogenous geometric Brownian
motion (IGBM), as studied in detail in the next sections.
Suppose that the investor starts with a flat position in both assets. When the spread
widens far from the equilibrium point, she naturally opens her trade by buying the under-
priced asset, and selling the overpriced one. Next, if the spread narrows, she closes her
trades, thus generating a profit. Such trading rules are quite popular in practice among
hedge funds managers with cutoff values determined empirically by descriptive statistics.
The main aim of this paper is to rationale mathematically these rules and find optimal
cutoffs, by means of a stochastic control approach. More precisely, we formulate the pairs
trading problem as an optimal switching problem with three regimes. Let {−1,0,1}be the
set of regimes where i= 0 corresponds to a flat position (no stock holding), i= 1 denotes
a long position in the spread corresponding to a purchase of Aand a sale of B, while i=
−1 is a short position in X(i.e. sell Aand buy B). At any time, the investor can decide
to open her trade by switching from regime i= 0 to i=−1 (open to sell) or i= 1 (open
to buy). Moreover, when the investor is in a long (i= 1) or short position (i=−1), she
3

can decide to close her position by switching to regime i= 0. We also assume that it is
not possible for the investor to switch directly from regime i=−1 to i= 1, and vice-versa,
without first closing her position. The trading strategies of the investor are modelled by a
switching control α= (τn, ιn)n≥0where (τn)nis a nondecreasing sequence of stopping times
representing the trading times, with τn→ ∞ a.s. when ngoes to infinity, and ιnvalued
in {−1,0,1},Fτn-measurable, represents the position regime decided at τnuntil the next
trading time. By misuse of notations, we denote by αtthe value of the regime at any time
t:
αt=ι01{0≤t<τ0}+X
n≥0
ιn1{τn≤t<τn+1}, t ≥0,
which also represents the inventory value in the spread at any time. We denote by gij(x)
the trading gain when switching from a position ito j,i, j ∈ {−1,0,1},j6=i, for a spread
value x. The switching gain functions are given by:
g01 (x) = g−10 (x) = −(x+ε)
g0−1(x) = g10 (x) = x−ε,
where ε > 0 is a fixed transaction fee paid at each trading time. Notice that we do not
consider the functions g−11 and g11 since it is not possible to switch from regime i=−1 to
i= 1 and vice-versa. By misuse of notations, we also set g(x, i, j) = gij (x).
Given an initial spread value X0=x, the expected reward over an infinite horizon
associated to a switching trading strategy α= (τn, ιn)n≥0is given by the gain functional:
J(x, α) = EhX
n≥1
e−ρτng(Xx
τn, ατ−
n, ατn)−λZ∞
0
e−ρt|αt|dti.
The first (discrete sum) term corresponds to the (discounted with discount factor ρ > 0)
cumulated gain of the investor by using pairs trading strategies, while the last integral term
reduces the inventory risk, by penalizing with a factor λ≥0, the holding of assets during
the trading time interval.
For i= 0,−1,1, let videnote the value functions with initial positions iwhen maximizing
over switching trading strategies the gain functional, that is
vi(x) = sup
α∈Ai
J(x, α), x ∈(`−,∞), i = 0,−1,1,
where Aidenotes the set of switching controls α= (τn, ιn)n≥0with initial position α0−=
i, i.e. τ0= 0, ι0=i. The impossibility of switching directly from regime i=±1 to ∓1 is
formalized by restricting the strategy of position i=±1: if α∈ A1or α∈ A−1then ι1=
0 for ensuring that the investor has to close first her position before opening a new one.
3 PDE characterization
Throughout the paper, we denote by Lthe infinitesimal generator of the diffusion process
X, i.e.
Lϕ(x) = µ(L−x)ϕ0(x) + 1
2σ2(x)ϕ00(x).
4

The ordinary differential equation of second order
ρφ − Lφ= 0,(3.1)
has two linearly independent positive solutions. These solutions are uniquely determined
(up to a multiplication), if we require one of them to be strictly increasing, and the other
to be strictly decreasing. We shall denote by ψ+the increasing solution, and by ψ−the
decreasing solution. They are called fundamental solutions of (3.1), and any other solution
can be expressed as their linear combination. Since `+=∞is a natural boundary, and `−
∈ {−∞,0}is either a natural or non attainable boundary, we have:
ψ+(∞) = ψ−(`−) = ∞, ψ−(∞) = 0.(3.2)
We shall also assume that
lim
x→`−
x
ψ−(x)= 0,lim
x→∞
x
ψ+(x)= 0.(3.3)
We refer to the classical handbook [2] for the existence, uniqueness, and properties of such
functions ψ+and ψ−.
Canonical examples
Our two basic examples in finance for Xsatisfying the above assumptions are
•Ornstein-Uhlenbeck (OU) process (`−=−∞) :
dXt=−µXtdt +σdWt,(3.4)
with µ,σpositive constants. In this case, `+=∞,`−=−∞ are natural boundaries,
the two fundamental solutions to (3.1) are given by
ψ+(x) = Z∞
0
tρ
µ−1exp −t2
2+√2µ
σxtdt, ψ−(x) = Z∞
0
tρ
µ−1exp −t2
2−√2µ
σxtdt,
and it is easily checked that condition (3.3) is satisfied.
•Inhomogeneous Geometric Brownian Motion (IGBM) (`−= 0) :
dXt=µ(L−Xt)dt +σXtdWt, X0>0,(3.5)
where µ,Land σare positive constants. In this case, `+=∞is a natural boundary,
`−= 0 is a non attainable boundary, and the two fundamental solutions to (3.1) are
given by
ψ+(x) = x−aU(a, b, c
x), ψ−(x) = x−aM(a, b, c
x).(3.6)
where
a=pσ4+ 4(µ+ 2ρ)σ2+ 4µ2−(2µ+σ2)
2σ2>0,
b=2µ
σ2+ 2a+ 2, c =2µL
σ2,(3.7)
5

and Mand Uare the confluent hypergeometric functions of the first and second kind.
Moreover, by the asymptotic property of the confluent hypergeometric functions (see
[1]), the fundamental solutions ψ+and ψ−satisfy condition (3.3), and
ψ+(0+) = 1
ca.(3.8)
In this section, we state some general PDE characterization of the value functions by
means of the dynamic programming approach. We first state a linear growth property and
Lipschitz continuity of the value functions.
Lemma 3.1 There exists some positive constant r(depending on σ) such that for a dis-
count factor ρ>r, the value functions are finite on R. In this case, we have
0≤v0(x)≤C(1 + |x|),∀x∈(`−,∞),
−λ
ρ≤vi(x)≤C(1 + |x|),∀x∈(`−,∞), i = 1,−1,
and
|vi(x)−vi(y)| ≤ C|x−y|,∀x, y ∈(`−,∞), i = 0,1,−1,
for some positive constant C.
Proof. The arguments are rather standard and the proof is rejected into the appendix. 2
In the sequel, we fix a discount factor ρ > r so that the value functions viare well-
defined and finite, and satisfy the linear growth and Lipschitz estimates of Lemma 3.1. The
dynamic programming equations satisfied by the value functions are thus given by a system
of variational inequalities:
min ρv0− Lv0, v0−max v1+g01 , v−1+g0−1 = 0,on (`−,∞),(3.9)
min ρv1− Lv1+λ , v1−v0−g10 = 0,on (`−,∞),(3.10)
min ρv−1− Lv−1+λ , v−1−v0−g−10 = 0,on (`−,∞).(3.11)
Indeed, the equation for v0means that in regime 0, the investor has the choice to stay in
the flat position, or to open by a long or short position in the spread, while the equation
for vi,i=±1, means that in the regime i=±1, she has first the obligation to close her
position hence to switch to regime 0 before opening a new position. By the same argument
as in [12], we know that the value functions vi, i = 0,1,−1 are viscosity solutions to the
system (3.9)-(3.10)-(3.11), and satisfied the smooth-fit C1condition.
Let us introduce the switching regions:
•Open-to-trade region from the flat position i= 0:
S0=nx∈(`−,∞) : v0(x) = max v1+g01 , v−1+g0−1(x)o
=S01 ∪ S0−1,
6

where S01 is the open-to-buy region, and S0−1is the open-to-sell region:
S01 =nx∈(`−,∞) : v0(x)=(v1+g01 )(x)o,
S0−1=nx∈(`−,∞) : v0(x)=(v−1+g0−1)(x)o.
•Sell-to-close region from the long position i= 1:
S1=nx∈(`−,∞) : v1(x) = (v0+g10 )(x)o.
•Buy-to-close region from the short position i=−1:
S−1=nx∈(`−,∞) : v−1(x) = (v0+g−10 )(x)o,
and the continuation regions, defined as the complement sets of the switching regions:
C0= (`−,∞)\ S0=nx∈(`−,∞) : v0(x)>max v1+g01 , v−1+g0−1(x)o,
C1= (`−,∞)\ S1=nx∈(`−,∞) : v1(x)>(v0+g10 )(x)o,
C−1= (`−,∞)\ S−1=nx∈(`−,∞) : v−1(x)>(v0+g−10 )(x)o.
4 Solution
In this section, we focus on the existence and structure of switching regions, and then we
use the results on smooth fit property, uniqueness result for viscosity solutions of the value
functions to derive the form of value functions in which the optimal cut-off points can be
obtained by solving smooth-fit condition equations.
Lemma 4.1
S01 ⊂− ∞,µL −`0
ρ+µ∩(`−,∞),S0−1⊂µL +`0
ρ+µ,∞,
S1⊂µL −`1
ρ+µ,∞∩(`−,∞),S−1⊂− ∞,µL +`1
ρ+µ∩(`−,∞),
where
0< `0:= λ+ρε, `1:= λ−ρε ∈(−`0, `0).
Proof. Let ¯x∈ S01 , so that v0(¯x)=(v1+g01 )( ¯x). By writing that v0is a viscosity
supersolution to: ρv0− Lv0≥0, we then get
ρ(v1+g01 )(¯x)− L(v1+g01 )(¯x)≥0.(4.1)
Now, since g01 +g10 =−2ε < 0, this implies that S01 ∩ S1=∅, so that ¯x∈ C1. Since v1
satisfies the equation ρv1− Lv1+λ= 0 on C1, we then have from (4.1)
ρg01 (¯x)− Lg01 (¯x)−λ≥0.
7

Recalling the expressions of g01 and L, we thus obtain: −ρ(¯x+ε)−µ¯x−λ+Lµ ≥0, which
proves the inclusion result for S01. Similar arguments show that if ¯x∈ S0−1then
ρg0−1(¯x)− Lg0−1(¯x)−λ≥0,
which proves the inclusion result for S0−1after direct calculation.
Similarly, if ¯x∈ S1then ¯x∈ S0−1or ¯x∈ C0: if ¯x∈ S0−1, we obviously have the inclusion
result for S1. On the other hand, if ¯x∈ C0, using the viscosity supersolution property of
v1, we have:
ρg10 (¯x)− Lg10 (¯x) + λ≥0,
which yields the inclusion result for S1. By the same method, we shows the inclusion result
for S−1.2
We next examine some sufficient conditions under which the switching regions are not
empty.
Lemma 4.2 (1) The switching regions S1and S0−1are always not empty.
(2)
(i) If `−=−∞, then S−1is not empty
(ii) If `−= 0, and ε < λ
ρ, then S−16=∅.
(3) If `−=−∞, then S01 is not empty.
Proof. (1) We argue by contradiction, and first assume that S1=∅. This means that once
we are in the long position, it would be never optimal to close our position. In other words,
the value function v1would be equal to ˆ
V1given by
ˆ
V1(x) = Eh−λZ∞
0
e−ρtdti=−λ
ρ.
Since v1≥v0+g10 , this would imply v0(x)≤ −λ
ρ+ε−x, for all x∈(`−,∞), which obviously
contradicts the nonnegativity of the value function v0.
Suppose now that S0−1=∅. Then, from the inclusion results for S0in Lemma 4.1, this
implies that the continuation region C0would contain at least the interval (µL−`0
ρ+µ,∞)∩
(`−,∞). In other words, we should have: ρv0− Lv0= 0 on ( µL−`0
ρ+µ,∞)∩(`−,∞), and so
v0should be in the form:
v0(x) = C+ψ+(x) + C−ψ−(x),∀x > µL −`0
ρ+µ∨`−,
for some constants C+and C−. In view of the linear growth condition on v0and condition
(3.3) when xgoes to ∞, we must have C+= 0. On the other hand, since v0≥v−1+g0−1,
and recalling the lower bound on v−1in Lemma 3.1, this would imply:
C−ψ−(x)≥ −λ
ρ+x−ε, ∀x > µL −`0
ρ+µ∨`−.
8

By sending xto ∞, and from (3.2), we get the contradiction.
(2) Suppose that S−1=∅. Then, a similar argument as in the case S1=∅, would imply
that v0(x)≤ −λ
ρ+ε+x, for all x∈(`−,∞). This immediately leads to a contradiction
when `−=−∞ by sending xto −∞. When `−= 0, and under the condition that ε < λ
ρ,
we also get a contradiction to the non negativity of v0.
(3) Consider the case when `−=−∞, and let us argue by contradiction by assuming
that S01 =∅. Then, from the inclusion results for S0in Lemma 4.1, this implies that the
continuation region C0would contain at least the interval (−∞,µL+`0
ρ+µ). In other words, we
should have: ρv0− Lv0= 0 on (−∞,µL+`0
ρ+µ), and so v0should be in the form:
v0(x) = C+ψ+(x) + C−ψ−(x),∀x < µL +`0
ρ+µ,
for some constants C+and C−. In view of the linear growth condition on v0and condition
(3.3) when xgoes to −∞, we must have C−= 0. On the other hand, since v0≥v1+g01 ,
recalling the lower bound on v1in Lemma 3.1, this would imply:
C+ψ+(x)≥ −λ
ρ−(x+ε),∀x < µL +`0
ρ+µ.
By sending xto −∞, and from (3.2), we get the contradiction. 2
Remark 4.1 Lemma 4.2 shows that S1is non empty. Furthermore, notice that in the case
where `−= 0, S1can be equal to the whole domain (0,∞), i.e. it is never optimal to stay
in the long position regime. Actually, from Lemma 4.1, such extreme case may occur only
if µL −`1≤0, in which case, we would also get µL −`0<0, and thus S01 =∅. In that
case, we are reduced to a problem with only two regimes i= 0 and i=−1. 2
The above Lemma 4.2 left open the question whether S−1is empty when `−= 0 and ε
≥λ
ρ, and whether S01 is empty or not when `−= 0. The next Lemma (and Remark 4.2)
provides sufficient condition under which these sets are not empty in the case of IGBM
process.
Lemma 4.3 Let Xbe governed by the Inhomogeneous Geometric Brownian motion in
(3.5), and set
K0(y) := ( c
y)−a1
U(a, b, c
y)(y−ε+λ
ρ)−(λ
ρ+ε), y > 0,
K−1(y) := ( c
y)−a1
U(a, b, c
y)(y−ε−λ
ρ)+(λ
ρ−ε), y > 0,
where a,band care defined in (3.7). If there exists y∈(0,µL+`0
ρ+µ)(resp y > 0) such that
K0(y)(resp. K−1)>0, then S01 (resp. S−1) is not empty.
Proof. See Appendix. 2
9

Remark 4.2 The above Lemma 4.3 gives a sufficient condition in terms of the function
K0and K−1, which ensures that S01 and S−1are not empty when the spread is an IGBM
process. Let us discuss how it is satisfied. From the asymptotic property of the confluent
hypergeometric functions, we have: limz→∞ zaU(a, b, z ) = 1. Then by sending Lto infinity
(recall that c=2µL
σ2), and from the expression of K0and K−1in Lemma 4.3, we have:
lim
L→∞ K0(y) = lim
c→∞ K0(y) = y−2ε= lim
L→∞ K−1(y).
This implies that for Llarge enough, one can choose 2ε < y < µL+`0
ρ+µso that K0(y)
>0. Notice also that K0is nondecreasing with Las a consequence of the fact that
∂
∂z zaU(a, b, z) = aU (a+1,b,z )(a−b+1)
z<0. In practice, one can check by numerical method
the condition K0(y)>0 for 0 < y < µL+`0
ρ+µ. For example, with µ= 0.8, σ= 0.5 , ρ= 0.1,
λ= 0.07, ε= 0.005, and L= 3, we have µL+`0
ρ+µ= 2.7450, and K0(1) = 0.9072 >0.
Similarly, for Llarge enough, one can find y > 2εsuch that K−1(y)>0 ensuring that S−1
is not empty. 2
We are now able to describe the complete structure of the switching regions.
Proposition 4.1 1) There exist finite cutoff levels ¯x01 ,¯x0−1,¯x1,¯x−1such that
S1= [¯x1,∞)∩(`−,∞),S0−1= [¯x0−1,∞),
S−1= (`−,−¯x−1],S01 = (`−,−¯x01],
and satisfying ¯x0−1≥µL+`0
ρ+µ,¯x1≥µL−`1
ρ+µ,−¯x−1≤µL+`1
ρ+µ,−¯x01 ≤µL−`0
ρ+µ. Moreover, −¯x01
<¯x1, i.e. S01 ∩ S1=∅and ¯x0−1>−¯x−1, i.e. S0−1∩ S−1=∅.
2) We have ¯x1≤¯x0−1, and −¯x01 ≤ −¯x−1i.e. the following inclusions hold:
S0−1⊂ S1,S01 ⊂ S−1.
Proof. 1) (i) We focus on the structure of the sets S01 and S−1, and consider first the case
where they are not empty. Let us then set −¯x01 = sup S01 , which is finite since S01 is not
empty, and is included in (`−,µL−`0
ρ+µ] by Lemma 4.1. Moreover, since S0−1is included in
[µL+`0
ρ+µ,∞), it does not intersect with (`−,−¯x01 ), and so v0(x)>(v−1+g0−1)(x) for x <
−¯x01, i.e. (`−,−¯x01 )⊂ S01 ∪ C0. From (3.9), we deduce that v0is a viscosity solution to
min ρv0− Lv0, v0−v1−g01 = 0,on (`−,−¯x01).(4.2)
Let us now prove that S01 = (`−,−¯x01]. To this end, we consider the function w0=v1+g01
on (`−,−¯x01]. Let us check that w0is a viscosity supersolution to
ρw0− Lw0≥0 on (`−,−¯x01 ).(4.3)
For this, take some point ¯x∈(`−,−¯x01 ), and some smooth test function ϕsuch that ¯xis
a local minimum of w0−ϕ. Then, ¯xis a local minimum of v1−(ϕ−g01 ) by definition of
10

w0. By writing the viscosity supersolution property of v1to: ρv1−Lv1+λ≥0, at ¯xwith
the test function ϕ−g01 , we get:
0≤ρ(ϕ−g01 )(¯x)− L(ϕ−g01 )(¯x) + λ
=ρϕ(¯x)− Lϕ(¯x)+(ρ+µ)(¯x+`0−µL
ρ+µ)
≤ρϕ(¯x)− Lϕ(¯x),
since ¯x < −¯x01 ≤µL−`0
ρ+µ. This proves the viscosity supersolution property (4.3), and
actually, by recalling that w0=v1+g01,w0is a viscosity solution to
min ρw0− Lw0, w0−v1−g01 = 0,on (`−,−¯x01).(4.4)
Moreover, since −¯x01 lies in the closed set S01, we have w0(−¯x01) = (v1+g01 )(−¯x01 )
=v0(−¯x01). By uniqueness of viscosity solutions to (4.2), we deduce that v0=w0on
(`−,−¯x01], i.e. S01 = (`−,−¯x01]. In the case where S01 is empty, which may arise only
when `−= 0 (recall Lemma 4.2), then it can still be written in the above form (`−,−¯x01 ]
by choosing −¯x01 ≤`−∧(µL−`0
ρ+µ).
By similar arguments, we show that when S−1is not empty, it should be in the form:
S−1= (`−,−¯x−1], for some −¯x−1≤µL+`1
ρ+µ, while when it is empty, which may arise only
when `−= 0 (recall Lemma 4.2), it can be written also in this form by choosing −¯x−1≤0
∧(µL+`1
ρ+µ).
(ii) We derive similarly the structure of S0−1and S1which are already known to be non
empty (recall Lemma 4.2): we set ¯x0−1= inf S0−1, which lies in [ µL+`0
ρ+µ,∞) since S0−1is
included in [µL+`0
ρ+µ,∞) by Lemma 4.1. Then, we observe that v0is a viscosity solution to
min ρv0− Lv0, v0−v−1−g0−1= 0,on (¯x0−1,∞).(4.5)
By considering the function ˜w0=v−1+g0−1, we show by the same arguments as in (4.4)
that ˜w0is also a viscosity solution to (4.5) with boundary condition ˜w0(¯x0−1) = v0(¯x0−1).
We conclude by uniqueness that ˜w0=v0on [¯x0−1,∞), i.e. S0−1= [¯x0−1,∞). The same
arguments show that S1is in the form stated in the Proposition.
Moreover, from Lemma 4.1 we have : ¯x0−1≥µL+`0
ρ+µ>µL+`1
ρ+µ≥ −¯x−1and ¯x1≥µL−`1
ρ+µ
>µL−`0
ρ+µ≥ −¯x01 .
2) We only consider the case where −¯x−1<¯x1, since the inclusion result in this proposition
is obviously obtained when −¯x−1≥¯x1from the above forms of the switching regions. Let
us introduce the function U(x)=2v0(x)−(v1+v−1)(x) on [−¯x−1,¯x1]. On (−¯x−1,¯x1), we
see that v1and v−1are smooth C2, and satisfy:
ρv1− Lv1+λ= 0, ρv−1− Lv−1+λ= 0,
which combined with the viscosity supersolution property of v0, gives
ρU − LU= 2(ρv0− Lv0)+2λ≥0 on (−¯x−1,¯x1).
11

At x= ¯x1we have v1(x) = v0(x) + x−εand v0(x)≥v−1(x) + x−εso that 2v0(x)≥
v1(x) + v−1(x), which means U(¯x1)≥0. By the same way, at x=−¯x−1we also have
2v0(x)≥v1(x) + v−1(x), which means U(−¯x−1)≥0. By the comparison principle, we
deduce that
2v0(x)≥v1(x) + v−1(x) on [−¯x−1,¯x1].
Let us assume on the contrary that ¯x1>¯x0−1. We have v0(¯x0−1) = v−1( ¯x0−1) + ¯x0−1−ε
and v1(¯x0−1)> v0(¯x0−1) + ¯x0−1−ε, so that (v−1+v1)( ¯x0−1)>2v0(¯x0−1), leading to a
contradiction. By the same argument, it is impossible to have −¯x−1<−¯x01 , which ends
the proof. 2
Remark 4.3 Consider the situation where `−= 0. We distinguish the following cases:
(i) λ > ρε. Then, we know from Lemma 4.2 that S−16=∅. Moreover, for Lsmall enough,
namely L≤`0/µ, we see from Proposition 4.1 that −¯x01 ≤0 and thus S01 =∅.
(ii) λ≤ρε. Then `1≤0, and for Lsmall enough namely, L≤ −`1/µ, we see from
Proposition 4.1 that −¯x−1≤0, and thus S−1=∅and S01 =∅.
2
The next result shows a symmetry property on the switching regions and value functions.
Proposition 4.2 (Symmetry property) In the case `−=−∞, and if σ(x)is an even
function and L= 0, then ¯x0−1= ¯x01 ,¯x−1= ¯x1and
v−i(−x) = vi(x), x ∈R, i ∈ {0,−1,1}.
Proof. Consider the process Yx
t=−Xx
t, which follows the dynamics:
dYt=−µYtdt +σ(Yt)d¯
Wt,
where ¯
W=−Wis still a Brownian motion on the same probability measure and filtration
of W, and we can see that Yx
t=X−x
t. We consider the same optimal problem, but we use
Ytinstead of Xt, we denote
JY(x, α) = EhX
n≥1
e−ρτng(Yx
τn, ατ−
n, ατn)−λZ∞
0
e−ρt|αt|dti,
For i= 0,−1,1, let vY
idenote the value functions with initial positions iwhen maximizing
over switching trading strategies the gain functional, that is
vY
i(x) = sup
α∈Ai
JY(x, α), x ∈R, i = 0,−1,1.
12

For any α∈ Ai, we see that g(Yx
τn,−ατ−
n,−ατn) = g(Xx
τn, ατ−
n, ατn), and so JY(x, −α) =
J(x, α). Thus, vY
−i(x)≥JY(x, −α) = J(x, α), and since αis arbitrary in Ai, we get: vY
−i(x)
≥vi(x). By the same argument, we have vi(x)≥vY
−i(x), and so vY
−i=vi,i∈ {0,−1,1}.
Moreover, recalling that Yx
t=X−x
t, we have:
v−i(−x) = vY
−i(x) = vi(x), x ∈R, i ∈ {0,−1,1}.
In particular, we v−1(−¯x1) = v1(¯x1) = (v0+g10 )(¯x1)=(v0+g−10 )(−¯x1), so that −¯x1∈ S−1.
Moreover, since ¯x1= inf S1, we notice that for all r > 0, ¯x1−r6∈ S1. Thus, v−1(−¯x1+r)
=v1(¯x1−r)>(v0+g10 )(¯x1−r) = (v0+g−10 )(−¯x1+r), for all r > 0, which means that
−¯x1= sup S−1. Recalling that sup S−1=−¯x−1, this shows that ¯x1= ¯x−1. By the same
argument, we have ¯x0−1= ¯x01 .2
To sum up the above results, we have the following possible cases for the structure of
the switching regions:
(1) `−=−∞. In this case, the four switching regions S1,S−1,S01 and S0−1are not
empty in the form
S1= [¯x1,∞),S0−1= [¯x0−1,∞),
S−1= (−∞,−¯x−1],S01 = (−∞,−¯x01 ],
and are plotted in Figure 1. Moreover, when L= 0 and σis an even function, S1=
−S−1and S01 =−S0−1.
(2) `−= 0. In this case, the switching regions S1and S0−1are not empty, in the form
S1= [¯x1,∞)∩(0,∞),S0−1= [¯x0−1,∞),
for some ¯x1∈R, and ¯x0−1>0 by Proposition 4.1. However, S−1and S01 may
be empty or not. More precisely, for the case of IGBM process, we have the three
following possibilities:
(i) S−1and S01 are not empty in the form:
S−1= (0,−¯x−1],S01 = (0,−¯x01 ],
for some 0 <−¯x01 ≤ −¯x−1by Proposition 4.1. Such cases arises for example
when Xis the IGBM (3.5) and for Llarge enough, as showed in Lemma 4.3 and
Remark 4.2. The visualization of this case is the same as Figure 1.
(ii) S−1is not empty in the form: S−1= (0,−¯x−1] for some ¯x−1<0 by Proposition
4.1, and S01 =∅. Such case arises when λ > ρε, and for L≤(λ+ρε)/µ, see
Remark 4.3(i). This is plotted in Figure 2.
(iii) Both S−1and S01 are empty. Such case arises when λ≤ρε, and for L≤
(ρε−λ)/µ, see Remark 4.3(ii). This is plotted in Figure 3. Moreover, notice that
in such case, we must have λ≤ρε by Lemma 4.2(2)(ii), and so by Proposition
4.1, ¯x1≥µL−`1
ρ+µ>0, i.e. S1= [ ¯x1,∞).
13

Figure 1: Regimes switching regions in cases (1) and (2)(i).
Figure 2: Regimes switching regions in case (2)(ii).
Figure 3: Regimes switching regions in case (2)(iii).
The next result provides the explicit solution to the optimal switching problem.
14

Theorem 4.1 •Case (1): `−=−∞. The value functions are given by
v0(x) = 




A1ψ+(x)−λ
ρ+g01 (x), x ≤ −¯x01 ,
A0ψ+(x) + B0ψ−(x),−¯x01 <x<¯x0−1,
B−1ψ−(x)−λ
ρ+g0−1(x), x ≥¯x0−1,
v1(x) = (A1ψ+(x)−λ
ρ, x < ¯x1,
v0(x) + g10 (x), x ≥¯x1,
v−1(x) = (v0(x) + g−10 (x), x ≤ −¯x−1,
B−1ψ−(x)−λ
ρ, x > −¯x−1,
and the constants A0,B0,A1,B−1,¯x01 ,¯x0−1,¯x1,¯x−1are determined by the smooth-fit
conditions:
A1ψ+(−¯x01 )−λ
ρ+g01 (−¯x01 ) = A0ψ+(−¯x01 ) + B0ψ−(−¯x01 )
A1ψ0
+(−¯x01 )−1 = A0ψ0
+(−¯x01 ) + B0ψ0
−(−¯x01 )
B−1ψ−(¯x0−1)−λ
ρ+g0−1(¯x0−1) = A0ψ+(¯x0−1) + B0ψ−(¯x0−1)
B−1ψ0
−(¯x0−1) + 1 = A0ψ0
+(¯x0−1) + B0ψ0
−(¯x0−1)
A1ψ+(¯x1)−λ
ρ=A0ψ+(¯x1) + B0ψ−(¯x1) + g10 (¯x1)
A1ψ0
+(¯x1) = A0ψ0
+(¯x1) + B0ψ0
−(¯x1)+1
B−1ψ−(−¯x−1)−λ
ρ=A0ψ+(−¯x−1) + B0ψ−(−¯x−1) + g−10 (−¯x−1)
B−1ψ0
−(−¯x−1) = A0ψ0
+(−¯x−1) + B0ψ0
−(−¯x−1)−1.
•Case (2)(i): `−= 0, and both S−1and S01 are not empty. The value functions have the
same form as Case (1) with the state space domain (0,∞).
•Case (2)(ii): `−= 0,S−1is not empty, and S01 =∅. The value functions are given by
v0(x) = (A0ψ+(x),0<x<¯x0−1,
B−1ψ−(x)−λ
ρ+g0−1(x), x ≥¯x0−1,
v−1(x) = (v0(x) + g−10 (x),0< x ≤ −¯x−1,
B−1ψ−(x)−λ
ρ, x > −¯x−1,
v1(x) = (A1ψ+(x)−λ
ρ,0<x<max(¯x1,0),
v0(x) + g10 (x), x ≥max(¯x1,0),
15

and the constants A0,A1,B−1,¯x0−1>0,¯x1,¯x−1<0are determined by the smooth-fit
conditions:
B−1ψ−(¯x0−1)−λ
ρ+g0−1(¯x0−1) = A0ψ+(¯x0−1)
B−1ψ0
−(¯x0−1) + 1 = A0ψ0
+(¯x0−1)
A1ψ+(¯x1)−λ
ρ=A0ψ+(¯x1) + g10 (¯x1)
A1ψ0
+(¯x1) = A0ψ0
+(¯x1)+1
B−1ψ−(−¯x−1)−λ
ρ=A0ψ+(−¯x−1) + g−10 (−¯x−1)
B−1ψ0
−(−¯x−1) = A0ψ0
+(−¯x−1)−1.
•Case (2)(iii): `−= 0, and S−1=S01 =∅. The value functions are given by
v0(x) = (A0ψ+(x),0<x<¯x0−1,
−λ
ρ+g0−1(x), x ≥¯x0−1,
v1(x) = (A1ψ+(x)−λ
ρ, x < ¯x1,
v0(x) + g10 (x), x ≥¯x1,
v−1=−λ
ρ,
and the constants A0,A1,¯x0−1>0,¯x1>0, are determined by the smooth-fit conditions:
−λ
ρ+g0−1(¯x0−1) = A0ψ+(¯x0−1)
1 = A0ψ0
+(¯x0−1)
A1ψ+(¯x1)−λ
ρ=A0ψ+(¯x1) + g10 (¯x1)
A1ψ0
+(¯x1) = A0ψ0
+(¯x1)+1.
Proof. We consider only case (1) and (2)(i) since the other cases are dealt with by similar
arguments. We have S01 = (`−,−¯x01 ], which means that v0=v1+g01 on (`−,−¯x01 ].
Moreover, v1is solution to ρv1−Lv1+λ= 0 on (`−,¯x1), which combined with the bound
in the Lemma 3.1, shows that v1should be in the form: v1=A1ψ+−λ
ρon (`−,¯x1). Since
−¯x01 <¯x1, we deduce that v0has the form expressed as: A1ψ+−λ
ρ+g01 on (`−,−¯x01 ].
In the same way, v−1should have the form expressed as B−1ψ−−λ
ρon (−¯x−1,∞) and v0
has the form expressed as B−1ψ−−λ
ρ+g0−1on [¯x0−1,∞). We know that v0is solution
to ρv0− Lv0= 0 on (−¯x01 ,¯x0−1) so that v0should be in the form: v0=A0ψ++B0ψ−
on (−¯x01 ,¯x0−1). We have S1= [¯x1,∞), which means that v1=v0+g10 on [¯x1,∞) and
S−1= (`−,−¯x−1], which means that v−1=v0+g−10 on (`−,−¯x−1]. From Proposition
4.1 we know that ¯x1≤¯x0−1, and −¯x01 ≤ −¯x−1and by the smooth-fit property of value
function we obtain the above smooth-fit condition equations in which we can compute the
cut-off points by solving these quasi-algebraic equations. 2
16

Remark 4.4 The cases (2)(i)-(iii) of Theorem 4.1 imply that one needs to establish the
emptiness or non-emptiness of the sets S01 and S−1when `−= 0 before finding the cut-off
points. This issue is covered when the spread is IGBM and both of the conditions on K0
and K−1of Lemma 4.3 are satisfied, in which case one can apply case (2)(i) of the previous
Theorem. However, when `−= 0, we do not know which case (2) of Theorem 4.1 is relevant
when the spread is not IGBM or when it is IGBM but either K0and K−1are nonpositive.
2
Remark 4.5 1. In Case (1) and Case(2)(i) of Theorem 4.1, the smooth-fit conditions
system is written as:





ψ+(−¯x01 ) 0 −ψ+(−¯x01 )−ψ−(−¯x01 )
0ψ−(¯x0−1)−ψ+(¯x0−1)−ψ−(¯x0−1)
ψ+(¯x1) 0 −ψ+(¯x1)−ψ−(¯x1)
0ψ−(−¯x−1)−ψ+(−¯x−1)−ψ−(−¯x−1)




×




A1
B−1
A0
B0





=




λρ−1−g01 (−¯x01 )
λρ−1−g0−1(¯x0−1)
λρ−1+g10 (¯x1)
λρ−1+g−10 (−¯x−1)




(4.6)
and





ψ0
+(−¯x01 ) 0 −ψ0
+(−¯x01 )−ψ0
−(−¯x01 )
0ψ0
−(¯x0−1)−ψ0
+(¯x0−1)−ψ0
−(¯x0−1)
ψ0
+(¯x1) 0 −ψ0
+(¯x1)−ψ0
−(¯x1)
0ψ0
−(−¯x−1)−ψ0
+(−¯x−1)−ψ0
−(−¯x−1)




×




A1
B−1
A0
B0





=




1
−1
1
−1





.(4.7)
Denote by M(¯x01 ,¯x0−1,¯x1,¯x−1) and Mx(¯x01 ,¯x0−1,¯x1,¯x−1) the matrices:
M(¯x01 ,¯x0−1,¯x1,¯x−1) = 




ψ+(−¯x01 ) 0 0 −ψ−(−¯x01 )
0ψ−(¯x0−1)−ψ+(¯x0−1) 0
ψ+(¯x1) 0 0 −ψ−(¯x1)
0ψ−(−¯x−1)−ψ+(−¯x−1) 0





,
Mx(¯x01 ,¯x0−1,¯x1,¯x−1) = 




ψ0
+(−¯x01 ) 0 0 −ψ0
−(−¯x01 )
0ψ0
−(¯x0−1)−ψ0
+(¯x0−1) 0
ψ0
+(¯x1) 0 0 −ψ0
−(¯x1)
0ψ0
−(−¯x−1)−ψ0
+(−¯x−1) 0





.
Once M(¯x01 ,¯x0−1,¯x1,¯x−1) and Mx(¯x01 ,¯x0−1,¯x1,¯x−1) are nonsingular, straightforward com-
putations from (4.6) and (4.7) lead to the following equation satisfied by ¯x01 , ¯x0−1, ¯x1, ¯x−1:
Mx(¯x01 ,¯x0−1,¯x1,¯x−1)−1




1
−1
1
−1





=M(¯x01 ,¯x0−1,¯x1,¯x−1)−1




λρ−1−g01 (−¯x01 )
λρ−1−g0−1(¯x0−1)
λρ−1+g10 (¯x1)
λρ−1+g−10 (−¯x−1)





.
This system can be separated into two independent systems:
"ψ0
+(−¯x01 )−ψ0
−(−¯x01 )
ψ0
+(¯x1)−ψ0
−(¯x1)#−1
×"1
1#=
"ψ+(−¯x01 )−ψ−(−¯x01 )
ψ+(¯x1)−ψ−(¯x1)#−1
×"λρ−1−g01 (−¯x01 )
λρ−1+g10 (¯x1)#(4.8)
17

and
"ψ0
−(¯x0−1)−ψ0
+(¯x0−1)
ψ0
−(−¯x−1)−ψ0
+(−¯x−1)#−1
×"−1
−1#=
"ψ−(¯x0−1)−ψ+(¯x0−1)
ψ−(−¯x−1)−ψ+(−¯x−1)#−1
×"λρ−1−g0−1(¯x0−1)
λρ−1+g−10 (−¯x−1)#(4.9)
We then obtain thresholds ¯x01 , ¯x0−1, ¯x1, ¯x−1by solving two quasi-algebraic system equa-
tions (4.8) and (4.9). Notice that for the examples of OU or IGBM process, the matrices
M(¯x01 ,¯x0−1,¯x1,¯x−1) and Mx(¯x01 ,¯x0−1,¯x1,¯x−1) are nonsingular so that their inverses are
well-defined. Indeed, we have: ψ00
+>0 and ψ00
−>0. This property is trivial for the case of
OU process, while for the case of IGBM:
d2ψ+(x)
dx2=d
dx a
xa+1 (−U(a+ 1, b, c
x)(a−b+ 1))
=a(a+ 1)
xa+2 U(a+ 2, b, c
x)(a−b+ 1)(a−b+ 2) >0,∀x > 0.
dψ−(x)
dx =−ax−a−2(bxM(a, b, c
x) + cM(a+ 1, b + 1,c
x))
b,∀x > 0.
Thus, ψ0
−is strictly increasing since M(a, b, c
x) is strictly decreasing, and so ψ00
−>0. More-
over, we have:
det M(¯x01 ,¯x0−1,¯x1,¯x−1)(4.10)
=ψ−(¯x−01 )ψ+(¯x1)−ψ−(¯x1)ψ+(¯x−01 )ψ−(¯x0−1)ψ+(¯x−1)−ψ−(¯x−1)ψ+(¯x0−1).
Recalling that −¯x01 <¯x1and ¯x0−1>−¯x−1(see Proposition 4.1), and since ψ+is a strictly
increasing and positive function, while ψ−is a strictly decreasing positive function, we
have: ψ−(¯x−01 )ψ+(¯x1)−ψ−(¯x1)ψ+( ¯x−01 )>0 and ψ−(¯x0−1)ψ+( ¯x−1)−ψ−(¯x−1)ψ+( ¯x0−1)<
0, which implies the non singularity of the matrix M(¯x01 ,¯x0−1,¯x1,¯x−1). On the other hand,
we have:
detMx(¯x01 ,¯x0−1,¯x1,¯x−1)(4.11)
=ψ0
−(¯x−01 )ψ0
+(¯x1)−ψ0
−(¯x1)ψ0
+(¯x−01 )ψ0
−(¯x0−1)ψ0
+(¯x−1)−ψ0
−(¯x−1)ψ0
+(¯x0−1).
Since ψ0
+is a strictly increasing positive function and ψ0
−is a strictly increasing function,
with ψ0
−<0, we get: ψ0
−(¯x−01 )ψ0
+(¯x1)−ψ0
−(¯x1)ψ0
+(¯x−01 )<0 and ψ0
−(¯x0−1)ψ0
+(¯x−1)−
ψ0
−(¯x−1)ψ0
+(¯x0−1)>0, which implies the non singularity of the matrix Mx(¯x01 ,¯x0−1,¯x1,¯x−1).
2. In Case (2)(ii) of Theorem 4.1, we obtain the thresholds ¯x0−1>0, ¯x−1<0 from the
smooth-fit conditions which lead to the quasi-algebraic system:
"−ψ0
−(¯x0−1)ψ0
+(¯x0−1)
−ψ0
−(−¯x−1)ψ0
+(−¯x−1)#−1
×"1
1#=
"−ψ−(¯x0−1)ψ+(¯x0−1)
−ψ−(−¯x−1)ψ+(−¯x−1)#−1
×"−λρ−1+g0−1(¯x0−1)
−λρ−1−g−10 (−¯x−1)#.(4.12)
18

The non singularity of the matrix above is checked similarly as in case (1) and (2)(i) for
the examples of the OU or IGBM process. Note that ¯x0−1, ¯x−1are independent from ¯x1,
which is obtained from the equation:
−λρ−1−g10 (¯x1)ψ0
+(¯x1) + ψ+(¯x1) = 0.(4.13)
When ¯x1≤0, this means that S1= (0,∞).
3. In Case (2)(iii) of Theorem 4.1, the threshold ¯x1>0 is obtained from the equation
(4.13), while the threshold ¯x0−1>0 is derived from the smooth-fit condition leading to the
quasi-algebraic equation:
λρ−1−g0−1(¯x0−1)ψ0
+(¯x0−1) + ψ+(¯x0−1)=0.(4.14)
2
5 Numerical examples
In this part, we consider OU process and IGBM as examples.
1. We first consider the example of the Ornstein-Uhlenbeck process:
dXt=−µXtdt +σdWt,
with µ,σpositive constants. In this case, the two fundamental solutions to (3.1) are given
by
ψ+(x) = Z∞
0
tρ
µ−1exp −t2
2+√2µ
σxtdt, ψ−(x) = Z∞
0
tρ
µ−1exp −t2
2−√2µ
σxtdt,
and satisfy assumption (3.3). We consider a numerical example with the following specifi-
cations: : µ= 0.8 , σ= 0.5 , ρ= 0.1 , λ= 0.07 , ε= 0.005 , L= 0.
Remark 5.1 We can reduce the case of non zero long run mean L6= 0 of the OU process
to the case of L= 0 by considering process Yt=Xt−Las spread process, because in this
case σis constant. Finally, we can see that, cutoff points translate along L, as illustrated
in figure 6. 2
We recall some notations:
S01 = (−∞,−¯x01 ] is the open-to-buy region,
S0−1= [¯x0−1,∞) is the open-to-sell region,
S1= [¯x1,∞) is Sell-to-close region from the long position i= 1,
S−1= (−∞,−¯x−1] is Buy-to-close region from the short position i=−1.
We solve the two systems (4.8) and (4.9) which give
¯x01 = 0.2094,¯x1= 0.0483,¯x−1= 0.0483,¯x0−1= 0.2094,
and confirm the symmetry property in Proposition 4.2.
19

0 1000 2000 3000 4000 5000 6000 7000 8000
−1
−0.5
0
0.5
1
1.5
Threshold and spread simulation
0 1000 2000 3000 4000 5000 6000 7000 8000
−1
−0.5
0
0.5
1
Strategy curve
Spread
−¯x01
¯x1
−¯x−1
¯x0−1
Figure 4: Simulation of trading strategies
−5 −4 −3 −2 −1 0 1 2 3 4 5
0
2
4
6
8
10
12
Initial spread
Value
v0
v1
v−1
Figure 5: Value functions
In figure 5, we see the symmetry property of value functions as showed in Proposition
4.2. Moreover, we can see that v1is a non decreasing function while v−1is non increasing.
The next figure shows the dependence of cut-off point on parameters
20

0 0.5 1
−0.5
0
0.5
µ
The impact of µ
0 0.5 1
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
σ
The impact of σ
0 0.005 0.01 0.015
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
ε
The impact of transaction fee ε
0 0.1 0.2
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
λ
The impact of penalty factor λ
0 0.5 1 1.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
L
The impact of L
−¯x01
¯x1
−¯x−1
¯x0−1
Figure 6: The dependence of cut-off point on parameters
In figure 6, µmeasures the speed of mean reversion and we see that the length of
intervals S01 ,S0−1increases and the length of intervals S1,S−1decreases as µgets bigger.
The length of intervals S01 ,S0−1,S1, and S−1decreases as volatility σgets bigger. Lis
the long run mean, to which the process tends to revert, and we see that the cutoff points
translate along L. We now look at the parameters that does not affect on the dynamic
of spread: the length of intervals S01,S0−1,S1, and S−1decreases as the transaction fee ε
gets bigger. Finally, the length of intervals S01,S0−1decreases and the length of intervals
S1,S−1increases as the penalty factor λgets larger, which means that the holding time in
flat position i= 0 is longer and the opportunity to enter the flat position from the other
position is bigger as the penalty factor λis increasing.
2. We now consider the example of Inhomogeneous Geometric Brownian Motions which
has stochastic volatility, see more details in Zhao [17] :
dXt=µ(L−Xt)dt +σXtdWt, X0>0,
where µ,Land σare positive constants. Recall that in this case, the two fundamental
solutions to (3.1) are given by
ψ+(x) = x−aU(a, b, c
x), ψ−(x) = x−aM(a, b, c
x),
21

where
a=pσ4+ 4(µ+ 2ρ)σ2+ 4µ2−(2µ+σ2)
2σ2>0,
b=2µ
σ2+ 2a+ 2, c =2µL
σ2,
Mand Uare the confluent hypergeometric functions of the first and second kind. We can
easily check that ψ−is a monotone decreasing function, while
dψ+(x)
dx =a
xa+1 (−U(a+ 1, b, c
x)(a−b+ 1)) >0,∀x > 0,
so that ψ+is a monotone increasing function. Moreover, by the asymptotic property of the
confluent hypergeometric functions (cf.[1]), the fundamental solutions ψ+and ψ−satisfy
the condition (3.3).
•Case (2)(i): Both S−1and S01 are not empty. Let us consider a numerical example with
the following specifications: : µ= 0.8 , σ= 0.5 , ρ= 0.1 , λ= 0.07 , ε= 0.005 , and we set
L= 10. Note that, in this case the condition in Lemma 4.3 is satisfied, and we solve the
two systems (4.8) and (4.9) which give
¯x01 =−8.2777,¯x1= 9.3701,¯x−1=−8.4283,¯x0−1= 9.5336.
0 5 10 15 20 25
10
20
30
40
50
60
70
Initial spread
Value
v0
v1
v−1
Figure 7: Value functions
In the figure 7, we can see that v1is non decreasing while v−1is non increasing. More-
over, v1is always larger than v0, and v−1.
The next figure 8 shows the dependence of cut-off points on parameters (Note that the
condition in Lemma 4.3 is satisfied for all parameters in this figure).
22

0 0.5 1
4
5
6
7
8
9
10
µ
The impact of µ
0 0.5 1
8
8.5
9
9.5
10
σ
The impact of σ
0 0.005 0.01 0.015
8
8.5
9
9.5
10
ε
The impact of transaction fee ε
0 0.05 0.1 0.15 0.2
8
8.5
9
9.5
10
λ
The impact of penalty factor λ
5 10 15
4
6
8
10
12
14
16
L
The impact of L
−¯x01
¯x1
−¯x−1
¯x0−1
Figure 8: The dependence of cut-off point on parameters
We can make the same comments as in the case of the OU process, except for the
dependence with respect to the long run mean L. Actually, we see that when Lincreases,
the moving of cutoff points is no more translational due to the non constant volatility.
•Case (2)(ii):S01 is empty. Let us consider a numerical example with the following
specifications: : µ= 0.8, σ= 0.3, ρ= 0.1, λ= 0.35, ε= 0.55, and L= 0.5. We solve the
two systems (4.12) and (4.13) which give
¯x1= 0.1187,¯x−1=−0.8349,¯x0−1= 2.7504.
•Case (2)(iii): Both S−1and S01 are empty. Let us consider a numerical example with
the following specifications: : µ= 0.8, σ= 0.3, ρ= 0.2, λ= 0.05, ε= 0.65, and L= 0.1.
The two equations (4.14) and (4.13) give
¯x1= 0.4293,¯x0−1= 0.9560.
23

Appendix
A Proof of Lemma 3.1
The lower bound for v0and viare trivial by considering the strategies of doing nothing.
Let us focus on the upper bound. First, by standard arguments using Itˆo’s formula and
Gronwall lemma, we have the following estimate on the diffusion X: there exists some
positive constant r, depending on the Lipschitz constant of σ, such that
E|Xx
t| ≤ Cert(1 + |x|),∀t≥0,(A.1)
E|Xx
t−Xy
t| ≤ ert|x−y|,∀t≥0,(A.2)
for some positive constant Cdepending on ρ,Land µ. Next, for two successive trading
times τnand σn=τn+1 corresponding to a buy-and-sell or sell-and-buy strategy, we have:
Ehe−ρτng(Xx
τn, ατ−
n, ατn) + e−ρσng(Xx
σn, ασ−
n, ασn)i(A.3)
≤
Ehe−ρσnXx
σn−e−ρτnXx
τni≤EhZσn
τn
e−ρt(µ+ρ)|Xx
t|dti+EhZσn
τn
e−ρtµLdti,
where the second inequality follows from Itˆo’s formula. When investor is staying in flat
position (i= 0), in the first trading time investor can move to state i= 1 or i=−1, and
in the second trading time she has to back to state i= 0. So that, the strategy when we
stay in state i= 0 can be expressed by the combination of infinite couples: buy-and-sell,
sell-and-buy, for example: states 0 →1→0→ −1→0→ −1→0→1→0... it means:
buy-and-sell, sell-and-buy, sell-and-buy, buy-and-sell,.... We deduce from (A.3) that for any
α∈ A0,
J(x, α)≤EhZ∞
0
e−ρt(µ+ρ)|Xx
t|dti+µL
ρ.
Recalling that, when investor starts with a long or short position (i=±1) she has to close
first her position before opening a new one, so that for α∈ A1or α∈ A−1,
J(x, α)≤ |x|+EhZτ1
0
e−ρt(µ+ρ)|Xx
t|dti+EhZτ1
0
e−ρtµLdti
+EhZ∞
τ2
e−ρt(µ+ρ)|Xx
t|dti+EhZ∞
τ2
e−ρtµLdti
≤ |x|+EhZ∞
0
e−ρt(µ+ρ)|Xx
t|dti+µL
ρ,
which proves the upper bound for viby using the estimate (A.1). By the same argument,
for two successive trading times τnand σn=τn+1 corresponding to a buy-and-sell or sell-
and-buy strategy, we have:
Ehe−ρτng(Xx
τn, ατ−
n, ατn) + e−ρσng(Xx
σn, ασ−
n, ασn)
−e−ρτng(Xy
τn, ατ−
n, ατn)−e−ρσng(Xy
σn, ασ−
n, ασn)i
≤
Ehe−ρσnXx
σn−e−ρτnXx
τn−e−ρσnXy
σn+e−ρτnXy
τni
≤EhZσn
τn
e−ρt(µ+ρ)|Xx
t−Xy
t|dti,
24

where the second inequality follows from Itˆo’s formula. We deduce that
|vi(x)−vi(y)| ≤ sup
α∈Ai|J(x, α)−J(y, α)|
≤ |x−y|+EhZ∞
0
e−ρt(µ+ρ)|Xx
t−Xy
t|dti,
which proves the Lipschitz property for vi, i = 0,1,−1 by using the estimate (A.2). 2
B Proof of Lemma 4.3
Suppose that S01 =∅. Then, from the inclusion results for S0in Lemma 4.1, this implies
that the continuation region C0would contain at least the interval (0,µL+`0
ρ+µ). In other
words, we should have: ρv0− Lv0= 0 on (0,µL+`0
ρ+µ), and so v0should be in the form:
v0(x) = C+ψ+(x) + C−ψ−(x),∀0<x<µL +`0
ρ+µ,
for some constants C+and C−. From the bounds on v0in Lemma 3.1, and (3.2), we must
have C−= 0.
Next, for 0 < x ≤y, let us consider the first passage time τx
y:= inf{t:Xx
t=y}of the
inhomogeneous Geometric Brownian motion. We know from [17] that
Exe−ρτx
y=x
y−aU(a, b, c
x)
U(a, b, c
y)=ψ+(x)
ψ+(y).(B.4)
We denote by ¯v1(x;y) the gain functional obtained from the strategy consisting in changing
position from initial state xand regime i= 1, to the regime i= 0 at the first time Xx
thits
y(0 < x ≤y), and then following optimal decisions once in regime i= 0:
¯v1(x;y) = E[e−ρτx
y(v0(y) + y−ε)−Zτx
y
0
λe−ρtdt],0< x ≤y.
Since v0(y) = C+ψ+(y), for all 0 < y < µL+`0
ρ+µ, and recalling (B.4) we have:
¯v1(x;y) = E[e−ρτx
y(C+ψ+(y) + y−ε)−Zτx
y
0
λe−ρtdt]
=ψ+(x)
ψ+(y)(C+ψ+(y) + y−ε+λ
ρ)−λ
ρ
=v0(x) + ψ+(x)
ψ+(y)(y−ε+λ
ρ)−λ
ρ,∀0< x ≤y < µL +`0
ρ+µ.
Now, by definition of v1, we have v1(x)≥¯v1(x;y), so that:
v1(x)≥v0(x) + ψ+(x)
ψ+(y)(y−ε+λ
ρ)−λ
ρ,∀0< x ≤y < µL +`0
ρ+µ.
By sending xto zero, and recalling (3.6) and (3.8), this yields
v1(0+)≥v0(0+) + K0(y) + ε, ∀0< y < µL +`0
ρ+µ.
25

Therefore, under the condition that there exists y∈(0,µL+`0
ρ+µ) such that K(y)>0, we
would get:
v1(0+)> v0(0+) + ε,
which is in contradiction with the fact that we have: v0≥v1+g01 , and so: v0(0+)≥v1(0+)
−ε.
Suppose that S−1=∅, in this case v−1=−λ/ρ. By the same argument as the above
case, we have
v0(x)≥E[e−ρτx
y(v−1(y) + y−ε)] = E[e−ρτ x
y(−λ
ρ+y−ε)]
=−λ
ρ+y−εψ+(x)
ψ+(y).
by (B.4). By sending xto zero, and recalling (3.6) and (3.8), we thus have
v0(0+)≥ −λ
ρ+ε+K−1(y)y > 0.(B.5)
Therefore, under the condition that there exists y > 0 such that K−1(y)>0, we would get:
v0(0+)>−λ
ρ+ε,
which is in contradiction with the fact that we have: v−1≥v0+g−10 , and so: −λ
ρ=v−1(0+)
≥v0(0+)−ε.2
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