Oxf-Man 2011OMI01 Angoshtari
Transcript of Oxf-Man 2011OMI01 Angoshtari
Portfolio Choice with Cointegrated Assets _______________ Bahman Angoshtari The Oxford-Man Institute, University of Oxford Working paper, OMI11.01 January 2011
Oxford Man Institute of Quantitative Finance, Eagle House, Walton Well Road, Oxford, OX2 6ED Tel: +44 1865 616600 Email: [email protected] www.oxford-man.ox.ac.uk
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International Journal of Theoretical and Applied Financec© World Scientific Publishing Company
PORTFOLIO CHOICE WITH COINTEGRATED ASSETS
BAHMAN ANGOSHTARI
The Oxford-Man Institute of Quantitative Finance and the Mathematical Institute,
University of Oxford
Walton Well Road, OX2 6ED Oxford, UK
24 Jan. 2011
In portfolio management, there are specific strategies for trading between two assetsthat are cointegrated. These are commonly referred to as pairs-trading or spread-tradingstrategies. In this paper, we provide a theoretical framework for portfolio choice that jus-tifies the choice of such strategies. For this, we consider a continuous-time error correctionmodel to model the cointegrated price processes and analyze the problem of maximizingthe expected utility of terminal wealth, for logarithmic and power utilities. We obtainand justify an extra no-arbitrage condition on the market parameters with which oneobtains decomposition results for the optimal pairs-trading portfolio strategies.
Keywords: cointegrated assets; mean-reversion; portfolio management; relative valuetrading; pairs-trading; spread-trading; continuous-time error correction model; no-arbitrage condition.
1. Introduction
This paper is a contribution to portfolio management using assets whose price pro-
cesses are cointegrated. Such processes have the property that a linear combination
of them is stationary. Intuitively speaking, two cointegrated processes are tied to-
gether, will never go too far from each other and have a long-run equilibrium with
respect to each other. Many economic and financial data series are known to exhibit
these properties. Examples include interest rates, foreign exchange rates, stock price
indices, stock prices, future and spot prices, and commodities (see, among others,
[2], [4], [5], [8], [10], [21] and [27]).
In portfolio management, there are specific strategies for trading among assets
which have co-movement in their prices. These strategies are commonly referred
to as pairs-trading or spread-trading. Generally speaking, these strategies try to
exploit the relative mispricing of the stocks by taking a long position in the over-
priced asset and a short position in the under-priced one, while maintaining market-
neutrality by taking offsetting short/long positions. They have been around in one
form or another since the beginning of listed markets, but the hedge fund industry
has given a new face to these strategies as well as the specific vehicle needed to
demonstrate their successes and failures. We refer the reader to [7], [28] and [29]
1
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2 B. Angoshtari
for a detailed exposition on pairs-trading as well as on historical insights. Despite
being in widespread practice, perhaps for decades, the academic community has
just recently paid attention to study these strategies. Recent studies include [1],
[14], and [23].
Almost all of the quantitative analyses on the subject restrict the portfolio
strategies to pairs-trading strategies per se. Although this approach is intuitively
appealing and there are various good reasons that support it, there is, from a the-
oretical point of view, an unanswered fundamental question. How can one justify
this investment practice in a theoretical portfolio choice framework? In other words,
can one identify a market model and a preference criterion for the investor which
support pairs-trading? The answer to this question will be the main concern of this
paper. In other words, the main motivation is to provide a theoretical ground for
pairs-trading, without restricting the portfolio strategies a priori.
To provide such a framework, one needs to identify an appropriate market
model for cointegrated assets. To this end, a result from econometrics known as
the Granger representation theorem (see [8]) will be quite relevant. According to
this result, a pair of cointegrated processes can be represented by a so-called er-
ror correction model (ECM). Moreover, in this paper we take a continuous-time
framework. Therefore we need a continuous-time ECM. The work in [6], on spread
options valuation, provides such a model. This model can also be seen as a bivariate
generalization of the Schwartz exponential Ornstein-Uhlenbeck process (see [3] and
[26]).
With this market model at hand, one can readily apply the classical portfolio
choice approach. That is, one assumes an investment horizon and a utility function,
say the logarithmic or the power utility, at the end of the trading horizon. In turn,
one aims at maximizing the expected utility of terminal wealth and finds the optimal
admissible strategy. From the mathematical point of view, such results are not new,
and can be seen as a special case of the ones obtained, for example, in [20] or [25].
However, the results obtained in this way do not support the practice of pairs-
trading. Furthermore, they exhibit some unpleasant characteristics. For example, for
some investors who are less risk averse than an investor with logarithmic preferences,
the optimal expected terminal utility increases rapidly with the investment horizon
and approaches infinity at a finite critical horizon (these are the so-called nirvana
strategies, cf. [17]).
One of the main contributions herein is to provide and justify an extra no-
arbitrage condition (i.e. Condition 5.1) to the framework mentioned above. By
adding this condition, one obtains decomposition results for the optimal portfolio
strategies which justify pairs-trading. Furthermore, the unpleasant scenarios (i.e.
nirvana strategies) are, in turn, excluded.
The paper is organized as follows. In section 2, we introduce the market model.
In section 3, we explain the main ideas behind pairs-trading and the approach taken
by practitioners. In section 4, we pose the portfolio choice problem, and point out
the deficiencies in the associated optimal strategies, while in section 5 we introduce
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Portfolio Choice with Cointegrated Assets 3
an extra condition which amends the deficiencies and provides justification for pairs-
trading. In section 6, we provide a brief review of the cointegration property and of
error correction models. Finally, in section 7 we present numerical examples, and
conclude in section 8.
2. The market setting
The market consists of one riskless and two risky assets (stocks). The riskless asset,
the bank account Bt, t ≥ 0, offers constant interest rate r > 0. The stock price
processes, denoted by S1t and S2
t , t ≥ 0, satisfy
dS1t
S1t
=(
µ1 + δ1(
lnS1t + c lnS2
t
))
dt+ σ1dW1t , (2.1)
and
dS2t
S2t
=(
µ2 + δ2(
lnS1t + c lnS2
t
))
dt+ σ2ρdW1t + σ2
√
1− ρ2dW 2t , (2.2)
with Si0 = Si > 0, i = 1, 2. The process Wt =
(
W 1t ,W
2t
)⊤is a two dimensional
standard Brownian motion on a filtered probability space (Ω,F , (Ft) ,P) with Ft =
σ Ws : 0 ≤ s ≤ t.The coefficients µ1, µ2, δ1, δ2, c, σ1, σ2, and ρ are constants, with σ1, σ2 > 0,
|ρ| < 1 and c < 0. To ease the presentation, we introduce the notation
µ =
(
µ1
µ2
)
, δ =
(
δ1δ2
)
, and σ =
(
σ1 0
σ2ρ σ2
√
1− ρ2
)
. (2.3)
The central feature of the market model, represented by the price equations
(2.1) and (2.2), is that the process zt, t ≥ 0, defined as
zt = lnS1t + c lnS2
t , (2.4)
is enforced to be a stationary Ornstein-Uhlenbeck process. We state this simple, yet
important, result next.
Proposition 2.1. Let S1t and S2
t satisfy (2.1) and (2.2) and the process zt be given
by (2.4). Assume that the constants δ1, δ2, and c satisfy
δ1 + cδ2 < 0. (2.5)
Then zt, t ≥ 0, is a stationary Ornstein-Uhlenbeck process, solving
dzt = κ (z − zt) dt+ σzdWzt (2.6)
with
z0 = lnS10 + c lnS2
0 .
The constants κ, σz, and z are
κ = − (δ1 + cδ2) , σ2z = ‖(1, c)σ‖2 , (2.7)
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4 B. Angoshtari
and
z =1
κ
(
µ1 + cµ2 −1
2
(
σ21 + cσ2
2
)
)
, (2.8)
and the process W zt , t ≥ 0, is given by
W zt = 1
σz(1, c)σWt
= 1σz
(σ1 + cσ2ρ)W1t + cσ2
√
1− ρ2W 2t
.
(2.9)
Proof. Applying Ito’s formula to (2.1) and (2.2) yields
d lnS1t =
(
µ1 −1
2σ21 + δ1zt
)
dt+ σ1dW1,t
and
d lnS2t =
(
µ2 −1
2σ22 + δ2zt
)
dt+ σ2ρdW1,t + σ2
√
1− ρ2dW2,t.
It then follows that zt satisfies (2.6) and, hence, it is an Ornstein-Uhlenbeck process.
Moreover, assumption (2.5) implies that κ > 0 and the stationarity of zt follows.
This concludes the proof.
We recall that two stochastic processes are said to be cointegrated if a linear
combination of them is a stationary process. Hence, Proposition 2.1 implies that
the stock log-prices are cointegrated. For this reason, we will refer to inequality
(2.5) as the cointegration condition, and to c as the cointegration coefficient.
As it was mentioned in the introduction, more can be said about the connection
of the market model considered herein and the theory of cointegration. Indeed, as
shown in [6], the price dynamics given by (2.1) and (2.2) is the diffusion limit of a
so-called error correction model. These models are discrete-time representations of
systems of cointegrated processes. We present further details on cointegration and
error correction models in section 6 and we, also, refer the reader to [11], [13], and
[15], and the references therein, for a more detailed exposition on cointegration.
This connection with econometrics will be particularly useful when we estimate the
parameters in section 7.
3. Pairs-trading in investment practice
In portfolio management, there are specific strategies for trading among assets
which have co-movement in their prices. These strategies are commonly referred
to as pairs-trading or spread-trading (the difference will be clarified in a moment).
As mentioned earlier, the main motivation of this paper is to provide a theoretical
ground for these investment rules. Before we provide such a justification, we explain
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Portfolio Choice with Cointegrated Assets 5
the main ideas behind pairs-trading and, specifically, the approach taken by practi-
tioners. To make the arguments more precise, we assume that, in accordance to the
model introduced earlier, we have only one pair of assets, say S1 and S2. Moreover,
to keep the concepts intuitive, the arguments will be presented in an informal way.
We refer the reader to [7], [28] and [29] for a detailed exposition on pairs-trading.
The key step in pairs-(or spread-)trading strategies is to find a way to quantify
the relative price of the pair. Note that co-movement in prices implies that there
should be a way to combine the two prices to obtain a mean-reverting process in a
way highlighted in Proposition 2.1. It is this mean-reverting process that is used to
quantify the relative price of the pair.
There are two common assumptions regarding this relative price indicator:
i) A linear combination of the asset prices S1t and S2
t is mean-reverting. In other
words, there is a constant c < 0 such that the process st, t ≥ 0, given by
st = S1t + cS2
t ,
is mean-reverting.
ii) A linear combination of the logarithm of the prices is mean-reverting. In other
words, there is a constant c < 0 such that the process zt, t ≥ 0, defined as
zt = lnS1t + c lnS2
t , (3.1)
is mean-reverting.
To differentiate between the two cases, we will be referring to st as the spread
and to zt as the residual. In analogy, we will be referring to market settings with
assumption (i) as spread-trading, and those with assumption (ii) as pairs-trading.
In this paper, we work under assumption (ii), that is pairs-trading.
Next, we explain the practitioners’ approach. A pairs-trader starts by identifying
the residual zt of (3.1), ignoring the asset prices S1t and S2
t . Note that modeling ztis essentially equivalent to determining the price of one of the assets in terms of the
other. For this reason, any model for zt is often called a partial pricing model.
A benchmark partial pricing model is to assume that zt, t ≥ 0, is a stationary
Ornstein–Uhlenbeck process, given by
dzt = κ (z − zt) dt+ σzdWzt , (3.2)
with z0 = lnS10 + c lnS2
0 . Here (with a slight abuse of notation) κ, z, and σz are
constants, κ > 0, and W zt is a standard Brownian motion.
We clarify the difference between the partial pricing model (3.2) and the seem-
ingly identical model (2.6). The former is an assumption about the market per se,
while the latter is a direct consequence of the price equations (2.1) and (2.2).
As mentioned earlier, a pairs-trader does not model S1t and S2
t separately. In-
stead, he takes (3.2) as the partial market model.
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6 B. Angoshtari
After assuming such a model, the pairs-trader restricts the candidate market
strategies to the so-called pairs-trading strategies. These are any strategies with the
following two properties:
(1) They are market-neutral, namely,
(
the portfolio weight
of the second stock
)
= c×(
the portfolio weight
of the first stock
)
.
In section 4, we denote the portfolio weights of the risky assets by αt =(
α1t , α
2t
)
.
Therefore, any market-neutral strategy αt is given by
αt = αMNt (1, c) , (3.3)
for some scalar process αMNt .
(2) The strategies exploit the relative mispricing of the pair by maintaining a long
position in the over-priced stock and a short position in the under-priced one,
as indicated by the sign and size of the residual zt.
Remark 3.1. As it can be easily seen, the main idea behind pairs-trading is that
the trader tries to exploit the relative mispricing in the pair, while being protected
from the overall market movements by following a market-neutral strategy.
Although this approach is intuitively appealing and there are various good rea-
sons that support it, there is, from a theoretical point of view, an unanswered
fundamental question. How can one justify this investment practice in a theoretical
portfolio choice framework? In other words, can one identify a market model and a
preference criterion for the investor which support pairs-trading?
Remark 3.2. It is important to clarify the following issue: in practice there is no
claim that pairs-trading is the best strategy. On the contrary, the consensus is that
pairs-trading should be followed given that one only has a partial pricing model.
Therefore, to find a set of assumptions that support pairs-trading, we do not need to
find pairs-trading as the optimal strategy under some criterion but, rather, we only
need to show that if one only knows a partial pricing model, then a pairs-trading
strategy is the only part of the optimal strategy which can be fully identified.
This idea will be the theme for the rest of the paper. Specifically, we try to
justify pairs-trading in the sense mentioned above, without restricting the portfolio
strategies per se. This requires us to assume a full pricing model which implies
the partial pricing model (3.2). Note that according to Proposition 2.1, the market
setting of section 2 fulfills this requirement.
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Portfolio Choice with Cointegrated Assets 7
4. The Merton’s problem with homothetic utilities
We provide a theoretical framework for the investment practice of pairs-trading.
We do so by considering a risk preference criterion for the investor, analyzing the
associated maximal expected utility problem, and exploring the connection between
pairs-trading and the optimal investment strategies.
To this end, we consider the market setting introduced in section 2. Starting at
t = 0 with an initial endowment x > 0 and a trading horizon [0, T ], the investor
invests, at any time t ∈ [0, T ], in the three assets. We denote the portfolio weights of
the risky assets by αt =(
α1t , α
2t
)
. Then,(
1− α1t − α2
t
)
is the proportion of wealth
invested in the bank account.
The discounted value (with the bank account Bt as the numeraire) of the total
investment at time t ≥ 0 is denoted by Xα,xt . Whenever it is appropriate, we might
use the short-hand notations Xαt or Xt to refer to the wealth process. Using (2.1)
and (2.2), we have that Xα,xt satisfies
dXα,xt = Xα,x
t αt (µ+ δzt − 1r) dt+Xα,xt αtσdWt, (4.1)
with Xα,x0 = x, x ≥ 0. Here µ, σ and δ are as in (2.3), and zt, t > 0, is as in (2.4).
The investment policies α1t and α2
t , t ≥ 0, will play the role of control pro-
cesses and are taken to satisfy the usual admissibility assumptions of being self-
financing, Ft-progressively measurable, and satisfying the integrability condition
E
(
∫ T
0 ‖Xαt αtσ‖2 dt
)
< ∞, and the no bankruptcy condition Xαt ≥ 0, t ∈ [0, T ]. We
denote the set of admissible strategies by A.
We assume that the investor has an increasing and concave utility function
U : R+ −→ R at T . The value function V (t, x, z) : [0, T ]×R+ ×R 7→ R is given by
V (t, x, z) = supA
E (U (XαT ) |Xα
t = x, zt = z) . (4.2)
The goal is to find V (0, x, z) and the associated optimal portfolio strategy.
For tractability, we consider two specific choices for the utility function U (x) ,
the logarithmic and power utilities.
4.1. Logarithmic utility
Let
U (x) = lnx, x > 0.
The following Proposition gives the log-optimal portfolio strategya.
Proposition 4.1. Assume that the cointegration condition (2.5) holds. Let zt, t ≥0, be the residual process introduced in (2.4). Then, the value function, denoted by
aFor rigorous results on the existence and uniqueness of log-optimal strategies under more generalmarket setting, see [9], [16], and [19].
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8 B. Angoshtari
V log (t, x, z), is finite for all T > 0 and given by
V log (t, x, z) = lnx+ E
(
12
∫ T
t
∥
∥σ−1 (µ+ δzs − 1r)∥
∥
2ds|zt = z
)
. (4.3)
Furthermore, the log-optimal portfolio, denoted by αlogt is given by the vector
αlogt = αC,log + αR.V,log (zt) . (4.4)
Here, αC,log is a constant weight investment strategy given by
αC,log = (µ− 1r)⊤(σσ⊤)−1, (4.5)
and αR.V,log (zt) is a relative-value investment strategy given by
αR.V,log (zt) = ztδ⊤ (σσ⊤)−1
. (4.6)
Proof. See, Appendix A.1.
The well-known property of the log-optimal allocation is that it is myopic, in
that a long term strategy can be thought of as a sequence of short-term strategies
executed one after another (cf. [22]). Indeed, assume that there are two investors,
both with logarithmic utility but with two different time horizons, T ′ and T , where
T ′ < T (so the first investor is short-sighted or myopic compare to the second
investor). Note that the investment horizon does not appear in equation (4.4), i.e.
αC,log is a constant-weight portfolio and αR.V,log (z) is a time-independent function.
Therefore, both investors will follow precisely the same investment strategy on the
interval t ∈ [0, T ′). It follows that one can think of a long-term log-optimal strategy
as a sequence of short-term log-optimal strategies executed one after another.
It is important to observe that the optimal strategy of Proposition 4.1 does not
justify pairs-trading, in the sense discussed in Remark 3.2. Indeed, firstly, we see
that the relative-value portfolio αR.V,log is not market-neutral (cf. 3.3). Secondly,
it depends on δ and σ, and, hence, it cannot be identified if one only knows the
partial pricing model (2.6). We provide the remedy for these deficiencies in section
5.
4.2. Power utility
Let
U (x) =x1−γ
1− γ, x > 0, (4.7)
where γ ∈ (0, 1) ∪ (1,+∞) is the relative risk aversion parameter.
It is well known that the logarithmic utility can be considered as the limiting
case of the power utility when γ → 1. In turn, if γ > 1, respectively 0 < γ < 1,
the investor is more risk averse, respectively more risk seeking, than a log-utility
investor, while if γ = 0, the investor is risk neutral.
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Portfolio Choice with Cointegrated Assets 9
We consider the expected utility of terminal wealth problem (4.2) with U as in
(4.7)b. We denote the corresponding value function by V p (t, x, z). Propositions 4.2
and 4.3 bellow are the analogues of Proposition 4.1.
We introduce the constant γ0, given by
γ0 = 1− κ2
σ2z ‖σ−1δ‖2
, (4.8)
with σ and δ as in (2.3), and κ and σ2z as in (2.7), respectively. Note that γ0 ∈ [0, 1),
as it follows from the inequality
κ2 ≤ σ2z
∥
∥σ−1δ∥
∥
2
(see (A.16) for details).
As it is shown bellow, the optimal strategies under power utility can be classified
as follows:
(1) The well-behaved strategies, which correspond to the case γ ≥ γ0. In this case,
the value function is finite for any choice of time horizon T > 0.
(2) The so-called nirvana strategiesc, which correspond to the case γ < γ0. In this
case, the expected terminal utility increases rapidly with the investment horizon
and approaches infinity at a finite critical horizon, denoted by Tmax (cf. (4.21)).
The well-behaved and the nirvana strategies are described in Propositions 4.2
and 4.3, respectively.
Proposition 4.2. (Well-behaved case) Suppose that the cointegration condition
(2.5) holds and that γ ≥ γ0, with γ0 as in (4.8).
i) The value function is finite for all T > 0 and given by
V p (t, x, z) =x1−γ
1− γef(t)+g(t)z+ 1
2h(t)z2
, (4.9)
with h (t), g (t), and f (t) being the deterministic functions given below.
ii) The optimal portfolio, denoted by αpt , is given by the vector
αpt =
1
γαlogt +
(
g (t) + h (t) ztγ
)
(1, c) , (4.10)
where αlogt is the log-optimal portfolio (cf. (4.4)), and c is the cointegration
coefficient.
iii) Define the constants
a = −σ2z
γ, b = 2κ
γ, c = − 1−γ
γ
∥
∥σ−1δ∥
∥
2, (4.11)
bThe optimal investment problem with power utility has been studied in market settings that aremore general than what considered herein. Specifically, Propositions 4.2 and 4.3 can be seen as aspecial case of the results obtained in [20] or [25].cThe terminology is borrowed from [17].
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10 B. Angoshtari
and
∆ = b2 − 4ac =4σ2
z
∥
∥σ−1δ∥
∥
2
γ2(γ − γ0) . (4.12)
Then, the functions h and g appearing in (4.9) and (4.10) are given by
h (t) =
−2c
b+√∆coth
(√∆2 (T−t)
) , if γ > γ0,
b−2a
(
1− 2b(T−t)+2
)
, if γ = γ0,
(4.13)
and
g (t) = C1G1 (t) + C2G2 (t) , (4.14)
with
C1 =γκz + (1− γ) (1, c) (µ− 1r)
σ2z
, (4.15)
C2 =
κσ2z
(
κz + 1−γγ
(1, c) (µ− 1r))
+ 1−γγ
δ⊤(
σσ⊤)−1(µ− 1r)
,
(4.16)
G1 (t) =
2√∆
b+√∆
e
√∆2
(T−t)
e√
∆(T−t)− b−√
∆
b+√
∆
− 1, if γ > γ0,
2b(T−t)+2
− 1, if γ = γ0
(4.17)
and
G2 (t) =
2√∆
(
e√
∆2 (T−t) − 1
)
(
e
√∆2
(T−t)− b−√
∆
b+√
∆
e√
∆(T−t)− b−√
∆
b+√
∆
)
, if γ > γ0,
(T − t) b(T−t)+4
2b(T−t)+4, if γ = γ0.
(4.18)
Moreover, the function f appearing in (4.9), is given, in terms of h and g, by
f (t)
=∫ T
t
σ2z
2γ g2 (s) +
(
κz + 1−γγ
(1, c) (µ− 1r))
g (s) + 12σ
2zh (s)
ds
+ 1−γ2γ
∥
∥σ−1 (µ− 1r)∥
∥
2(T − t) .
(4.19)
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Portfolio Choice with Cointegrated Assets 11
Proof. See, Appendix A.2.
Proposition 4.3. (Nirvana case) Suppose that the cointegration condition (2.5)
holds and that γ < γ0, with γ0 as in (4.8).
i) The value function is finite if and only if
T < Tmax, (4.20)
with
Tmax =γ
σz ‖σ−1δ‖√γ0 − γ
(
arctan
(
κ
σz ‖σ−1δ‖√γ0 − γ
)
+π
2
)
. (4.21)
ii) If (4.20) holds, then the value function V p (x, z, t) is given by (4.9), the optimal
portfolio weights αpt is given by (4.10), and f (t) is given by (4.19). However,
h (t) and g (t) are now given by
h (t) =
√−∆
2atan
(
arctan
(
b√−∆
)
−√−∆
2(T − t)
)
− b
2a(4.22)
and
g (t) = C1G3 (t) + C2G4 (t) . (4.23)
Here, a, b, and ∆ are as in (4.11) and (4.12), and C1 and C2 are as in (4.15)
and (4.16). The functions G3 (t) and G4 (t) are given by
G3 (t) =cos(
arctan(
b√−∆
))
cos(
arctan(
b√−∆
)
−√−∆2 (T − t)
) − 1 (4.24)
and
G4 (t) =2√−∆
×
×sin(
arctan(
b√−∆
))
− sin(
arctan(
b√−∆
)
−√−∆2 (T − t)
)
cos(
arctan(
b√−∆
)
−√−∆2 (T − t)
) . (4.25)
Proof. See, Appendix A.2.
Equality (4.10) gives the form of the optimal allocation for both the well-behaved
and the nirvana strategies. It shows that moving from the logarithmic preference to
power utility will change the optimal portfolio in two ways:
i) the log optimal portfolio is divided by γ, and
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12 B. Angoshtari
ii) the investor also invests in a market-neutral strategy, denoted by αMN (t, zt),
and given by
αMN (t, zt) =1
γ(g (t) + h (t) zt) (1, c) . (4.26)
The market-neutral strategy, αMN (t, zt), acts against the change in the pairs-
trading part of the portfolio when the risk aversion changes. Specifically, it can be
shown that:
a) if γ > 1, then h (t) < 0, t ≥ 0. This implies that the market-neutral term is a
pairs-trade (i.e. it buys the under-priced stock and sells the over-priced one).
Hence, the market-neutral component mitigates the decrease in the pairs-trade
which comes from dividing the positions in the stocks by γ.
b) if γ < 1, then h (t) > 0, t ≥ 0. This implies that the market-neutral term is the
opposite of a pairs-trade (i.e. it sells the under-priced stock and buys the over-
priced one). Therefore, the market-neutral component mitigates the increase in
the pairs-trade resulting from dividing the positions in the stocks by γ.
Note that the optimal allocation for power utility is not myopic, as the functions
h (t) and g (t) depend explicitly on the time horizon T .
From a practical point of view, these results are not satisfactory. By the same
argument as in the logarithmic case, we deduce that they are not consistent with
practice of pairs-trading. Furthermore, the optimal policies might have unpleasant
properties (i.e. blow-ups for nirvana solutions). In the next section, we provide a
way to amend these deficiencies and find theoretical ground for pairs-trading.
5. The decomposition results and pairs-trading
Aiming at remedying the deficiencies of the strategies obtained in the previous
section, we introduce the following condition.
Condition 5.1. Assume that there exists a constant η such that
δ = η σσ⊤ (1, c)⊤. (5.1)
Equivalently
δ1δ2
=σ21 + cσ1σ2ρ
cσ22 + σ1σ2ρ
, (5.2)
or,
δ = − κ
σ2z
σσ⊤ (1, c) . (5.3)
Here, δ and σ are given by (2.3), and κ and σ2z are given by (2.7).
Remark 5.1. The facts that (5.1) and (5.2) are equivalent, and that (5.3) implies
(5.1), are straightforward. To see that (5.3) follows from (5.1), left-multiply (5.1)
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Portfolio Choice with Cointegrated Assets 13
by the vector (1, c) to obtain
−κ = (1, c) δ = η ‖(1, c)σ‖2 = ησ2z . (5.4)
This, in turn, yields
η = − κ
σ2z
, (5.5)
and (5.3) follows.
The following Proposition provides three seemingly different ways to obtain, and
interpret, this condition.
Proposition 5.1. Condition 5.1 is:
i) Equivalent to the so-called Novikov condition, namely
E
(
e12
∫
T
0‖λs‖2ds
)
< ∞, ∀T ∈ (0,∞) , (5.6)
where λt, t ∈ [0, T ], is the market price of risk, defined by
λt = σ−1 (µ+ δzt − 1r) . (5.7)
ii) The necessary and sufficient condition under which the optimal strategies of
section 4 justify pairs-trading, in the sense discussed in Remark 3.2.
iii) Necessary and sufficient in order to exclude nirvana strategies.
Proof. See, Appendix A.3.
The Novikov condition is a sufficient condition for the market to be arbitrage-
free. Therefore, from statement (i) above, imposing Condition 5.1 guarantees that
there is no arbitrage opportunity in the market.
From a theoretical point of view, it would be interesting to investigate whether
this condition is also necessary for the market to be arbitrage-free. In general, the
Novikov condition is a rather strong condition, and usually not necessary. Nonethe-
less, the appearance of nirvana solutions when the condition fails, might lead to
arbitrage. The necessity of Condition 5.1 is currently under investigation.
Next, we turn our attention to the optimal strategies obtained by imposing
Condition 5.1. Note that, by statements (ii) and (iii) in Proposition 5.1, Condition
5.1 is precisely what we need to remedy the deficiencies of the optimal strategies
obtained in section 4. For the case of logarithmic utility, the optimal strategy for the
investor is to divide her wealth into a constant-weight portfolio and a pairs-trading
portfolio. We state this decomposition result next.
Theorem 5.1. Suppose that both the cointegration condition (2.5) and Condition
5.1 hold. Then, the log-optimal portfolio weights of Proposition 4.1 can be decom-
posed as
αlogt = αC,log + αP.T,log (zt) , (5.8)
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14 B. Angoshtari
with αC,log as in (4.5), and
αP.T,log (zt) = − κ
σ2z
zt (1, c) . (5.9)
Proof. The results follow by imposing (5.3) on Proposition 4.1.
We continue with a discussion on the above policies. The constant weight com-
ponent of the log-optimal portfolio, αC,log, is the log-optimal portfolio if there was
no cointegration (i.e. δ1 = δ2 = 0 in (2.1) and (2.2)).
The relative value component of the log-optimal portfolio, αP.T,log (zt), is a
market-neutral strategy (cf. 3.3), with αMNt = κzt/σ
2z . Furthermore, the inequality
−κ/σ2z < 0 implies that this market-neutral strategy always shorts the over-priced
stock and longs the under-priced one. Therefore, this is a genuine pairs-trading
strategy. Note that the pairs-trading component depends solely on the parameters
c, κ, and σ2z . This, in turn, yields that in order to identify the log-optimal pairs-
trade, we only need to specify the partial pricing model (3.2).
We conclude that Theorem 5.1 gives a solid ground for choosing a pairs-trading
strategy. Indeed, the pairs-trading component (5.9) only depends on parameters
of the partial pricing model, while the constant weight portfolio component (4.5)
depends on the drift parameters, µ1 and µ2, which are quite hard to estimate in
practice. Hence, if we only know a partial pricing model, then the pairs-trading
component is the only part of the optimal strategy which we can fully identify.
Finally, the form of the log-optimal pairs-trade (5.9) is quite intuitive, and de-
serves attention of its own. It tells us that the long-short positions should be bigger
if the mean-reversion rate κ is bigger, and they should be smaller if the variance
rate of the residual, σ2z , is larger.
Next, we consider the case of power utility. As in the logarithmic case, much
more can be said if we, also, consider the market to be arbitrage-free. In particular
the optimal strategies will be well-behaved for all values of γ, and we would obtain
a decomposition result like (5.8).
We state these results next.
Theorem 5.2. Suppose that both the cointegration condition (2.5) and Condition
5.1 hold. Then, the value function V p (t, x, z), introduced in section 4.2, is finite for
all T > 0 and γ > 0.
Furthermore, the optimal strategy αpt (cf. (4.10)) can be decomposed as
αpt = αC + αP.T. (zt) + αT.V. (t) , (5.10)
where αC is a constant portfolio given by
αC =1
γ(µ− 1r)
⊤(σσ⊤)−1, (5.11)
αP.T. (zt) is a pairs-trading strategy given by
αP.T. (zt) =
(−κ
σ2z
)
h (t) zt (1, c) , (5.12)
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Portfolio Choice with Cointegrated Assets 15
with
h (t) =1 + 1√
γcoth
(
κ√γ(T − t)
)
1 +√γ coth
(
κ√γ(T − t)
) , (5.13)
and αT.V. (t) is deterministic, given by
αT.V. (t) = g (t) (1, c) , (5.14)
where
g (t) = κzσ2z
1γ
1 + ((1− β)− (2− β) γ)
(
1−√γ csch
(
κ√γ(T−t)
)
1+√γ coth
(
κ√γ(T−t)
)
)
−(1−γ)+
√γ csch
(
κ√γ(T−t)
)
1+√γ coth
(
κ√γ(T−t)
)
(5.15)
with
β =
(
σ21 + σ2
2
)
/2− (1 + c) r
κz. (5.16)
Proof. See, Appendix A.4.
Decomposition (5.10) says that it is optimal for the investor to divide her wealth
into a constant-weight portfolio αC , a pairs-trade αP.T. (zt), and a time-varying (but
deterministic) portfolio αT.V. (t).
As in the logarithmic case, the constant weight portfolio, i.e. αC of (5.11), is the
optimal portfolio if there was no cointegration (i.e. δ1 = δ2 = 0 in (2.1) and (2.2)).
The constant weight portfolio is the constant part of the log-optimal portfolio (i.e.
αC,log in (4.5)) divided by γ.
Moreover, as in the logarithmic case, the pairs-trading component, αP.T. (zt), has
the factor −κ/σ2z . In other words, the long-short positions are directly proportional
to the mean-reversion rate κ and inversely proportional to the variance rate σ2z .
Remark 5.2. The only difference between the pairs-trade for power utility (i.e.
αP.T. (zt) of (5.12)) and the pairs-trade for logarithmic utility (i.e. αP.T,log (zt) of
(5.9)) is the time-varying factor h (t) in (5.13). This adjustment factor has the
following properties:
i) limγ→1
h (t) = 1. Hence, the power-optimal pairs-trade becomes the log-optimal
pairs-trade when γ → 1, as it is expected.
ii) For γ > 1, the function h (t) is decreasing in t and satisfies
1√γ≤ h (t) <
1
γ< 1, for t ∈ (−∞, T ] . (5.17)
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16 B. Angoshtari
Therefore, more risk averse investors take smaller long-short positions if com-
pared to a log-utility investor. Furthermore, they tend to reduce the size of their
pairs-trade as time increases.
iii) For γ < 1, the function h (t) is increasing in t and satisfies
1 <1
γ< h (t) ≤ 1√
γ, for t ∈ (−∞, T ] . (5.18)
Therefore, less risk averse investors take larger long-short positions if compared
to a log-utility investor. Furthermore, they tend to increase the size of their
pairs-trade as time increases.
Remark 5.3. The time varying component αT.V. (t) is a market-neutral strategy.
Note that it does not depend on the residual zt, and it is fully known at the initial
time. It can be shown that the function g (t) of (5.15) vanishes as γ → 1. This fact
explains the absence of αT.V. (t) in the logarithmic case. Moreover, g (t) vanishes as
(T − t) → 0, and it is finite as (T − t) → +∞, with the limit given by
lim(T−t)→+∞
g (t) =1− β
γ+
1√γ− (2− β) .
We end this section by commenting on the practicality of the results. As in the
logarithmic case, we only need to know c, κ, and σ2z to identify the pairs-trading
component αP.T. (zt). On the other hand, to estimate the time varying component
αT.V. (t), we also need to know the parameters r, σ21 , σ
22 , and z, and to identify the
constant-weight component αC , we need to estimate µ1, µ2, and ρ. Hence, in the
power utility case, as in the logarithmic case, if we only know a partial pricing model,
then the pairs-trading component is the only part of the optimal strategy which we
can fully identify. Therefore, the decomposition result (5.10) justifies pairs-trading
for an investor with power utility.
6. A short review of cointegration and error correction models
We provide a brief review of cointegration and error correction models. We refer
the reader to [11], [13], [15] and the references therein for a detailed exposition on
the subject. For simplicity, we only consider processes that are discrete in time, i.e.
processes xt, t ∈ T , where T = ti, i ∈ Z+|0 = t0 < t1 < t2 < .... We first recall
the definition of a stationary process.
Definition 6.1. A discrete-time stochastic process xt, t ∈ T , is (variance) sta-
tionary, if there exist a constant µ < ∞ and a function σ : R → R such that,
∀t, s ∈ T ,
i) E (xt) = µ
and
ii) E ((xt − µ) (xs − µ)) = σ (s− t) .
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Portfolio Choice with Cointegrated Assets 17
For any process yt, we introduce the difference operator ∆yti = yti − yti−1 .
Then, a stochastic process yt is integrated of order 1, or simply yt ∈ I (1), if yt is
non-stationary and ∆yt is a stationary process.
Definition 6.2. Two discrete-time stochastic processes xt and yt are cointegrated
if the following two conditions hold:
i) each process is integrated of order 1,
and
ii) there exists a linear combination, say zt = xt+cyt, which is a stationary process.
The constant c is called the cointegration coefficient, and the stationary linear
combination zt is called the cointegration residual (or the residual, for short).
According to a result known as the Granger representation theorem (see [8]),
a pair of cointegrated processes xt and yt can be represented by a so-called error
correction model (ECM). Indeed, as it is shown in [6], the price dynamics given by
(2.1) and (2.2), is the continuous-time version of the following ECM:
∆ lnS1t =
(
µ1 −1
2σ21
)
∆t+ δ1∆tzt−1 + σ1
√∆tε1t (6.1)
and
∆ lnS2t =
(
µ2 −1
2σ22
)
∆t+ δ2∆tzt−1 + σ2
√∆tε2t , (6.2)
where µi, δi, and σi, i = 1, 2, are the constants introduced in section 2, zt−1 is given
by (2.4), ∆t is the time-increment, and(
ε1t , ε2t
)
is a two dimensional Gaussian white
noise with correlation ρ (|ρ| < 1).
For completeness on the subject, we also provide a brief discussion about the
statistical techniques used for cointegration analysis. To this end, suppose that we
have two processes, say xt and yt, t ∈ T , that are both I (1) (one can test this by
using various unit-root tests, for example the ADF (augmented Dicky Fuller) test).
The goal is then to determine whether they are cointegrated, and if so, estimate
the corresponding ECM.
There are two main approaches to this problem: the Engle-Granger two-step
procedure and the Johansen method. The Engle-Granger method works as follows:
i) One estimates the residual zt and the cointegration coefficient c by regressing
xt over yt, i.e.
xt = −cyt + zt.
At this point, one tests for cointegration. For this, there are various tests one
could apply. In this paper we chose the Phillips-Ouliaris variance ratio and trace
statistic tests (see [24]).
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18 B. Angoshtari
2001 2002 2003 2004 2005 2006 2007 2008 2009 20100
20
40
60
80
100
120
140Stock Prices
IBMMSFT
2001 2002 2003 2004 2005 2006 2007 2008 2009 2010−0.6
−0.4
−0.2
0
0.2The Residual
z = ln(MSFT) − 0.73 ln(IBM)
$
Fig. 1. Stock prices of Microsoft and IBM (top), and the associated residual (bottom) from Jan2001 to Dec 2009.
ii) If the tests did not reject cointegration, then one estimates the ECM by re-
gressing ∆xt and ∆yt over zt−1.
In the Johansen approach, one considers the ECM directly and simultaneously
tests for cointegration and estimates the ECM. For more information on the Jo-
hansen approach, we refer the reader to [13] and [15].
7. Numerical example
We provide an illustration using real market data in order to give some insights
on the ideas presented in the previous sections. We used the daily stock prices of
Microsoft and IBM for a period of nine years (from Jan. 2, 2001 to Dec. 31, 2009).
The data series are extracted from the CRSPd database and are adjusted for splits
and cash dividends. The top part of figure 1 shows these time series.
We propose a methodology to estimate the ECM of equations (6.1) and (6.2)
while imposing the no-arbitrage condition (5.3).
We take the Engle-Granger two-step procedure, as discussed in section 6. After
dSource: c©201010 CRSP R©, Center for Research in Security Prices. Booth School of Business,The University of Chicago. Used with permission. All rights reserved. www.crsp.chicagobooth.edu
January 24, 2011 14:58 WSPC/INSTRUCTION FILE PortfolioChoice-WithCoIntegratedAssets˙LaTex
Portfolio Choice with Cointegrated Assets 19
establishing that the processes are I(1) using the ADF test, we use the Phillips-
Ouliaris variance ratio (or Pu) and trace statistic (or Pz) tests for testing for coin-
tegration. If the tests imply cointegration, then c and the residual process zt can
be obtained by regressing lnS1t over lnS2
t . To find κ and σz , we fit a first order
autoregressive model (i.e. AR(1)) to the time series zt obtained by regression in the
previous step. To obtain σ1, σ2 and ρ, we regress ∆ lnS1t and ∆ lnS2
t over zt−1.
Then, δ1 and δ2 are calculated by inserting the estimates of σ1, σ2, ρ, κ and σz
in equation (5.3). Finally µi, i = 1, 2, are estimated by regressing ∆ lnSit − δizt−1,
i = 1, 2, over a constant.
To check the performance of the estimation method, we try it on simulated data,
generated by an Euler scheme from the SDE given by the price equations (2.1) and
(2.2), with the market parameters given in Table 1. The top part of figure 2 shows
ten simulated sample paths for each stock.
We calculate out-of-sample test statistics and estimates as follows. On each day
form Dec. 12, 2004 to Dec. 12, 2009, we take the last four years of data and run
the estimation process discussed above. The results are shown in figures 3 to 6. The
results suggest that the estimates for c, σz, σ1, σ2, and ρ are acceptable while the
estimates for κ, δ1 and δ2 are occasionally far off and the ones for µi, i = 1, 2, are
not robust at all (as is expected).
Next, we evaluate the portfolio value for the log-optimal strategy of equation
(4.4) using simulated data. We assume an initial wealth of $100. Figure 7 shows the
log-optimal portfolio value for each simulation using the real parameters (i.e. from
Table 1). In all scenarios except one, the portfolio does not lose more than half of its
initial value, while all scenarios end up with the terminal wealth of at least $1000.
Figure 8 shows the log-optimal portfolio value, for each simulation, by using the
out-of-sample estimates. The results are quite different from the case of using real
values of parameters. In five out of ten scenarios, it is observed that the portfolio
ends up losing more than 90% of its initial value at some point during the trading
horizon. This observation highlights the importance of having good estimates.
We have conducted the same procedure for the real data series of figure 1. The
results are shown in figures 9 to 12. Note that, as expected, the estimator for c is
quite robust, but the estimators for µi, i = 1, 2, are not robust at all. Moreover,
the test statistics imply that the cointegration relation ceased to exist somewhere
during the estimation period. The estimates for σz , σ1, σ2, and ρ suggest that these
parameters vary significantly during the estimation period.
Figure 13 shows the performance of the log-optimal strategy using real data of
figure 1. The portfolio weights are calculated by using the out-of-sample estimates
discussed above. We consider three scenarios with different assumptions on trans-
action costs and frequency of trade. In the first scenario, associated with the top
solid line, it is assumed that there are no transaction costs and that the investor
is adjusting his/her portfolio daily. In the second scenario, showed in the bottom
line, it is assumed that the investor is buying with the daily high price and selling
January 24, 2011 14:58 WSPC/INSTRUCTION FILE PortfolioChoice-WithCoIntegratedAssets˙LaTex
20 B. Angoshtari
with the daily low price. As it can be seen, this strategy is not profitable due to the
high transaction cost. Finally, the third scenario (the middle line) refers to the case
that there are transaction costs, but the investor adjusts his/her portfolio every two
weeks.
Table 1. Values of market parameters used for simulation.
c µ1 µ2 σ1 σ2 ρ δ1 δ2 σ2z
κ S10
S20
0.73 0.05 0.05 0.25 0.24 0.5 -4.49 1.33 0.0494 5.46 21.7 84.8
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Portfolio Choice with Cointegrated Assets 21
2001 2002 2003 2004 2005 2006 2007 2008 2009 20100
50
100
150
200
250$
Stock Prices
2001 2002 2003 2004 2005 2006 2007 2008 2009 2010−0.4
−0.2
0
0.2
0.4The Residual
z = ln(MSFT) − 0.73 ln(IBM)
Fig. 2. Ten sample paths of simulated stock prices of Microsoft/IBM pair (top), and the associatedresidual (bottom).
2005 2006 2007 2008 2009 20100
20
40
60
80
100
Pu Test
2005 2006 2007 2008 2009 20100
50
100
150
Pz Test
2005 2006 2007 2008 2009 2010−0.75
−0.73
−0.7c
10% critical level
10% critical level
Fig. 3. Phillips-Ouliaris Pu and Pz cointegration tests, run for ten simulated sample paths, andthe estimated cointegration coefficient c.
January 24, 2011 14:58 WSPC/INSTRUCTION FILE PortfolioChoice-WithCoIntegratedAssets˙LaTex
22 B. Angoshtari
2005 2006 2007 2008 2009 2010
3
5.46
8
11
κ
2005 2006 2007 2008 2009 2010
0.03
0.0494
0.07
σ2
z
Fig. 4. Estimation of mean reversion rate κ and variance rate σ2zof the residual, for ten sample
paths.
2005 2006 2007 2008 2009 2010
0.2
0.22
0.240.25
0.27
0.29
σ1 and σ2
2005 2006 2007 2008 2009 2010
0.4
0.5
0.6
ρ
Fig. 5. Estimation of volatilities σ1 and σ2 and correlation ρ, for ten sample paths.
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Portfolio Choice with Cointegrated Assets 23
−10
−4.448
−2
01.332
3
δ1 and δ2
2005 2006 2007 2008 2009 2010
−0.2
−0.1
00.05
0.1
0.2
0.3
µ1 and µ2
Fig. 6. Estimation of δ and µ for 10 sample paths.
2005 2006 2007 2008 2009 2010
10
50100
1,000
10,000
100,000
1,000,000
$
Portfolio value (using real parameters)
Fig. 7. Portfolio value of the log-optimal strategy for ten sample paths, using the real values ofthe parameters.
January 24, 2011 14:58 WSPC/INSTRUCTION FILE PortfolioChoice-WithCoIntegratedAssets˙LaTex
24 B. Angoshtari
2005 2006 2007 2008 2009 2010
10
50100
1,000
10,000
100,000
1,000,000
$
Portfolio value (using out of sample estimates)
Fig. 8. Portfolio value of the log-optimal strategy for ten sample paths, using out of sampleestimates of the parameters.
2005 2006 2007 2008 2009 20100
20
40
60
80
Pu Test
2005 2006 2007 2008 2009 20100
20
40
60
80
Pz Test
2005 2006 2007 2008 2009 2010
−0.74
−0.72
−0.7c
10% critical leveltest statistic
10% critical leveltest statistic
Fig. 9. Phillips-Ouliaris Pu and Pz cointegration tests, run on a period from Jan 2005 to Dec 2009(at each day the last four years of data is considered), and the estimated cointegration coefficientc.
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Portfolio Choice with Cointegrated Assets 25
2005 2006 2007 2008 2009 20100
2
4
6
8
10
12κ
2005 2006 2007 2008 2009 20100.03
0.04
0.05
0.06
0.07
0.08σ2
z
Fig. 10. Estimation of mean reversion rate κ and variance rate σ2zof the residual.
2005 2006 2007 2008 2009 2010
0.2
0.25
0.3
0.35σ1 and σ2
2005 2006 2007 2008 2009 20100.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7ρ
σ1
σ2
Fig. 11. Estimation of volatilities σ1 (for MSFT) and σ2 (for IBM) and correlation ρ.
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26 B. Angoshtari
2005 2006 2007 2008 2009 2010−10
−8
−6
−4
−2
0
2δ1 and δ2
2005 2006 2007 2008 2009 2010−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25µ1 and µ2
δ1
δ2
MSFTIBM
Fig. 12. Estimation of δ and µ.
2005 2006 2007 2008 2009 2010
10
50100
1,000
10,000
100,000
1,000,000
$
Portfolio value
No Transaction costBid/Ask, biweekly tradeBid/Ask, daily trade
Fig. 13. Portfolio value of the log-optimal strategy for the MSFT/IBM pair, assuming that: a)there is no transaction cost (black line). b) buying with the daily high price and selling with thedaily low price (red line). c) same as (b), but trading biweekly.
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Portfolio Choice with Cointegrated Assets 27
8. Conclusions and future research direction
We considered the problem of optimal investment in a market with two cointegrated
risky assets, with the motivation of finding a theoretical ground for the so-called
pairs-trading strategies. For this, we formulated the classical Merton problem of
expected utility of terminal wealth and investigated whether this model supports,
in terms of optimal choice, pairs-trading strategies. We focused on the class of
homothetic utilities and found that such models does not support, in general, pairs
trading policies. Moreover, the optimal policies might have unpleasant properties
(blow ups for ’nirvana solutions’).
Aiming at remedying these deficiencies, we introduced an extra condition on
the market coefficients. This condition, which is one of the main contribution of the
paper, can be obtained, and interpreted, in three seemingly irrelevant ways. Firstly,
it is equivalent to the so-called Novikov condition which guarantees that the market
is arbitrage-free. Secondly, it is the necessary and sufficient condition under which
the optimal portfolios in the underlying Merton problem indeed justify pairs-trading
policies. Thirdly, this condition is, also, necessary and sufficient in order to exclude
nirvana solutions.
We showed that, the optimal pairs-trading strategies obtained by imposing this
condition, have intuitive properties and transparent structure, and can be inter-
preted easily. We concluded with numerical examples including both simulated and
real data.
In terms of future research directions, several interesting questions arise. Specifi-
cally, a theoretical question is whether this condition is also necessary for the market
to be arbitrage-free. In more practical directions, one might generalize the market
model, with possible extensions including, among others, allowing for n risky assets,
stochastic volatility, jump-diffusion stock prices, and regime-switching. As a more
challenging task, but very relevant in practice, one might incorporate transaction
costs.
Other possible research directions include the development of robust estimators
for the market parameters and optimal strategies, and statistical tests for the va-
lidity of the no-arbitrage condition. Developing these tools will make it possible to
conduct empirical studies in order to check the relevance of the results obtained
herein.
Appendix A. Proofs
A.1. Proof of Proposition 4.1
The following argument is a direct adaptation of the one used in [12]. From (4.1),
for any t ≤ T , we have
XαT = Xα
t exp(
∫ T
t
(
− 12αuσσ
⊤α⊤u + αu (µ+ δzu − 1r)
)
du+∫ T
tαuσdWu
)
.
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28 B. Angoshtari
Then,
V log (t, x, z) = E (lnXαT |Xα
t = x, zt = z)
= lnx+ E
(
∫ T
t
(
− 12αuσσ
⊤α⊤u + αu (µ+ δzu − 1r)
)
du|zt = z)
.
(A.1)
The integrand on the right-hand-side can be rearranged as the following quadratic
form
−1
2αuσσ
⊤α⊤u + αu (µ+ δzu − 1r) = −1
2
∥
∥σ⊤α⊤u − λu
∥
∥
2+
1
2‖λu‖2 ,
with λt = σ−1 (µ+ δzt − 1r). Maximizing the above integrand yields
σ⊤(
αlogt
)⊤− λt = 0.
It, then, follows that the log-optimal portfolio weights are given by equation (4.4).
Substituting back into (A.1) yields the value function as in (4.3).
Finally, the finiteness of the value function follows from Propositions 4.2. Note
that, for any constant γ ∈ (0, 1), we have that lnx < 11−γ
(
x1−γ − 1)
. Therefore,
V log (t, x, z) < V p (t, x, z) ,
where
V p (t, x, z) = E
(
1
1− γ
(
(XαT )
1−γ − 1)
|Xαt = x, zt = z
)
.
On the other hand, since γ0 < 1 (cf. (4.8)), one can always find a constant γ ∈ (γ0, 1)
such that, by Proposition 4.2, the corresponding power utility has a finite value
function. We easily conclude.
A.2. Proof of Propositions 4.2 and 4.3
The associated HJB equation for this stochastic control problem is
Vt + κ (z − z)Vz +12σ
2zVzz
+supα
12x
2ασσ⊤α⊤Vxx + xασσ⊤ (1, c)⊤Vxz
+xα (µ− 1r + δz)Vx = 0, for (t, x, z) ∈ [0, T )× R+ × R,
V (x, z, T ) = x1−γ
1−γ, for (x, z) ∈ R
+ × R.
(A.2)
The candidate optimal control is given in the feedback form by
α∗ (x, z, t) =
(µ− 1r + δz)⊤(
σσ⊤)−1(
− Vx
xVxx
)
+ (1, c)(
− Vxz
xVxx
)
.
(A.3)
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Portfolio Choice with Cointegrated Assets 29
Substituting in the HJB equation yields
Vt − 12
∥
∥σ−1 (µ− 1r + δz)∥
∥
2V 2x /Vxx
− 12σ
2zV
2xz/Vxx + (κz − (1, c) (µ− 1r))VxVxz/Vxx
+κ (z − z)Vz +12σ
2zVzz = 0,
V (x, z, T ) = x1−γ
1−γ.
(A.4)
To solve the above terminal value problem, we take the ansatz
V p (t, x, z) =x1−γ
1− γef(t)+g(t)z+ 1
2h(t)z2
,
for some appropriate functions f , g, and h. Substituting it back into (A.4) yields
that these functions must satisfy
h′ = −σ2z
γh2 + 2κ
γh− 1−γ
γ
∥
∥σ−1δ∥
∥
2,
h (T ) = 0,
(A.5)
g′ + σ2zh−κ
γg +
(
κz + 1−γγ
(1, c) (µ− 1r))
h
+ 1−γγ
δ⊤(
σσ⊤)−1(µ− 1r) = 0,
g (T ) = 0,
(A.6)
and
f ′ +
σ2z
2γ g2 +
(
κz + 1−γγ
(1, c) (µ− 1r))
g
+ 12σ
2zh+ 1−γ
2γ
∥
∥σ−1 (µ− 1r)∥
∥
2
= 0
f (T ) = 0.
(A.7)
We need the following preliminary result. Its proof is immediate and, thus,
omitted.
Lemma Appendix A.1. Consider the Riccati equation,
h′ (t) = a h2 (t) + b h (t) + c, for t ∈ [0, T ) ,
h (T ) = 0,
(A.8)
where a, b, and c are constants. Also, define the constant
∆ = b2 − 4ac. (A.9)
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30 B. Angoshtari
i) if ∆ ≥ 0, the solution of (A.8) is finite for T > 0 and given by
h (t) =
−2c
b+√∆coth
(√∆2 (T−t)
) , if ∆ > 0,
b−2a
(
1− 2b(T−t)+2
)
, if ∆ = 0.
(A.10)
ii) if ∆ < 0, the solution of (A.8) is finite if and only if
T < Tmax =2√−∆
(
arctan
(
b√−∆
)
+π
2
)
(A.11)
and given by
h (t) =
√−∆
2atan
(
arctan
(
b√−∆
)
−√−∆
2(T − t)
)
− b
2a. (A.12)
We continue with the proof Propositions 4.2 and 4.3. Note that ODE (A.5) is
the Riccati equation introduced in (A.8), with the constants a, b, c, and ∆ as in
(4.11) and (4.12).
We identify the following two cases, which correspond to Propositions 4.2 and
4.3, respectively.
a) γ ≥ γ0 (cf. Proposition 4.2): By (4.12), we have ∆ ≥ 0. Using (A.10) yields
the function h as in (4.13). With h at hand, (A.6) is a first order ODE for g,
and solving it results in (4.14). Equation (4.19) for f is merely (A.7) in the
integral form. Hence, the candidate for the value function is given by (4.9), and
substituting for the value function in (A.3) yields the corresponding control as
in (4.10).
Finally, by Lemma Appendix A.1 and for ∆ ≥ 0, the function h is finite for
all T > 0. Therefore, by (A.6) and (A.7), g and f are, also, finite. It follows
from (4.10) and (4.9) that αpt and V p are finite as well.
b) γ < γ0 (cf. Proposition 4.2): By (4.12), we have ∆ < 0. Using (A.12), the
function h is as in (4.22). In analogy with the previous case, ODE (A.6) gives
g as in (4.23). Moreover, f , αpt and V p are obtained by the same argument as
in the previous case.
Finally, by using Lemma Appendix A.1 for ∆ < 0, the function h is finite
if and only if (A.11) holds. This, in turn, yields Condition (4.20). As in the
previous case, the finiteness of the value function and the control follows.
To finish the proof, we need to verify that the candidate optimal control is indeed
optimal. For the related regularity and verification results we refer the reader to [3]
and [18].
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Portfolio Choice with Cointegrated Assets 31
A.3. Proof of Proposition 5.1
Part (i):
From (5.7), we have
E
(
e12
∫
T
0‖λs‖2ds
)
= e12T‖σ−1(µ−1r)‖2
E
(
e12
∫
T
0
(
‖σ−1δ‖2z2s+2(µ−1r)⊤(σσ⊤)
−1δzs
)
ds
)
.
(A.13)
Define
v (t, z) = E
(
e12
∫
T
t
(
‖σ−1δ‖2z2s+2(µ−1r)⊤(σσ⊤)−1
δzs
)
ds|zt = z
)
,
and recall that the process zt follows (2.6) and (2.7). Then, the Novikov condition
(5.6) becomes
E
(
e12
∫
T
0‖λs‖2ds
)
= e12T‖σ−1(µ−1r)‖2
v (0, z0) < ∞, ∀T ∈ (0,∞) . (A.14)
Hence, we only need to show that v (0, z) is finite for all z and T > 0. Using the
Feynman-Kac formula, we deduce that the function v satisfies
vt + κ (z − z) vz +12σ
2zvzz
+ 12
(
∥
∥σ−1δ∥
∥
2z2 + 2 (µ− 1r)
⊤ (σσ⊤)−1
δz)
v = 0,
v (T, z) = 1.
(A.15)
To solve the above terminal value problem, we use the ansatz v (t, z) =
ef(t)+g(t)z+ 12h(t)z
2
, for some appropriate functions f , g, and h. Substituting back
into (A.15) results in three ODE for these functions. The equations for f and g are
quite similar to (A.7) and (A.6) above. It can be easily shown that f and g are
finite if and only if h is finite. Therefore, one only needs to find conditions under
which h is finite for all T > 0.
The function h solves the Riccati equation (A.8), with the coefficients a = −σ2z ,
b = 2κ, and c = −∥
∥σ−1δ∥
∥
2. On the other hand,
∆ = 4(
κ2 − σ2z
∥
∥σ−1δ∥
∥
2)
,
with ∆ as in (A.9). Moreover, by (2.7), κ = − (1, c) δ = − (1, c)σ σ−1δ. Then, the
Cauchy–Schwarz inequality yields
|κ| ≤ ‖(1, c)σ‖∥
∥σ−1δ∥
∥ = σz
∥
∥σ−1δ∥
∥ , (A.16)
with equality if and only if there exists a constant η such that
σ−1δ = η ((1, c)σ)⊤ .
Therefore, ∆ ≤ 0, with ∆ = 0 if and only if Condition 5.1 holds. On the other hand,
by Lemma Appendix A.1, h is finite for all T > 0 if and only if ∆ ≥ 0. Hence, the
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32 B. Angoshtari
only possibility is that ∆ = 0, and we deduce that the Novikov condition holds if
and only if Condition 5.1 holds. This concludes the proof of part (i).
Part (ii):
The sufficiency follows from Theorems 5.1 and 5.2.
For the necessity, note that the results of section 4 supports pairs-trading, only
if, the relative-value policy αR.V,log, introduced in (4.6), is a market-neutral strategy
(cf. (3.3)). This, in turn, implies that there should be a constant η such that
δ⊤(
σσ⊤)−1= η (1, c) . (A.17)
It follows that
δ = η σσ⊤ (1, c)⊤, (A.18)
which is Condition 5.1, and we easily conclude.
Part (iii): If Condition 5.1 holds, then there cannot be any nirvana solutions,
since the value function (4.2) is finite for any utility function U (x) and all T > 0.
To see this, note (5.3) implies that in equation (4.8) we have γ0 = 0. Therefore,
Proposition 4.2 holds for γ = 0, i.e. supA
E (XαT ) < ∞ for all T > 0. Finally, for any
increasing and concave function U (x), Jensen’s inequality yields
supA
E (U (XαT )) ≤ sup
AU (E (Xα
T )) ≤ U
(
supA
E (XαT )
)
< ∞. (A.19)
Conversely, if Condition 5.1 fails, then nirvana solutions exist. To see this, note
that if (5.3) does not hold, then
0 < |κ| < σz
∥
∥σ−1δ∥
∥ (A.20)
(see (A.16)). Therefore, equation (4.8) yields that γ0 > 0, and Proposition 4.3 yields
nirvana solutions for any constant γ ∈ (0, γ0). This concludes the proof of part (iii).
A.4. Proof of Theorem 5.2
Condition (5.3) is equivalent to κ2 = σ2z
∥
∥σ−1δ∥
∥
2(cf. (A.16)). By (4.8) we, then,
deduce
γ0 = 1− κ2
σ2z ‖σ−1δ‖2
= 0. (A.21)
Therefore, for any power utility, we have that γ ≥ γ0. By setting
δ = − κ
σ2z
σσ⊤ (1, c)⊤
in Proposition 4.2, one would obtain Theorem 5.2. We easily conclude.
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Portfolio Choice with Cointegrated Assets 33
Acknowledgments
This work is part of the author’s D.Phil. dissertation under advising of T. Za-
riphopoulou. The author would like to thank her as well as T. Schoeneborn for
their invaluable suggestions and comments. Financial support from the Oxford-Man
Institute of Quantitative Finance is acknowledged.
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