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

mailto:[email protected]

January 24, 2011 14:58 WSPC/INSTRUCTION FILE PortfolioChoice-WithCoIntegratedAssets˙LaTex

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

[email protected]

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


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


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)


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



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.


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.



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].


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.



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].


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)




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)


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


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)



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.


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)


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)



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)


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)


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) .



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]).


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



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


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



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.


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.



−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.


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.



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.


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.



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

)

.


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)



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)


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].



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


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.



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.

References

[1] M. Avellaneda and J.-H. Lee, Statistical arbitrage in the US equities market, Quan-

titative Finance, 10(7) (2010) 761–78.[2] R. T. Baillie and T. Bollerslev, Common stochastic trends in a system of exchange

rates, The Journal of Finance 44(1) (1989) 167–181.[3] F. E. Benth and K. H. Karlsen, A note on Merton’s portfolio selection problem for

the Schwartz mean-reversion model, Stochastic Analysis and Applications 23 (2005)687–704.

[4] R. J. Brenner and K. F. Kroner, Arbitrage, cointegration, and testing the unbiased-ness hypothesis in financial markets, Journal of Financial and Quantitative Analysis

30(1) (1995) 23–42.[5] M. Cerchi and A. Havenner, Cointegration and stock prices: the random walk on Wall

Street revisited, Journal of Economic Dynamics and Control 12 (1988) 333–346.[6] J-C. Duan and S. R. Pliska, Option valuation with co-integrated asset prices, Journal

of Economic Dynamics and Control 28 (2004) 727–754.[7] D.S. Ehrman, The handbook of pairs trading, John Wiley & Sons 2006.[8] R. Engle and C. Granger, Co-integration and error correction, representation, esti-

mation and testing, Econometrica 55 (1987) 251–276.[9] T. Goll and J. Kallsen, A complete explicit solution to the log-optimal portfolio

problem, The Annals of Applied Probability 13.2 (2003) 774–799.[10] A. Hall, H. Anderson and C. Granger, A cointegration analysis of treasury bill yields,

Review of Economics and Statistics 74 (1992) 116–126.[11] J. D. Hamilton, Time Series Analysis, Princeton University Press 1994.[12] F. Jamshidian, On utility of wealth maximization, Unpublished note 2009.[13] S. Johansen, Likelihood-based inference in cointegrated vector autoregressive models,

Advanced Texts in Econometrics, Oxford University Press Oxford 1995.[14] J. W. Jurek and H. Yang, Dynamic portfolio selection in arbitrage, European Finance

Association 2006 meetings paper,Available at SSRN: http://ssrn.com/abstract=882536

[15] K. Juselius, The cointegrated VAR model: methodology and applications, AdvancedTexts in Econometrics, Oxford University Press Oxford 2006.

[16] I. Karatzas and C. Kardaras, The numeraire portfolio in semimartingale financialmodels, Finance Stoch. 11 (2007) 447–493.

[17] S. K. Kim and E. Omberg, Dynamic nonmyopic portfolio behavior, The Review of

Financial Studies 9(1) (1996) 141–161.[18] R. Korn and H. Kraft, A stochastic control approach to portfolio problem with

stochastic interest rates, SIAM J. Control Optim. 40(4) (2001) 1250–1269.[19] D. Kramkov and W. Schachermayer, The asymptotic elasticity of utility functions

and optimal investment in incomplete markets, The Annals of Applied Probability

9(3) (1999) 904–950.[20] J. Liu, Portfolio selection in stochastic environments, The Review of Financial Studies

20(1) (2007) 1–39.


34 B. Angoshtari

[21] R. MacDonald and M. Taylor, Metals prices, efficiency and cointegration: some ev-idence from the London metal exchange, Bulletin of Economic Research 40 (1988)235-239.

[22] J. Mossin, Optimal multiperiod portfolio policies, The Journal of Business 41(2)(1968) 215–229.

[23] S. Mudchanatongsuk, J. A. Primbs and W. Wong, Optimal pairs trading: a stochasticcontrol approach, Proceedings of the American Control Conference, Seattle, Washing-

ton (2008) 1035–1039.[24] P. C. B. Phillips and S. Ouliaris, Asymptotic properties of residual based tests for

cointegration, Econometrica 58 (1990) 165–193.[25] M. Schroder and C. Skiadas, Optimal consumption and portfolio selection with

stochastic differential utility, Journal of Economic Theory 89 (1999) 68–126.[26] E. S. Schwartz, The stochastic behaviour of commodity prices: Implications for val-

uation and hedging, The Journal of Finance 52(3) (1997) 923–973.[27] M. P. Taylor and I. Tonks, The internationalisation of stock markets and the abolition

of U.K. exchange control, The Review of Economics and Statistics 71(2) (1989) 332–336.

[28] G. Vidyamurthy, Pairs trading: quantitative methods and analysis, John Wiley &

Sons Inc. 2004.[29] M. Whistler, Trading pairs: capturing profits and hedging risk with statistical arbi-

trage strategies, John Wiley & Sons Inc. 2004.

Founded in 2007, the Oxford-Man Institute of Quantitative Finance is the home of interdisciplinary research in quantitative aspects of finance at the University of Oxford. Drawing researchers and students from many Departments in the University, it has particular strengths in computational finance, financial econometrics, hedge fund research, and quantitative aspects of law and finance. The Institute works closely with colleagues in the Finance Group at the Said Business School and the Mathematical and Computational Finance Group in the Mathematics Institute. The University teaching and research in finance is coordinated through our umbrella organisation Oxford Finance. The core funding for the Institute and its activities is provided by Man Group plc, which is a leading global provider of alternative investment products and solutions. In addition individual faculty and students at the Institute are supported by a variety of grants from research councils and charities. Further, the Institute benefits from the support companies provide to Oxford Finance.

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

mailto:[email protected]

Oxf-Man 2011OMI01 Angoshtari

Documents

Transcript of Oxf-Man 2011OMI01 Angoshtari