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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
HOLDING COSTS AND EQUILIBRIUM ARBITRAGE
BY
BRUCE TUCKMAN AND JEAN-LUC VILA
LATEST VERSION: JUNE 1992
WORKING PAPER NO: 3437-92-EFA
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
HOLDING COSTS AND EQUILIBRIUM ARBITRAGE
BY
BRUCE TUCKMAN AND JEAN-LUC VILA
LATEST VERSION: JUNE
WORKING PAPER NO:
1992
3437-92-EFA
AUG
2 7
\
HOLDING COSTS AND EQUILIBRIUM ARBITRAGE
Bruce
TUCKMAN
Stem School of Business, New York
University,
Jean-Luc
New
York,
NY 10006, USA
VILA
Sloan School of Management, Massachusetts Institute of Technology, Cambridge,
June, 1992
MA
02139,
USA
Abstract
In a world
were trading
If arbitrageurs face unit
is
costless, assets with identical
cash flows must have identical prices.
time costs, or holding costs, the prices of these assets need not be equal,
i.e
the assets can be relatively mispriced. This paper constructs a dynamic model of the equilibrium
determination of prices under costly arbitrage.
(i)
Our
analysis reveals that:
Mispricing and arbitrage can exist in a market equilibrium.
(ii)
(iii)
Riskless arbitrage arguments
may not provide
tight
bounds around observed market
Arbitrage activity reduces equilibrium mispricing and
liquidity
shocks are transient and conditionally
is
particularly effective
prices.
when
volatile.
We would like to thank John Heaton for helpful comments and suggestions. Also
Jean-Luc Vila wishes to acknowledge flnancial support from the International Financial Services
Research Center at the Sloan School of Management.
Acknowledgments
:
1
1.
Introduction
The notion of
arbitrage
is
extremely useful in markets without
frictions.
Researchers define
arbitrage as a set of transactions which costs nothing and yet risklessly provides positive cash flows.
They then assume
that arbitrage opportunities never exist
and derive precise implications about
relative prices.
The presence of market
frictions,
however, reduces the usefulness of arbitrage arguments. In
when market
the simplest of models, trading costs lower arbitrage profits. Therefore,
admit
riskless profit opportunities inclusive
prices
do admit such opportunities, arbitrageurs stake
will
eventually
come back
of trading costs, arbitrageurs do nothing.
all
they have
prices
do not
When
market
on the sure proposition
that prices
into line.
Richer models of arbitrage
activity
recognize that arbitrageurs
may
take positions even
simple, buy-and-hold arbitrage strategy does not furnish riskless profit opportunities. In
model,'
Tuckman and
if
the
one such
Vila (1992a) constrain market mispricings so that they never violate the riskless
arbitrage bounds. Nevertheless, risk averse arbitrageurs facing unit-time costs, or holding costs, take
a finite, risky position
if
and only
if
mispricings are large enough. This result raises the possibility that
arbitrageurs routinely find and take advantage of market mispricings,
i.e.
arbitrage
may
exist in a
market equilibrium.
This paper embeds the
in
Tuckman and
Vila (1992a) model in a dynamic, equilibrium setting
order to explore the effects of market frictions and arbitrage activity on equilibrium prices. The
model assumes that there are two
Each bond
issue trades in
respects, they are
its
segmented
distinct
own
bond
issues
which generate identical cash flows over time.
market. Furthermore, while the two markets are identical in
in that investors in
these assumptions the two bonds would
sell for
one market cannot trade
the same price, adding
Brennan and Schwartz (1990) and Hodges and Neuberger (1989) are other examples.
in the other.
random
all
While under
liquidity
shocks to
2
each market drives the prices of the bonds apart.
Finally, risk-averse arbitrageurs facing
holding costs
trade across the two markets to take advantage of any price deviations which do occur.
Many
lessons
emerge from the equilibrium
prices determined through this model. First,
because arbitrageurs do not take large enough positions to bring prices immediately back into
line,
mispricings and arbitrage activity are consistent with market equilibrium. Second, because arbitrageurs
take positions even
when no
riskless profit opportunities are available, equilibrium prices are
well within the riskless arbitrage bounds. This finding implies that models based
arguments, like
many
in the
option replication literature,'
observed prices. Third, arbitrage
activity
may not provide
on
riskless arbitrage
usefully tight
bounds on
reduces both the magnitude and unconditional variance of
the mispricing process.' Fourth, arbitrageurs are most effective in eliminating mispricings
shocks are transient and conditionally
liquidity
often
volatile.
This
last
when
the
finding serves as a microeconomic
foundation for the recent literature on the connection between investor impatience and market
mispricings.
De Long et al.
(1990) and Lee
et al.
{\99\) have hypothesized that investors' short term
horizons allow for persistent deviations from fundamental values. Shleifer and Vishny (1990) argue
that
when
mispricings increase in asset maturity, risk averse corporate managers will pursue short
term objectives. The present analysis reveals that the nature of holding costs cause arbitrageurs to
react
most forcefully to mispricings stemming from causes which tend to disappear
relatively quickly.
This paper contributes not only to the arbitrage pricing literature, but also to the growing
literature
costs,^
on imperfect
While the impact of
frictions
costs,^
^Scc Consiantinides (1986), Davis and
aL (1989),
Tuckman and
is
holding
relatively
Leiand (1985) and Boyle and Votst (1992).
See Grossman and Miller (1988), FremauU (1991) and Holden (1990) for similar
'Sec
such as trading
borrowing constraints,' or market incompleteness' on dynamic investment strategy
^Sec, for atampic,
^
financial markets.
Vila (1992).
Norman
results in a sutic
framework.
(1990), Davis, Panas and Zariphopoulou (1991), Duffie and
Sun (1989), Fleming
a
3
well understood,
little is
known about
the impact of these frictions
While some recent papers have addressed
this question,
on equilibrium
price processes.
they focus on trading costs rather than
holding costs/
Finally, this
paper has important welfare implications. Because most investors face
relatively
high transaction costs, their activities alone cannot ensure that marginal rates of return are equated
across markets.
While arbitrageurs are better suited
Therefore,
paper's analysis can be viewed as a study which relates a
this
to the ineffiencies
Section
I
it
11
describes holding costs and
many
The
common market
how
they affect arbitrage
activity.
The
first
The second subsection develops the equilibrium model without
example which explores the model's implications. Section
model
in the
presence of arbitrageurs.
*See Grossman and Vila (1992), Vila and Zariphopoulou (1990),
IV
presents a numerical
V concludes and suggests avenues for future
research.
(1991 a and
argues that
subsection discusses the essential elements of any
third subsection develops the equilibrium
He and Pearson
imperfection
activity. It also
Section HI demonstrates the solution method for the model. Section
^See
frictions as well.
different arbitrage contexts.
describes the model.
model of equilibrium arbitrage
arbitrageurs.
some
generates.
holding costs are important in
Section
for this function, they face
He and
Peareon (1991 a and
b).
'See Ayagari and Gertler (1991), Heaton and Lucas (1992) and Vayanos and Vila (1992).
b).
2.
Holding Costs and Risky Arbitrage
Markets provide many examples of
Some common
distinct portfolios
which generate identical cash
flows.
instances are a forward or futures contract vs. a levered position in the spot asset, a
bond denominated
in
one currency vs.
forward contract, and a coupon bond
a
bond denominated
vs. a
another currency plus a cross-currency
in
cash-flow matched portfolio of other coupon bonds.
Qansider two such portfolios, one "red" and one "green." Although they generate identical
cash flows, assume that, for
amount A. Assuming
that
some
reason, the market prices of the portfolios, Pr and Pq, differ by an
Pr > Pg, arbitrageurs facing no
purchase the green portfolio, realizing a
riskless profit
In reality, short-sale agreements are
arbitrageur must
will
demand
arranged
borrow the
securities
frictions will short the red portfolio
of A.
more complex.
from someone
In order to short the red portfolio, an
willing to lend them.
Furthermore, the lender
collateral in order to protect his position. Therefore, the arbitrage transaction will
in the following
manner: lend Pr to the holder of the securities
equivalently, post an interest-bearing security
worth
be
in the red portfolio (or,
Pr), take the securities as collateral
and
sell
them
purchase the green portfolio, and borrow P^. This set of transactions generates no cash flows
for Pr,
today, but assures the realization of the future value of
If
markets were
arbitrageurs
change
would take
this story in
frictionless, the equilibrium
infinite quantities
two important ways.
a risky investment opportunity
A upon
value of
closing the position.
A must
equal
0.
Were
maximum
out riskless arbitrage requires only that
when confronted with smaller values of A
not vanish quickly enough, the total holding costs will
Holding costs appear
in
many
not
so,
A be
less
possible holding costs. Second, arbitrageurs face
:
if
A
vanishes quickly enough,
the costs of the arbitrage will be small and the transaction will have proved profitable.
A does
it
of the arbitrage position described above. Holding costs
First, ruling
than or equal to the present value of the
hand,
and
arbitrage contexts.
First,
swamp
If,
on the other
the realized gains.
shorting any spot security or
5
commodity
least
will
often result in unit time costs since arbitrageurs must usually sacrifice the use of at
some of the
large client bases
holding costs
short sale proceeds.
It is
true that, in the case of stocks, brokerage houses with
assume short positions almost
when short-selling stocks. Second,
since part of margin deposits
often charge a per
annum
may not earn
costlessly. Nevertheless, smaller arbitrageurs
futures market positions
interest. Third,
may generate
banks making markets
rate over the life of the contract. Fourth,
when
in
do incur
unit time costs
forward contracts
collateral requirements
cause deviations from desired investment strategies, these requirements are essentially generating unit
time
3.
costs.'
The Model
Preliminaries
3.1.
Any model of equilibrium
arbitrage must 1) posit the existence of two distinct assets which
generate identical cash flows, 2) assume some forces which cause the prices of these assets to
and 3)
restrict investor
differ,
and arbitrageur behavior so that price differences do not vanish as soon
as
they appear.
As
discussed in the previous section, there are
many examples of
distinct assets
which
generate identical cash flows. This model assumes that there exist two distinct bond issues with
identical
coupons and
maturities.
For ease of exposition, one of the
while the other will consist of "green" bonds. Not
in
much
effort
issues will consist of "red" bonds,
would be required to
recast the
model
terms of the other mentioned examples.
The
prices of the red
and green bonds may tend to
differ for a
number of
reasons. For
A
See Tuckman and Vila (1992) for references supporting these
bond market.
the magnitude of holding cxsts in the U.S. Treasury
institutional details.
Also see that paper for a detailed description of
6
example, the bonds might be traded by different clienteles with different valuation
of
this sort
may have developed
for historical reasons or
may
exist for institutional reasons.
reason for price differences across these markets might be temporary supply and
due
rules. Clienteles
to microstructure imperfections. In any case, because this
paper aims
Another
demand imbalances
at explaining the process
which constrains these price differences, no serious effort has been made to model the source of price
differences. Instead, the
markets with buy and
model assumes that noise traders occasionally shock the red and green bond
assumed that the demand
that these shocks affect
bond
for the individual
Because close substitutes
demand
To ensure
sell orders.
exist for
for individual financial assets
is
issues
most financial
same
to a
will
buy
assets, the
it is
furthermore
elastic.
more usual assumption
become important. For example,
asset differently, thus generating a
curve. Similarly, to the extent that an asset
low transaction costs
not perfectly
prices,
is
that the
perfectly elastic. But, in the presence of market frictions,
considerations other than the existence of substitutes
different tax brackets value the
is
market
is
at higher prices
investors in
downward-sloping demand
purchased after the sale of other
assets, investors
with
than investors with high transaction costs, again leading
downward-sloping demand curve. This model employs the tax motivation because, as shown
below, the functional form of the
sloping
demand curve
demand curve can be
will translate liquidity
Finally, price differences
segmentation of the markets;
if
And
these migrations
any downward-
shocks into price differences across markets.
must not vanish as they appear. This certainly requires some
one bond market can
investors in
price differences will result in migrations
market.
easily derived. But, note that
from the
will, in turn,
relatively
easily
purchase bonds
in the other,
dear market to the relatively cheap
equalize the red and green
bond
prices.
While market
segmentation seems a reasonable assumption from the point of view of many investors, arbitrageurs
can usually trade across markets. Nevertheless, arbitrageur activity might not be sufficient to force
prices
back into
line.
Possible assumptions which limit arbitrageur positions are exogenous position
7
limits (see
Brennan and Schwartz
Holden
(1990)), oligopolistic behavior (see
(1990)), or non-
synchronous trading across the two markets (see Fremault (1990) and (1991)). Here, arbitrageurs are
assumed to be risk-averse and
to face holding costs
when
As
shorting bonds.
in
Tuckman and
(1992a), these assumptions imply that optimal arbitrage positions are not necessarily large
Vila
enough
to
eliminate price differences across markets.
3.2.
Equilibrium pricing without arbitrageurs
Turning to the model of this paper, begin by assuming that there are two
one red and one green, trading
and
entitle holders to
$d
at the
coupon payments are taxable
Finally, investors
in
two
Both bond
distinct markets.
issues
who buy bonds
own
to
$l+d
issues,
tax rates, but there are
no
at maturity.
The
capital gains taxes.
plan to hold them until maturity.'* Then, letting
r
discount rate for after-tax cash flows, the value of a bond to an investor with tax rate
market,
bond
mature n periods from now
end of each period before maturity and
to investors at their
distinct
denote the
t, in
either
is
V(x)=lilll2[l-—i_].-J_
(l*r)"'
'
(1-r)"
^^^
=
For a given bond
price, ?„,
V(0)
-
2i[l-__L_]
an investor with tax rate t
.
will
want
to
buy bonds
if
P„
<
V„(t).
Solving (1) for t shows that investors with tax rate t will want to buy bonds so long as
This assumes that investors are myopic
price to their
a sale
in the
the analogy
own
expectation that prices will
is
Tuckman and
in the following sense.
When
valuation under a buy and hold strategy. But this strategy
misleading: an investor
rise.
who
is
sell
While one might be tempted to think of the decision to
sells, i.e.
exercises, can repurchase the
show that the myopic strategy is sometimes optimal.
context of the present model will be left as a subject for future research.
Vila (1992b)
deciding to buy or
a bond, they
compare the market
not necessarily optimal, since investors
bond
In
sell in
may
prefer to delay
terms of exercising an option,
later and sell yet again. In fact, in a related context,
any case, careful modeling of investor decisions in the
^ r V(Q)-Pn
^ d
1
,_
(l*rr
,
'
To
the
derive a
number of
given by ot.
demand curve
for the bonds,
investors with tax rates
From
assume
below some
these assumptions and (2), the
t in
(2)
•
that
investors
i)
buy
at
most
1
bond and
ii)
each of the red and green bond markets
demand
function in each market, D„(P„),
is
is
given
by
D
(P
)
=
ar
f^
V(0)-P
-±1
1
.,s
.
'(l*r)-
Assume
that the quantity of each
bond
noise traders, the market price in each
issue available for investor trading
bond market would be the P„ which
is
Q. In the absence of
solves D„(P„)
presence of noise traders, however, can cause the prices of the red and green bonds to
Noise traders enter the markets to
sell
=
Q. The
differ.
or to buy. Let Lr,„ and Lq „ be the cumulative amount
of noise trading in the red and green bond markets, respectively. By convention, positive quantities
denote supply shocks while negative quantities denote demand shocks.
there are Lr_„-Lr^„+i sellers in the market for red bonds.
Lr_„^,-Lr_„ buyers.
As
in
other papers,''
no
explicit
If,
If,
for instance, Lr.„>Lr^o+i
on the other hand,
LR.n<LR,n+i, there are
model of the sources of noise trading
will
be
presented. Finally, note that noise trading shocks affect the quantity of bonds available for investor
trading: the supply of red
shock
is
Q
+
bonds
after the
shock
is
Q
+
Lr^,
while the supply of green bonds after the
Lo„.
In the absence of arbitrageurs, the equilibrium prices in the red
Po.n,
respectively, are
"Sec
and green markets,
determined by the following equations:
for instance, Kyle (1985),
DeLong
et. al.
(1990), Fremault (1990 and 1991) and
Holden (1990).
Pr_„
and
ar
V (0)
-J
^_
- P^„
/
^
-O^W.
^
and
(43)
'(1-rr
T
^_
^
1
^
^«-
•
(4b)
Solving for the prices,
PR.n
= V„(0)-e„(Q + L^J
(5a)
PG.„
= V„(0)-e„(Q +
(5b)
LG,„)
where
6„ =
"
Of particular
bonds. Letting A„
=
interest
Prj,
-
is
A[1-_L_].
ar
'
the difference between the price of red bonds and the price of green
and using the pricing equations
Pg,„
(6)
(l+r)"
A„
(5),
the relative mispricing equals
= e„L„
(7)
with
Equilibrium pricing with arbitrageurs
3.3.
Arbitrageurs can
now be
introduced into the model. Let x„ be the number of green bonds
bought by arbitrageurs and the number of red bonds sold by arbitrageurs. The optimal choice of
will
An
be discussed below. For now, consider the effect of arbitrage on the mispricing
> 0,
i.e. if
positive.
P^o
>
Pon, arbitrageurs will
Furthermore,
x„ will
want
to sell red
A„. If L„
bonds and buy green bonds, so
>
x„
and
x^ will
be
be added to the supply of red bonds and added to the demand of green
bonds. Adjusting equations (5a) and (5b) accordingly and subtracting (5b) from (5a) to obtain A„ for
this case,
A„
=
e„(L„-2x„). Notice that arbitrage
activity lowers the relative mispricing.
10
If
<0 and A„ <
L„
0, i.e. P|^„
<
want to buy red bonds and
Pg„, arbitrageurs will
be added
bonds, so x„ will be negative. Furthermore,
x„ will
supply of green bonds. In
= 8„(L„-2x„) and
case also A„
this
to the
demand
for red
sell
green
bonds and to the
arbitrage activity reduces the relative
mispricing.
Summarizing
this discussion, adjusting
supply and
demand
in the
bond markets
to account for
arbitrageur activity changes the mispricing A^ from (7) to
A„
By
model
definition, x„
is
the
sum of
consists of a strategy {x„}
strategy given the evolution of
=
8„(L„-2x„),
(8)
positions across arbitrageurs. Therefore, an equilibrium in this
and a process
{Aj, and
{
A„} such that
1)
each arbitrageur chooses an optimal
2) the resulting {x„}, in turn, generates the process {An}
given by (8).
For simplicity, assume that
they maximize their expected
utilities
exists a representative arbitrageur
terminal wealth.
To
all
arbitrageurs have negative exponential utility functions
and that
of wealth as of the date the bonds mature. In that case, there
with negative exponential
solve for equilibrium prices, then,
utility
who maximizes expected
utility
of
one must solve the investment problem of the
representative arbitrageur.
Begin with the value of an arbitrage position from one period to the next. Assume for the
moment
that
A„
>
0, i.e.
Pn„
>
Pg,,.
As
discussed in the previous section, the arbitrageur will buy x„
green bonds, borrow x„PG.n dollars, short
position
is
c
is
red bonds, and lend x„Pr,„ dollars,
x„
>
0.
Next period the
worth
x„PG,n.i-x„PR.„.,
where
x„
+ x„PR.„(l+r)-x„PG.„(l+r)-cix,| =x„[A„(l+r)-A„.,]-cK|,
the holding cost incurred for maintaining a short position over the period.
If
(9)
A„ <
0,
the
expression does not change, but x„ will be negative: the arbitrage position entails buying green bonds
and shorting red bonds.
11
Let
W„ be the wealth of the representative arbitrageur following the strategy
are n periods to maturity.
From
the above discussion, his wealth one period
later,
{x^}
W„.„
when
there
is
= W„(l+r)+x.[A„(l + r)-A„..]-c|x„|.
W„.,
DeGning w„ = W„(l+r)°,
8„
=
= w„ +
w„.,
=
A„(l+r)", and c„
x„
[
5„-
5„.,
]
(10)
c(l+r)°, (10) can be rewritten as
-c„.,lx„|
(11)
.
Equation (11) and the objective, discussed above, to maximize the expected value of
-exp(-Awo),
given
A>0, completes
the specification of the representative arbitrageur's investment problem
{AJ.
The model can be completed by
liquidity shocks, {L„}.
For
simplicity
value with n periods to maturity
7r(n,L„)
3.4.
and a value
L„.,
=
L„
-
is
it
L„,
will
specifying an exogenous stochastic process for the net
be assumed that {L^} evolves as a binomial process:
then
it
will
take
on the value
L„.,
=
L„
+ u
if its
with probability
u with probability l-ji(n,L„).
Model Solution
This section begins by solving the representative arbitrageur's investment problem. Let
V(w„,L„,n)
= max
maturity and the
E„[-exp(-Awo)] where E„ denotes the expectation
maximum
is
over the strategy
{x„}.
By the
when
there are n periods to
principle of optimality in dynamic
programming,
V(w
,L ,n) = max
{ir(n,L)V(w+xr5(L)-5 ,(L+u)l-c ,|x|,L
+u,n-l) +
(l-7r(n,L))V(w„.xJ5„(L)-V,(L-u)]-c„JxJ,L-u,n-l)}
Also, because the mispricing must vanish at the maturity date, the
is
V(wo,L^O)
=
initial
condition of the problem
-exp(-Awo).
Because of the special form of the
utility
function, V(w„,L„,n)
is
separable in wealth and the
12
and can be written
arbitrage opportunity
as exp{-Aw„}J(L„,n).
Using
this fact, (12)
J(L,n) =max{jr(n,L)e"^""'*-"-'"*"^'^*"""^-'"""j(L +u,n-l)
becomes
*
(l-«(n,L))e-^"'"'*"'^'-*"^"--""-'"^"-"j(L -u,n-l)}
with
condition J(Lo,0)
initial
To
=
solve for the optimal strategy as a function of 5„, begin as follows. If S„
increases with a
move
to fi„.,(Ln+u) while the mispricing decreases with a
the per period arbitrage profits
+ c„.„
of
the position
i.e. if
8„, x„
= 0. On
profitable
is
-1.
is
is
x„ (8„-ftn-i-Cn-i). the position will
not profitable even
the other hand,
even when the
fi„
<
when
is
falls.
the mispricing
to 6„.,(Ln-u). Since
never be profitable
the relative mispricing
+ c„.,
5„.,(L„+u)
move
> 0,
if fi„
<
ft„.i(L„-u)
So, for these value
inconsistent with equilibrium;
if
the position
relative mispricing rises, prices furnish a risldess arbitrage inclusive of
holding costs and the optimal x„ would equal +». In the intermediate range,
8„.,(L„+u)
Xn
5„
>
fi„.,(L„-u)
+ c„.,
can be found by solving the optimization problem (13) to obtain
^
Ln
=
X
A[(5
,(L +u)-6 ,(L -u)]
where (H)* = Max {H,
If fin
8„
+ c„., >
<
<
0, similar
J(k-".»-n
^'^^"'^"^ 'Sn(k)-'S„-l(k-")-Cn-»
{
(14)
J
7r(n,L
,+6 ,(L +u)-6 (L
c
)
J(L +u,n-l)
)
0}.
arguments reveal that x„=0 when
fi„
>
fin.,(Ln+u) -Co., while x„
=
-«>
when
5^,(L^,-u)-c^,. In the intermediate range,
>
fi^,(L„+u)-c„.,
the optimal x„
is
>
fi„.,(L„-u)-c„.,
given by
1
l-7r(n,L)
A(6 n-l,(L
+u)-6
,(L
*
'
'J
n
n-l^ n -u)l
7r(n,L
^
n')
I
where
fi„
'
5
(L )-5 ,(L -u)+c
,
J(L -u,n-l)
,
(\s\
(L ) J(L +u,n-l)'
-c n-1 +6 n-1^
,(Ln +u)-(5
n^n'^'n
'
(H)=Min{H;0}
To
solve for the equilibrium values of x„ and
fi„,
rewrite the equilibrium condition (8) in terms
13
of5„:
8„
= e„(l+rr(L„-2x„).
Then, solve (16) simultaneously with (14) or
illustrates the
The dotted
solution
(16)
(15), as appropriate.
For the case 5„ >
0, Figure
optimal position size and the equilibrium condition as a function of the mispricing
line represents the mispricing which generates infinite arbitrage activity.
1
b„.
The simultaneous
given by the intersection of the two functions.
is
INSERT
Given the solution technique
provide the solution for
all n.
The
for any
initial
HGURE
1
n given the values
at n-1,
backward induction
will
condition of the problem gives J(-,0). This allows for the
solution of X, and 5, along the lines described above. Then, substituting these values into (13) yields
J(-,l).
Proceeding
in this fashion
produces the entire mispricing process and the accompanying
arbitrage strategy.
4.
A
Numerical Ejtample
In order to illustrate the insights of the model, this section presents the solution of the
with the following parameter values:
d
= 8%
(annual bond coupon rate)
r
= 8%
(annual after-tax discount factor)
=
a
c
100 (number of investors
= .5%
in
each market)
(annual holding cost)
A=
.0001 (representative arbitrageur's coefficient of absolute risk aversion)
T=
5 years (maturity of the
Lr
=
(initial
bond
as of the starting date)
net liquidity shock)
Next, adjust the parameters to allow for different trading increments:
h
=
length of the trading period
model
14
N = ^ = number
=
(l+r)''-l
dh
=
d tJt
=
per period coupon rate'^
c^
=
c rjr
=
per period holding cost.
Fh
The
discretization
=
of periods to maturity as of starting date
per period interest rate
above preserves the present value of the coupon and of the holding cost
in the
sense that
l/h
^
d
(l^Tj
1-r
l/h
and
p
E
{l^Tj
t-l
1-r
Finally, define the stochastic evolution
y^
= o
Uj_
jr(n,L
)
of the liquidity shock, L„, by setting
and
=max
{
min{ Ici-P^ryiT); -1
The
process L„
dL,
where
db;
=
is
a
-
is
a discretized version of the
pLjdt
+
}
;
1
}
Omstein-Uhlenbeck process
odb,
Brownian Motion. In
particular,
it
can be shown that the conditional expectation of
the incremental liquidity shock (L„.,-L„) equals -p L„h+o(h) while
o'h+o(h)." For the analysis that follows
Uhlenbeck process has
Hence
it
is
useful to recall that,
a stationary distribution which
is
the unconditional volatility of the liquidity shock
standard deviation o and the persistence measure 1/p.
sources of unconditional
rhis
assume
that
p>0.
;
o
2
its
if
p
is
positive, the
normal with mean
comes from two
As shown below,
volatility.
coupons are paid each period.
'^For convergence results on this discretization scheme see Nelson and
conditional variance equals
Ramaswamy
(1990).
Omstein-
and variance o^/2p.
sources, the conditional
arbitrage activity affects both
15
The
rest
of the bonds.
its
effects
of
The
this section
first
is
devoted to the effects of arbitrage
on the
relative mispricing
subsection presents basic properties of the optimal arbitrage strategy x„ and
on the mispricing
process.
dynamics of the mispricing process,
4.1.
activity
The second
i.e.
subsection analyzes the effects of arbitrage
on the conditional moments of the
on the
mispricing.
Basic results
For the purpose of
this subsection,
net liquidity shock L„ follows a
equals
4%
it is
sufficient to consider the case
random walk, namely p =0.
Also, set h
=
1
where the cumulative
day o
= 4.
(This volatility
of the investor population.)
Figure 2 shows the optimal position size taken by arbitrageurs as a function of the mispricing
when
the bonds mature in two years. For relatively small levels of mispricing, arbitrageurs
do not take
any position: the potential profits are not large enough relative to the potential accumulation of
holding costs. For larger levels of mispricing, optimal position size increases with the mispricing.
reader can consult
Tuckman and
Vila (1992a) for further discussion
The
on the properties of optimal
arbitrage positions.
INSERT
HGURE 2
Figure 3 shows the equilibrium mispricing with and without arbitrageurs as a function of the
net liquidity shock. Again, the bonds have two years
INSERT
For
left to
maturity.
HGURE 3
relatively small net liquidity shocks, in absolute value, the difference
between the red and
green bond prices increases with the absolute value of the net liquidity shock. Furthermore, the
mispricing
is
the
same whether
because, as seen from figure
2,
arbitrageurs exist or not. Arbitrageurs
they do not take positions
when
make no
the mispricing
is
difference here
relatively small.
16
For larger net
liquidity shocks, in absolute value, the mispricing
absolute value of the net liquidity shock, but arbitrage activity
makes
continues to increase with the
a difference. Arbitrage activity
lowers the mispricing substantially below that which would exist were there no arbitrageurs. Note that
the mispricing with arbitrageurs flattens out at the
arbitrage opportunities. This
maximum
mispricing consistent with no riskless
bound equals the present value of the holding
cost incurred by
maintaining an arbitrage position until maturity.
Two
is
lessons
emerge from
figure 3. First,
when
the equilibrium mispricing without arbitrageurs
not zero, the equilibrium mispricing with arbitrageurs
is
also not zero.
Due
to holding costs
and
risk
aversion, arbitrageurs bring prices closer into line but never eliminate mispricing altogether. Second,
arbitrageurs often reduce mispricing to a level below that which
transactions. Therefore,
riskless arbitrage
Table
bounds
market prices
will
will
commonly
trigger riskless arbitrage
reflect these smaller mispricings
and the no-
not be binding.
summarizes the effects of arbitrage
1
would
activity
on
relative mispricings. Conditional
on
having five years to maturity, one can calculate the expected absolute mispricing and the standard
deviation of the mispricing at the two year maturity mark.
maximum
absolute price deviation consistent with no-riskless arbitrage
As reported
is
With two years
in table
only .0055. In
riskless arbitrage
1,
fact,
is
left to
maturity, the
.0089 per dollar face value.
however, the average absolute price deviation in the presence of arbitrageurs
the
mean
absolute price deviation without arbitrageurs
is
also
below the no-
bound. In short, the no-riskless arbitrage condition may not be very useful
describing prices in markets with frictions.
INSERT TABLE
1
in
17
4.2.
Dynamics of the mispricing process
This subsection studies the effects of arbitrageurs on the dynamics of the mispricing process.
Define the process Z„ as
Z„ = L„
Note
that
Z„ represents the
arbitrageurs. Since
processes L„ and Z„
This comparison
is
=
A„
is
liquidity
2x„.
-
(17)
imbalance between the two markets
=
9„L„ without arbitrageurs and A„
in the
presence of
6„Z„ with arbitrageurs, comparing the
equivalent to comparing the mispricing process with and without arbitrageurs.
easier than studying the mispricing directly because L„ has very simple conditional
moments:
E
These moments reveal
(dL„|L„)
=
-pL„dt and Var (dL„|L„)
that the effect of arbitrage activity can
dZ
Var
E[-.^|LJ
Z dt
[
o^dt.
be seen by comparing
dZ
I
L
]
and
'
'
=
"
dt
N
with p and o respectively.
First set the
parameter values h
=
1 day,
p
=
mean
remaining maturity of two years, figure 4 graphs the
E[^
and o
.5
dZ
1|L
Z dt
'
]
^
=
4.
For the Gve year bond with a
reversion as a function of L„,
i.e
.
"
n
INSERT
For large absolute values of
L„, the equilibrium mispricing
bounds, as shown earlier in figure
of L„. Consequently, Z„
is
Z„=L„ and mean
3. Similarly,
A„ approaches the
mean
reversion
for small absolute values of L„, arbitrageurs
reversion equals p=.5.
riskless arbitrage
X„ approaches some bound for large absolute values
almost constant in these regions and
At the other extreme,
so
HGURE 4
is
very small.
do not take any
positions,
18
As L„ approaches
in the
the value at which arbitrageurs will take a position, there
second derivative of the
size of the arbitrage position. This kink causes
is
mean
a discontinuity
reversion to be
very large.
To summarize
the insights provided by figure
4,
when
mispricings are very small, the presence
of arbitrageurs has no effect on the rate at which mispricings vanish. For larger, but relatively small
mispricings, arbitrage activity increases the rate at
which mispricings vanish.
Finally, for relatively large
which mispricings vanish.
mispricings, arbitrage activity reduces the rate at
Figure 5 plots the function
Var [dZ IL
1
>
dt
INSERT
HGURE 5
Notice that arbitrage reduces the conditional variance of the mispricing process. Since A^ approaches
when
the risldess arbitrage bounds
Ln
large in absolute value, the conditional variance of the
is
mispricing and of Z^ must be particularly low. In other words, the reduction of conditional variance
due
to the activity of arbitrageurs
The
discussion
now
is
most pronounced
at high levels
turns to the average values of the conditional
values of p and o. Mathematically, define
E
p'
and
-
[
n I
dZ
:.
L
z dt
setting h
=
1
]
I
"
day and o
I
(18a)
L =0
(18b)
]
L.
=0
dt
= 4,
shows
p'
and
INSERT TABLE
As can be seen from
for different
'^
E.,
N
2,
moments
o' as
Var [dZ IL
=
a'
Table
of mispricings.
this table, arbitrageurs
a' for different
values of p.
2
cause the liquidity imbalance process Z„ to be
less
19
conditionally volatile and
a'
does not change
p.
Hence
if
much
liquidity
Table
3,
more mean
with
p,
it
reverting than the cumulative liquidity shock process L„. While
can be seen that
shocks are transitory, arbitrage
setting h
=
= t
day and p
1
,
p'
can be very sensitive to increasing values of
will
presents
eliminate
p'
INSERT TABLE
Table 3 reinforces the results
reduces average conditional
signiflcant only
when
in table 2,
volatility.
liquidity
and
them
quickly.
a' for different
values of o.
3
namely, that arbitrage increases average
Table 3 adds the insight that
mean
reversion and
be economically
this effect will
shocks are sufficiently volatile.
Tables 2 and 3 show that arbitrageurs facing holding costs are particularly effective
if
liquidity
shocks are conditionally volatile and transient. Arbitrageurs like conditionally volatile markets since
this volatility creates potential arbitrage profits. Also, arbitrageurs like
mean
reversion because of the
nature of holding costs: transient mispricings allow for profit realization before holding costs have a
chance to accumulate.
V. Conclusion
Many
financial
models assume that markets are
opportunities do not exist.
frictions,
They then derive
however, the assumption that
frictionless
and that
riskless
arbitrage
precise, relative pricing implications. In the presence of
riskless arbitrage opportunities
do not
exist only
bounds
market mispricings. This paper suggests a research agenda which seeks to find the equilibrium
mispricing in a market with costly arbitrage.
The model of
equilibrium arbitrage developed here assumes that noise trading in two
segmented markets causes price deviations from fundamental values. Risk-averse arbitrageurs facing
holding costs reduce these deviations, but do not eliminate them completely. Furthermore, because
arbitrageurs will take risky positions in the hopes of capitalizing
on market
mispricings, riskless
20
arbitrage arguments
may not provide
The equilibrium
tight
bounds around observed market
prices generated by the
prices.
model provide a number of insights
into the
way
that
arbitrage activity absorbs market imbalances and eliminates deviations from fundamental values. In
particular, arbitrage activity reduces mispricings
and
is
most effective
in
doing so when the underlying
disturbances are transient and conditionally volatile.
While
much work
traders
this
paper has taken a step
remains. Theoretically, the model would be
were more
rational in their
be applied to particular markets
and
in the direction
into the process
in
of modeling equilibrium arbitrage
more
satisfying
if
activity,
the investors and noise
market actions. Empirically, the model or
its
descendants might
order to gain insights into the magnitude of market mispricings
from which these magnitudes emerge.
21
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A
and M. Gertler, (1991), "Asset Returns with Transactions Costs and Uninsured Individual
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mimeo.
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Time
Setting"
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Discrete-Continuous
Zariphopoulou, (1989), "Optimal Portfolio Rebalancing with
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A., (1991), "Stock Index Futures
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S.,
and M.
The Case of Stock Index Arbitrage,"
and Index Arbitrage
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(4), 523-547.
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and Market Structure," Journal of Finance,
43,
617-633.
Grossman, S., and J.-L. Vila, (1992), "Optimal Dynamic Trading with Leverage Constraints," Journal
of Financial and Quantitative Analysis, (forthcoming).
He, H. and N. Pearson, (1991a) "Consumption and Portfolio Policies with Incomplete Markets and
Short Sales Constraints: The Infinite Dimensional Case," Journal of Economic Theory, 54, 259-304.
He, H. and N. Pearson, (1991b) "Consumption and Portfolio Policies with Incomplete Markets and
Short Sales Constraints: The Finite Dimensional Case," Mathematical Finance,
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22
Heaton.
J.,
and D. Lucas, (1992), "Evaluating the Effects of Incomplete Markets on Risk Sharing and
Asset Pricing" mimeo, Massachusetts Institute of Technology.
and A. Neuberger, (1989), "Optimal Replication of Contingent Claims Under
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Hodges,
S..
Holden, C, (1990), "A Theory of Arbitrage Trading
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(1990) "Simple Binomial Processes as Diffusion Approximations in
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and T. Zariphopoulou, (1990), "Optimal Consumption and Investment with Borrowing
Constraints," mimeo, Massachusetts Institute of Technology.
Vila, J.-L.
23
Table
1
Table 1: The effects of arbitrage activity on mispricing. Table values are per dollar face value when
the bonds have 2 years to maturity. The expected absolute mispricings and the unconditional standard
deviation are conditional
on having
5 years to maturity.
24
Table 2
Table
2:
Average mean reversion and average conditional
conditional volatility of the cumulative liquidity shock, L„
is
volatility for different values
set
equal to
4.
of
p.
The
25
Table 3
Table
of
3:
mean
Average mean reversion and conditional
volatility for different values
reversion of the cumulative liquidity shock, p,
is
set equal to
.5.
of o.
The
coefficient
Figure
1
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LU
c
o
C
o
Q.
+-•
CD
C
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Q
8Z|S UOIJlSOd
Figure 2
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