HD28
•M414
no.
9/
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
ARBITRAGE WITH HOLDING COSTS:
A UTILITY-BASED APPROACH
by
Bruce
Latest
Tuckman and Jean-Luc
Revision:
Working
Paper
Vila
December 1991
No. 3364-91-EFA
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
ARBITRAGE WITH HOLDING COSTS:
A UTILITY-BASED APPROACH
by
Bruce
Latest
Tuckman and Jean-Luc
Revision:
Working
Paper
Vila
December 1991
3364-91-EFA
No.
M l.T.
LIBRARIES
1
1M*
Arbitrage With Holding Costs:
A
Utility-Based
Approach
by
Bruce Tuckman
Stern School of Business,
New York
University
and
Jean-Luc Vila
Sloan School of Management, M.I.T.
This version:
We
would
like to
participants in the 1990
and suggestions.
in the N.Y.U. and Cornell Finance Seminar Series,
and AFFI meetings, and an anonymous referee for helpful comments
thank participants
AFA
December, 1991
Abstract
Unit time costs, or holding
costs, are
incurred in
many
arbitrage contexts. Well
examples include losing the use of short sale proceeds and lending funds
reverse repurchase agreements.
arbitrageur
who
faces
at
below market
known
rates in
This paper analyzes the investment problem of a risk averse
holding costs.
The proposed model
allows prices to deviate from their
"fundamental" values without allowing for riskJess arbitrage opportunities. After characterizing an
arbitrageur's optimal strategy, the
model
is
examined
in
the context of the Treasury
analysis reveals that holding costs are an important friction in this
expected to be a significant determinant of arbitrageur behaviour.
market and
bond market. The
that they
can be
Arbitrage With Holding Costs:
Financial economists have
made
A
Utility-Based
Approach
great use of the notion of arbitrage in frictionless markets.
Researchers define arbitrage as a set of transactions which costs nothing and yet
They then assume
positive cash flows.
that arbitrage opportunities never exist
riskJessly provides
and derive precise
implications about the relative prices of traded securities.
The presence of market
make
it
of models, trading costs
impossible to riskJessly exploit small price deviations from arbitrage-free relationships.
Therefore,
When
frictions complicates the story. In the simplest
when market
do not admit
prices
riskless arbitrage opportunities, arbitrageurs
market prices do admit such opportunities, arbitrageurs bet
all
do nothing.
they have on the sure
proposition that prices will be back in line at the maturity date of the underlying securities.
While
this
simple model can explain the empirical regularity of price deviations from
1
"fundamental value," more careful portrayals of market frictions
activity.
Brennan and Schwartz (1990),
stock index futures market can
may be worthwhile
at
to
open
make
for
it
a position
make
for richer
models of arbitrage
example, show that trading costs and position
valuable to close a position before maturity.
even when the costs of opening
it
limits in the
As
a result,
it
plus the costs of closing
it
maturity exceed the price deviation from fundamental value. This paper contributes to the theory
of arbitrage pricing with market frictions in two ways. First,
of true-to-life arbitrage
which rules out
mispricings.
finite, risky
a result, risk averse arbitrageurs
position
is
if
is
large
without removing
do not
sit idly
enough and
all
its
aspect
Second,
model
it
builds a
incentives to exploit market
by or bet
adjust
somewhat ignored
all
the have.
They take
a
size as the mispricing changes.
consistent with the casual empirical observation that professional arbitrageurs
The phrase "fundamental
lhat the price, in the
the mispricing
focuses on a
unit time costs, or holding costs.
riskless arbitrage opportunities
As
This behavior
namely
activity,
it
value'
presence of
is
10
frictions,
mean
the arbitrage-tree price
when
should equal us lundamental value
there are no market frictions.
In
no sense do we mean
2
routinely find
and take advantage of deviations from fundamental values.
Section
I
describes
how
holding costs can transform riskless arbitrage opportunities into
potentially profitable but risky investment projects.
in
many
II
presents a model of arbitrage activity.
and an exogenous mispricing process
opportunities.
The paper then
arbitrageur's optimal,
is
dynamic
and 3) arbitrageurs
strategy.
will
since any future losses will be
Section
that,
in
is
set with a holding cost structure
combination, do not admit
if
For the case of negative exponential
and only
if
riskless
arbitrage
utility
it
is
shown
that
the potential gains are large enough, 2) the position
take a position even with an instantaneously negative expected return
accompanied by greater arbitrage opportunities.
examines the model's assumptions
III
The scene
derives the partial differential equation governing a risk averse
arbitrageurs will hold a position
finite,
then argues that holding costs are important
different arbitrage contexts.
Section
1 )
It
in the
context of the U.S. Treasury bond
market, estimates the parameters of the model, and numerically solves the partial differential equation
of section
II
using those parameters.
The data support
the contention that holding costs are quite
important, relative to trading costs, in preventing riskless arbitrage.
The
data also support the
assumption of an autoregressive process for the deviation of prices from fundamental value.
Finally,
the numerical solutions reveal an "option effect" and a "risk effect" in the determination of position
size
and allow
for
an assessment of the relative importance of the two
effects.
Section IV concludes and discusses avenues for future research. In particular,
be viewed as a
first
of holding costs.
this
work can
step towards building an equilibrium model of arbitrage activity in the presence
Such
a
model holds the promise of being able
around fundamental value than those supplied by
riskless arbitrage
to provide tighter price
arguments.
bounds
3
Holding Costs and Risky Arbitrage
I.
A
Consider two portfolios,
time of the
last
identical cash flows over time and, at the
cash flow, must have identical market prices. Nevertheless, for
market prices of the portfolios, P
in frictionless
and B, which provide
markets
A and P B
will short portfolio
,
an amount
differ by
A and
x.
Assuming
some
reason, the
PA > P B
that
,
investors
purchase portfolio B, realizing x today and incurring
no future cash inflows or outflows.
Although
this
just described, the
sale
paper could have assumed that institutions were such
model
to be presented in section
II
builds
on
a
more
as to allow the arbitrage
realistic description
of short
agreements. In order to short portfolio A, an investor must borrow the securities from
somewhere. Furthermore, the lender of the
securities will require collateral in order to ensure the
eventual return of his securities. Therefore, the arbitrage must be arranged as follows: lend P
to the holder of the securities (or, equivalently. post
securities in
more
A
as collateral, sell these securities for
realistic setting, the investor neither
A
dollars
A
an interest-bearing security worth P ), take the
P
A borrow P B and purchase
,
,
pays nor receives any
money when
portfolio B. In this
establishing his position,
but realizes the future value of x upon closing the position.
In
frictionless
conclusion that,
in
markets,
investors
in equilibrium, x
two important ways.
First,
would wish
must equal
0.
to
arbitrage
ad
infinitum,
leading to the
Introducing holding costs changes arbitrage behavior
the no-riskless arbitrage condition only requires that x be no larger than
the present value of the accumulated holding costs, from the opening of the position to the time
when
the two portfolio prices must be equal. Second, for smaller
opportunity:
if
x falls to
will
investors face a risky investment
quickly enough, then the costs of effecting the arbitrage will be small and
the transaction will have proved profitable.
accumulated costs
x,
If,
wipe out the realized
Holding costs appear
in
many
on the other hand,
x does not
fall
quickly enough, the
gains.
arbitrage contexts. First, the short sale of any spot security or
4
commodity will
usually incur unit time costs since investors
of the short sale proceeds. 2 In the case of stocks,
it
is
must often
sacrifice the use of at least part
true that institutions with large client bases
can take short positions almost costlessly. Nevertheless, smaller firms engaged
incur holding costs
when
short-selling stocks.
3
Second, trading
holding costs since at least part of margin deposits
markets
forward contracts often charge a per
in
in futures
may not earn
annum
The model
costs.
meet
do
markets may generate
4
Third, banks making
rate over the life of the contract.
deviations from desired investment strategies caused by having to
be thought of as unit time
interest.
in arbitrage activity
5
Fourth,
collateral requirements can
6
of this paper focuses on holding costs and ignores trading costs for a
reasons. First, despite their ubiquitousness, holding costs have not received
Second, trading costs would obscure the qualitative nature of
much
number of
scholarly attention.
this paper's results.
Trading costs
transform strategies which continuously adjust holdings into strategies which discretely adjust holdings,
where the frequency of adjustment decreases
elsewhere,
7
there
is little
in
the trading costs. Since this effect has been studied
danger that focusing on holding costs
will
prove misleading. Third, while
the trading costs of professional traders and arbitrageurs can be extremely small, the above examples
and the discussion to follow reveal that their holding costs need not be
One
of the easiest contexts
to trading costs
"
4
5
6
which to establish both the importance of holding costs
and the absolute magnitude of holding costs
For the case of stocks,
We would
in
like to
see. for
thank Billy
trivial.
is
the Treasury
relative
bond market. The major
example. Cox and Rubinstein (1985), pp. 98-103.
Rao
of
Susquehanna Investment Group for
this
comment.
See. for example. Duffie (1989). pp. 58-68.
See. for example.
Anderson (1987),
We
like to
thank Peter Carr for
this suggestion.
See. for example.
Dumas and Luciano
(1990).
(1989)
would
p.
198.
Hodges and Neuberger (1989), and Fleming. Grossman.
Vila,
and Zanphopoulou
5
trading costs in this market, namely bid-ask spreads and brokerage fees, are typically small, totalling
8
about 3/32 of one percent and 1/128 of one percent, respectively. The major holding costs
government bond market are reverse spreads. Recall
repurchase agreements
as collateral.
which the short
reverse spread
and the lending
collateral
available
The
in
rate
is
on
when any Treasury bond
Treasury bond
will
specific collateral. In other words, the reverse spread
will
same bond
translates into an opportunity loss
bond and the
likely
is
usually positive
9
is
the rate
when
only a specific
because the
difficulties in
serve as collateral minus the rate available
owner of
.25%
takes the security he wants to short
defined as the difference between the lending rate on general
finding the
reports average spreads of
bonds are sold short through reverse
money and
seller lends
serve as collateral. This spread
a particular
that
in the
attempts of other would-be shorters to find the
on the money
lent
through reverses. Stigum (1983)
.65%. 10 Now, to compare the magnitudes of trading costs and
to
holding costs, consider shorting $100 face value of a par bond and maintaining the position until
maturity.
The
trading costs
come
to
about 10 cents and, with
exceed the trading costs for maturities longer than about 2
section
III
show
that, for all
1/2
a
spread of .5%, the holding costs
months. The price data presented
in
but the shortest maturities, holding costs do more than trading costs to
inhibit riskless arbitrage activity.
Two
recent empirical papers serve as excellent motivations for the present analysis. Cornell
and Shapiro (1989) document both the persistent mispricing of
a particular
attempts of arbitrageurs, facing holding costs, to profit from that mispricing.
Treasury bond and the
Amihud and Mendelson
(1990), studying the effects of liquidity by comparing the prices of matched-maturity Treasury
and notes, find that notes are cheap
8
9
See Stigum (1990). pp. 649-650 and
See Stigum (1990). pp. 596-597.
10
p.
414.
p.
669.
relative to bills
bills
and proceed to examine whether these price
6
differences can be exploited. Considering only trading costs, as captured by the bid-ask spread and
brokerage
of these
costs, riskless arbitrage opportunities
seem rampant. Adding holding
costs,
however, most
money machines break down. 11
Another motivation
for the analysis of holding costs
about the source of market mispricings.
Some
authors
stems from the relatively recent literature
(e.g.
De Long
et al.
(1990) and Lee
et al.
(1991)) have hypothesized that investors' short term horizons allow for persistent deviations from
fundamental values. This paper reveals that holding
costs, in effect,
generate myopia. Even though
arbitrageurs have an investment horizon equal to the maturity of the underlying securities, holding
costs discourage the
viewed
maintenance of long-term arbitrage positions. Therefore, holding costs may be
as a particular
way
to
endogenize investor impatience.
II.
The Model
This section models the behavior of risk averse arbitrageurs
observe prices that deviate from fundamental value.
To
through their
common
maturity date, T.
incur holding costs and
allow for the possibility of arbitrage
opportunities, begin by assuming that there exist two portfolios.
identical cash flows
who
i:
A and
B,
which are characterized by
Let their date-t prices be P* and P^,
respectively.
Let
it
would be
x,
= P*-P^ denote
ideal to
The authors
endogenize the stochastic process governing the evolution of x
actually consider the bid-ask spread and then
brokerage costs are so small
While
allows the
position.
this
model
the difference between the prices of these portfolios
in this
market, the result stated
assumption seems innocuous enough,
to abstract
from the portfolio
effects
it
in
add both brokerage costs and holding
,
t
on date
this
t.
While
paper follows
costs. Nevertheless,
because
the text holds as well.
does rule out concurrent investments
in
other nsky assets. This simplification
which would arise from holding other nsky assets
in
addition to an arbitrage
7
others
13
process as exogenous in order to focus on the investment problem facing
in talcing the
The assumed
arbitrageurs.
1) x T
2) x,
=
is
should exhibit the following properties:
x,
the price deviation disappears at maturity.
0, i.e.
never so large as to admit
riskless arbitrage opportunities.
tends towards zero.
3)
x,
4)
The
The
evolution of
larger the absolute value of
last
two properties attempt
x,,
the faster x tends toward zero.
t
to infuse the process with
an equilibrium flavor. Property 3)
captures the idea that deviations are transient. Property 4) reflects the notion that larger deviations
will
be exploited more
readily, and. as a result,
tend to vanish at a faster rate than smaller deviations.
In order to find a process which satisfies (1) and (2). begin by determining the permissible
range of x
Along the
.
t
lines of the motivations discussed in section
unit time per portfolio unit.
14
Also,
I.
let
c>0 be
denote the arbitrageur's cost of
let r
value cost of maintaining a unit short position from date
t
until maturity
the holding cost per
capital.
Then, the present
is
T
ce- r(s
- l)
ds
.
J"
t
Letting t
=
T-t. the
time to maturity, and denoting
s(t)= f
If
this cost
function by s(t),
(l-e"")
(1)
the absolute value of x exceeds s(t) at any time, then a riskless profit could be
made through
arbitrage described at the start of the previous section. Therefore, to guarantee that
generate the process
See. for example.
x, in
|x,|
<
the
s(t).
the following way:
Brennan and Schwartz (1990). Recently. Holden (1990) has endogenized
market and that each segment is periodically hit by liquidity shocks.
a
mispncing process by assuming
thai
a clientele effect splits the
14
It
In the context of the U.S.
seems
best,
however,
to
Treasury market, the holding cost can change as the arbitrageurs
postpone modelling c
until
it
is
endogenized
roll
over expiring repurchase agreemenis
Furthermore, in
to reflect the collective activity of arbitrageurs.
order to model the nskless arbitrage boundary in a simple way. it has further been assumed that the holding cost
the U.S. Treasury market, however, the cost is usually per dollar of portfolio value.
is
per portfolio
unit. In
8
=
x,
where
an underlying state variable and
z, is
the sign of z and increases in
The
process for
autoregressive process.
Notice that
this
is
(2)
a function from
(-°o,
+ «>)
to (-1,1) which preserves
setup also ensures that x T
this
can be made to
x,
To
z.
<J>
s(t)4>(z,)
and 4)
satisfy properties 3)
=
since s(0)
listed
above
=
if
0.
zt
is
an
end, assume that the state variable evolves according to an Ornstein-
Uhlenbeck process
=
dz,
where db,
is
Now
example, he
convenience,
will short
I, is
some
who
quantity,
+
odb,
notices that x
is
not equal to zero.
I
,
of A, lend I,P* borrow
dW,
= rW
I
t
W, can be
t
P^, and buy
P^ exceeds
I,
units of B.
when P^ exceeds P* then
P^, for
If,
for
the evolution
written as
+
I,
[rP?
= rW dt +
I,
[rx,dt
dt
t
If
t
defined as the negative of the position size
of the investor's wealth.
(3)
Brownian motion.
the increment of a standard
consider an investor
-pz, dt
rP?
-
-
-
dP* + dP*]
-
c|I t |dt
dx,] -c|I,|dt.
(4)
t
The
first
term represents the interest received from previously accumulated wealth. The second term
gives the interest gain plus the capital gain or loss
the holding cost incurred for shorting
Assuming
I,
units of portfolio
that the arbitrageur will
completes the model's specification.
from the arbitrage position. The
More
A
or B.
last
term reflects
15
maximize the expected
utility
of
his
terminal wealth
formally, he maximizes
e ru(wT)]
While equation (4) assumes
thai the arbitrageur
be set up so that the arbitrageur had a higher cost.
a different
holding cost.
s
An
holding cost equals that which sets the nskless arbitrage bounds, the problem could
equilibrium model, then, might include several classes of arbitrageurs, each with
9
some von Neumann-Morgenstern
for
The
equation (4)
and
i,
first
in
function, U.
utility
16
step in solving for the optimal investment policy {I,}
terms of the underlying state variable,
= I,s(t)(|)'(z)e
rT
(4)
,
is
to rewrite the wealth
Using two transformed variables,
z.
w
=
t
W^
17
can be rewritten as
dw =
[u(z,t)dt
i,
t
+ odb
t
(5)
]
where
= pz
H(z,t)
and
e
is
the sign of
i
,
i.e.
1
if
Let V(w,z.t)
of optimality
in
>0,
i
= max E T
Ito's
o
=0, and
i
:
<(>7<l>'
+
c(<t>-€)/s4>'
(6)
-1 if i,<0.
t
{
U(W T
)
},
where E T denotes the expectation
at t.
By the
principle
dynamic programming,
V(w,z,t)
By
if
t
t
+
-
=
max,
E T [V(w+dw,z+dz.T+dT)]
(7)
Lemma,
V(w+dw,z+dz,t+dT) = V(w,z,t) +
V w dw + V dz + V T dr
z
+ T V^ldw) 2 + + V^dz) 2 + V^dwdz
where subscripts denote
partial
derivatives.
Substituting (3),
(5)
(8)
and
(8)
into
(7)
and taking
expectations gives
=
max,
{
Solving (9) numerically
(uV^V^i+jV^on
is
by no means a
V(w, + oc,-c)=V(w,-oo,t) = U( + ») does
conditions for large
One might argue
that
|z|
not
2
}
-
pzV z
trivial exercise.
characterize
the
-
V T + +V H o 2
(9)
Imposing the boundary conditions
function
V(-)
because
have not been specified. Furthermore, for arbitrary functions
U
because an arbitrage position based on deviations from fundamental value carries no market
Two
growth
the
and
risk,
4>,
it
is
discounted
Even if investors are risk neutral with respect to the
seem appropriate:
risk of this type of arbitrage, leverage constraints can induce risk aversion. Grossman and Vila (199H show that the possibility that an
investors future opportunity set will be limned by borrowing restrictions causes nsk neutral investors to display some risk aversion. Since
leverage constraints probably apply to most potential arbitrageurs, this consideration could justify the assumption of nsk aversion. 2) The
most important arbitrageurs are probably judged on ihe pertormance of their arbitrage portfolios, as opposed to the performance ot some
expected return evaluates the opportunity correctly.
replies
diversified portfolio of longer-term holdings. In that case, thev will be
.ire
only a lew arbitrage opportunities
dunng most time penods.
1 )
nsk averse with respect to the nsks of arbitrage so long
as there
10
not
at all clear that
for the case of a
exists.
there exists a bounded solution for V. Appendix
CARA utility
function and
<J>(z)
U =
-
e"
aw
for
some
growth conditions
proves that a bounded solution
positive constant a implies that
V(w,z,t)
some function
derivatives of
F.
F and
17
= U(«)
V(w,-oo, T )
should approach
max,
{
(
U(w) =
-e"
u
+ o 2 Fz )ai-+o 2 (ai) 2
V
in
terms of the partial
aw
,
so F(z,0)
maximum. Since
Equation (11)
will
-Vo^-F, 2
pzFz +
-
}
(11)
)
conditions for equation (11) can be derived as follows.
since, as the mispricing
its
(10)
substituting into (9) yields the following partial differential equation:
The appropriate boundary
V(w,z,t) must equal
= **"*<**)
Using (10) to calculate the partial derivatives of
FT =
far
finds the
This section continues, therefore, by specializing the model to this case.
Letting
for
= z/(l+z 2 ) 1/2 and
1
this
=
0.
maximum
in
is
riskless arbitrage
zero, F(«,t)
=
F(-»,t)
the following section.
presence of holding costs. These properties are presented
The
and
its
1:
An
in proposition
=
bound, the value function
=
<*>.
analysis has
proceeded
enough, however, to derive some properties of an arbitrageur's optimal investment policy
PROPOSITION
x=0.
Furthermore, as mentioned above, V(w, + oo,-c)
approaches the
be solved numerically
When
in the
1.
arbitrageur following the optimal dynamic strategy implied by equation (11)
boundary conditions
i)
will take a position if
ii)
iii)
and only
will take a finite position,
may
if
the mispricing,
x, is
large enough,
and
take a position even
if
the instantaneous expected return from the position
is
negative.
Because the
opportunities (see
utility
function exhibits constant absolute nsk aversion,
Hodges and Neuberger (1989)
it
is
for a similar simplification).
separable
in
the value of wealth and the value of arbitrage
11
Proof: See appendix
The
first
1.
and second part of proposition
1
tell
a story consistent with casual empiricism about
arbitrage activity. While price deviations from fundamental values encourage arbitrage activity, risk
aversion and holding costs prevent investors from taking infinite positions. Consequently, the arbitrage
activity
of these investors might not be great enough to force prices back into
possibility that consistent arbitrage activity
The
The
third part of proposition
1
can be sustained
in equilibrium.
reveals an interesting fact about the optimal investment policy.
intuition parallels that of Merton's (1973) intertemporal
CARA
This raises the
line.
CAPM
with opportunity set changes.
investors like to hold assets which are negatively correlated with favorable changes in their
opportunity
set.
Here, arbitrageurs do lose
if
the absolute value of x
rises,
but then they have a better
arbitrage opportunity available to them. Consequently, they are willing to take an arbitrage position
even
if it is
expected to lose over the next instant.
III.
An
Application to the Treasury Bond Market
There are many redundant
securities in the Treasury
whose cash flows can be replicated by buying and holding
such redundancies
c
:
.
and
c3
,
with c x
is
>
a portfolio
c2
>
c3
,
there are
of other bonds.
If their
bond
exactly replicates the cash flows of
one
many bonds
One
class of
coupons are
then a portfolio composed of (Cj^VCCj-Cj) units of the
first
Cj,
bond and
unit of the second
18
All Treasury triplets which traded from January 1960 to
18
i.e.
formed by three Treasury bonds of the same maturity.
(c 1 -c 2 )/(c 1 -c 3 ) units of the third
bond.
bond market,
Note
that the accrued interest
on the portfolio also matches the accrued
December 1990 were
interest
on the second bond.
identified
from
12
the
CRSP
data
files.
three bonds of the
Those containing
same maturity traded on
selected as the triplet for that date.
Appendix
callable
bonds or flower bonds were discarded.
of the 42
list
triplets
more than
most recently issued were
a particular date, the three
A detailed
If
forming the data set
given
is
in
2.
Month-end
bid prices
were obtained from
CRSP
for
all triplets.
of the second bond in the triplet and P R to be the price of
mispricing,
x,
as in the text,
i.e.
its
19
Define P 2 to be the price
replicating portfolio. Define the
P 2 -Pr- For the opportunity cost of funds,
r,
this
paper uses a 3-year
Treasury rate as reported by Moody's, but the results would not substantially change for other choices
of
r.
Without data available
for holding costs,
it
seems reasonable
to
choose
a
number from
the
range .25% to .65%, cited above. According to the theory described in the previous section, holding
costs should
it
be large enough so that no price deviations exceed the
turns out, there are
some
violations in the data set
even
at
riskless arbitrage
c=.65%. Table
I
bound,
reports the
and percentage of observations which violates the no arbitrage conditions for different
s(t).
As
number
levels of the
holding cost.
INSERT TABLE
Table
I
reveals that a
trading costs alone.
model of holding
The brokerage
costs alone
I
compares most favorably with
costs estimate of 1/128%, for both the long
arbitrage, plus a typical bid-ask spread of
3/32%
cited above,
would
result in 419, or
deviations which exceed the level of trading costs. So, holding costs
relative to trading costs, in foiling riskless arbitrage activity.
Of
and short
seem
to
a
model of
legs of the
35.39%, price
be quite important,
course, a model which incorporates
both holding costs and trading costs does best. With c = .006 and these trading costs, for example, only,
only 4 observations, or .34%, violate the riskless arbitrage bounds.
19
The
analysis
was also done using the average of
bid
and ask
prices.
The
results
were qualitatively
identical
and quantitatively similar
13
Returning to the model of
is
in
this
it
seemed reasonable
on
to settle
c
=
.006. This level
the range of prior beliefs and generates relatively few violations of the riskless arbitrage bounds.
Using
this level
of holding costs and ignoring trading costs, figure
as a function of the riskless arbitrage
The
plot provides
arbitrage activity. In the
some
bound,
The two
s(t).
plots the individual mispricings
1
rays are the lines
x=s and
figure
.31 for
1.
this
seems
Ks<2,
model presented
The
to be the case.
.29 for
earlier, mispricings
d)(z)
More
2<s<3, and
=
z/(l+z
underlying state variable,
z.
precisely, the standard deviation of x
2 1/2
)
,
one can use
Then, changes
21
(3).
in z
The
is
is
to estimate
and
(2) to transform x
can be regressed on z
s.
Looking
at
about .09 for s<l,
its
parameters p and
values into values of the
s
in
order to estimate the
results of the regression are reported in table
INSERT TABLE
of p
is
in
s>3.
.43 for
parameters of the z process given by
The estimate
:o
can take on any value between s(t) and
next step in applying the model to the data at hand
Recalling that
x=-s.
additional evidence that holding costs are important inhibitors of
Consequently, one would expect the variability of the mispricings to increase
-s(t).
o.
paper,
II.
II
significantly negative, confirming the autoregressive nature of the
deviation process."
The
partial differential
equation
in
(11) can
now be
solved numerically. Converting the
parameters estimated from monthly data to an annual basis gives values of about 5.42 and .88 for p
and
a. respectively.
funds,
r.
was taken
The holding
to
be 9%,
cost, c,
was
set to .6%, as discussed above.
a value close to the
sample average.
The opportunity
cost of
Finally, the original maturity of
in
'
Figure
1
also shows that the price deviations are
usually exceeds the price of the middle bond. This
Ingersoll (1984). Lilzenberger
"'*
The observations
for
— Regressing changes
more
often negative than not. In other words, the price of the replicating portfolio
consistent with a tax timing option or a tax clientele effect. See Constanunides and
and Rolfo (1984), and Schaefer (1982).
which
in x
is
on
the autoregressive nature of the
|x|
>s were dropped from the sample since
x also gives a significantly negative coefficient
z
process.
z is
undefined
in
these cases.
Therefore, the functional form of $
is
not responsible for
14
the triplet
was
The
set at 5 years.
relevant range for the underlying state variable,
other words, values of z which are larger
than those which appear in the data
in
z,
turns out to be well within (-2,2). In
absolute value correspond to mispricings which are larger
set.
Figure 2 graphs the optimal position size as a function of z for two different times to maturity.
As predicted by Proposition
mispricing
is
1,
part
i),
for small mispricings
work
not optimal to take a position.
large enough, however, position size increases in the absolute value of
Figure 2 reveals that position size
at
it is
If
the
z.
not monotonic in time to maturity. There are two effects
is
here. First, since positions of potentially longer maturities are likely to incur larger holding
costs, investors
would take on smaller
positions. This might
be called a
risk effect.
Second, since
potentially longer horizons are likely to provide better arbitrage opportunities in the future, investors
would be
willing to take
on
larger positions. This might be called an option effect,
the hedging properties of an arbitrage position raised by claim
current arbitrage profits
figure,
relatively
should dominate.
risk effect
relatively
is
When
more important and
when
|z| is
3.
z
is
z
is
|z|
is
related to
mispricings, so the
however, the potential of future arbitrage profits
small,
the option effect should dominate. Consequently, as
When
is
large, realization of
more important than the promise of future
large the position size with 3.04 years to maturity
with 5 years to maturity.
When
and
is
shown
in
is
the
greater than the position size
small the position size with 5 years to maturity
is
the larger
of the two.
Although not shown
in
the figure, the hedging effect described in part
iii)
of Proposition
1
can be examined through the numerical solutions. With 3.04 years to maturity for example, an
arbitrageur will take a position even
From equation
generality, a
was
(11).
one only need solve
set equal to
1
for
when
ai.
|z|<.04.
where
a
is
The instantaneous expected
the coefficient of absolute
return from the
nsk aversion. Therelore. without
loss ot
15
position
will
is
positive,
however, only when |z|>.2. In terms of the dollar mispricing,
x,
the arbitrageur
take a position even for a mispricing of less than 6 cents for every $100 face value bought or
The expected
shorted.
return over the next instant
is
positive,
however,
only
when
the mispricing
exceeds 31 cents on the $100 position.
A similar comparative static might have arisen with
respect to o. Larger values of o
arbitrage position riskier but raise the likelihood of future profit opportunities.
as before, that the risk effect
would dominate
dominate for small mispricings. As
One
make
the
might reason,
for large mispricings while the option effect
would
turns out, however, the parameters determined by the data
it
generate position sizes which increase
in
a: the
option effect dominates for
all
relevant levels of
mispricing.
The
become an
certainty equivalent of arbitrage activity can be defined as the
arbitrageur in a particular
triplet.
of z for three different times to maturity.
absolute value of
z.
More
24
amount one would pay
Figure 3 graphs this certainty equivalent as a function
As expected,
the certainty equivalent increases
interestingly, the certainty equivalent increases in time to maturity.
have been the case that the
risk effect
would cause the certainty equivalent
to
fall
maturity since longer maturities run the chance of incurring larger holding costs.
however, for the parameters
in
rv.
to
It
in
the
might
with time to
Once
again,
the Treasury market, the option effect reigns.
Conclusion and Suggested Extensions
This paper models the effect of holding costs on the activities of risk averse arbitrageurs.
exogenous mispricing process was chosen
to allow prices to deviate
The
from fundamental value without
allowing for riskless arbitrage opportunities. Solving for the optimal investment policy of arbitrageurs
Note
thai the certainty equivalent
is
measured
in
daie-T dollars and
is
therefore comparable across maturities
16
in
such markets reveals behavior consistent with the casual empirical observation that
professional
arbitrageurs consistently take limited arbitrage positions. Furthermore, a risk effect and an option
effect
were
identified as driving the comparative statics of optimal position size
and of the value of
being an arbitrageur. Data from the Treasury bond market supported the importance of holding costs
in
arbitrage activity, provided parameter estimates of the model, and led to
relative
importance of the
Two
risk
and option
some
insights
effects.
paths of future research emerge from this analysis.
First,
there
is
a
need
data about the size of holding costs in the various markets mentioned in section
analysis of this
would be
to
about the
to collect better
I.
Second, the
paper concentrated on the investment problem facing arbitrageurs. The next step
model the
factors generating price
deviations from fundamental value and thus
endogenize the deviation process. This avenue of research would clearly benefit from some
preliminary progress in the
first.
17
Appendix
PROPOSITION
Proof of
Recall that the Bellman equation associated with the
F T = max,
{
(u + o
2
1
1
CARA
Fz )ai4o 2 (ai) 2
}
arbitrageur's optimization
pzFz +
-
lo^-F, 2
problem
is
(11)
)
with u given by
= pz
u(z,t)
and with e being the sign of
The boundary conditions
-
4-o
2
Unfortunately, condition (A2)
z. It is
may be
the
=
(Al)
0;
F(-«,t)
=
+».
(A2)
not precise enough to characterize the value function F(v).
possible, a priori, that the arbitrageur reaches an
F can be bounded by
explicit solution. This will
moment, assume
increasing
+ oo, T ) =
is
therefore necessary to show that
which admits an
that the function
mapping from
The symmetry of
(6)
are
F(
it
c(<(>-€)/s4>'
i.
F(z,0)
Indeed
+
<t>74>'
(-°°.«>)
be done
F and
its
later
unbounded value of F
for every
the value function of a control problem
by the choice of a particular function
derivatives are well defined
and that
<\>
is
a
4>.
For
smooth
into (-1. + 1) such that 4>(-z) = -<J>(z).
the problem yields that
F(-z,t)
=
(A3)
F(z,t)
so that
Fz (0,t) =
We
can
Proof of
now proceed
it:
with the proof of Proposition
From equation
(9) the
(A4)
for every t.
1.
optimal investment
i(z,t)
is
given by
18
L
i(z,x)
=
i
i(z,t)
=
i
i(z,t)
=
s
L
(z,t)
if
i
(z,t)
if
i
s
(z,t)>0
(A5a)
(z,-c)<0
(A5b)
(A5c)
otherwise
with
L
i
=I-£.z-I*_-i.J_i^
(z,t)
a a2
s
i
2
a o2
From
(A4),
it
tf
-1*1
=I-P-z
(z,t)
2
s(t)
2
s(t)
2
<j/
F7
i>0;
(A5d)
wheni<0.
(A5e)
when
4/
^_L1^
+
+
*F
<j/
follows that
L
i
<
(0,t)
s
0;
i
(0,-r)
(A6)
>0.
Therefore i(0,t)=0 and, by continuity, the optimal investment decision for small mispricings
no
position. This proves
follows from (A5) above.
ii): ii)
Proof of
Hi):
as
arbitrageurs are better off with large
From
the symmetry of F(z,t) with respect to z
i
can be positive even
Boundedness of F(z,
critical for
if
r)
u
is
As
|z|.
=
u/o
2
it
follows that
Fz (z,t)
has the same sign
a result,
+ Fz
negative and vice versa.
and growth
\
As mentioned above, growth conditions
conditions:
the numerical analysis.
i
u(z,t)
=
If
4>(z)=z/(l+z
pz
+
3
2
-a'
2
To
to take
i).
Proof of
z. i.e.
is
2 1/2
z
)
then
1+z
r
2
1+z
l-e
_rT
f.
A7
(A7)
,
.
2
for large z are
.
2
provide the bounds and growth conditions of the value function, consider a class of control
problems which admits an
explicit solution.
More
precisely, for every function
M(t). consider the
19
value function
VM
=
-
<
>(w,z,t)
max,
E[U(WT)]
subject to the dynamics
dw,
=
(M(t)z,dt
i
odb,) and
-
dz,
=
+ odb,
-pz,dt
t
V M( )
The function
can be
explicitly derived
Vm<>(w,z,t)
where F M()
is
quadratic with respect to
F m «(z.t)
By symmetry Kj(t)=0. The functions
dK«
J^
i
=
dt
and are easy
'
and
=
-
of the form
is
M
exp {-aW-F ()(z,t)}
z:
1k2 (t)z -K
:
=
K ; (t)
a
differential equations
,
= _L
dt
2
(t)z ^KoCt).
i
—:
and
1
and K^t) solve the ordinary
dIC
,
o'K.
tsT.
M(x)2
*
-
(A8)
2(M(t)- P ))IC,(t)
to calculate.
Returning to the original problem, observe that there
|u(z,t)| s M|z|, for every
Recalling that
i,
and
z,
must have the same
dw =
i,
sign,
[u(z,,i)dt
it
exists a positive constant
M, such
that
(A9)
z.
follows that for every policy
i
t
-
odb
(
t
]
<.
Mi
z,dt
-
db
oi
t
t
(A10)
.
t
Hence
V(w,z.t) ^
It
V m (w.z,t)
< U(«).
follows that the control problem faced by the arbitrageur
is
bounded and admits
a
smooth
value function. (See Vila and Zariphopoulou (1989) on proving the regularity of the value function
in a similar context.)
Growth conditions can be derived by considering
From (A7)
it
follows that for large
z,
u
is
u(z,t) -
the dominant term in u(z,t) for large
approximately linear
(p-
L_)z
l-e~
rT
=
z.
in z i.e.
R(t)z
.
(All)
20
Let t" be such that R(t')=0.
for every -est*
and every
Then under
z the
expected gain u(z,t)i
follows that the function F(z,t)
considering the function
certain parameter restrictions, easily satisfied by the data,
= 0,
V M(T) (w,z,t)
for
is
nonpositive for every arbitrage position
=
max {R(t);0}.
Therefore the numerical analysis used the growth condition
1
F(z,t) -
It
tst*. For t;>t\ growths conditions are obtained by
with
M(t)
i.
-K^tJz
->
for large z.
21
22
References
Amihud,
and Mendelson,
Y.,
Securities," Journal
Andersen,
T., (1987),
of Finance, 46
1411-1425.
(4), pp.
Currency and Interest-Rate Hedg in g
Brennan, M., and Schwartz, M., (1990), "Arbitrage
63(1),
pt. 2,
.
New York
of Financial Economics,
Stock Index Futures," Journal of Business,
in
Bond Trading
(1984), "Optimal
J.,
13, pp.
.
De Long,
J. B.,
A Shleifer, L.H. Summers, and R.
Economy
markets. Journal of Political
Dumas,
and Luciano,
B.,
vol.
with Personal
E., (1991),
Taxes." Journal
299-335.
and Rubinstein, M., (1985), Options Markets Prentice-Hall,
J.,
Institute of Finance.
pp. s7-31.
Constantinides, G. and Ingersoll,
Cox.
and the Yields on U.S. Treasury
H., (1991), "Liquidity, Maturity,
98,
J.
Waldmann,
Inc.
1990, Noise trader risk in financial
703-738.
"An Exact Solution
to the Portfolio
Choice Problem Under
Transactions Costs," Journal of Finance, 46(2), pp. 577-595.
Duffle, D., (1989), Futures Markets Prentice-Hall. Inc.
,
Fleming, W., Grossman,
S.,
Vila, J.L.,
Zariphopoulou, T, (1989), "Optimal Portfolio Rebalancing with
Transaction Costs," unpublished manuscript.
Grossman,
S.,
and
Vila, J.L., (1989),
"Optimal Dynamic Trading with Leverage Constraints." Journal
of Financial and Quantitative Analysis, (forthcoming).
Hodges,
and
S..
Neuberger,
A,
(1989)
"Optimal
Replication
Transactions Costs," The Review of Futures Markets,
Holden. C, (1990), "A Theory of Arbitrage Trading
Paper #478, Indiana University.
Lee,
CM.,
A Shleifer and R. H. Thaler.
of Contingent
Under
222-242.
vol. 8(2), pp.
in Financial
Claims
Market Equilibrium," Discussion
1991, Investor Sentiment and the
Closed-End Fund Puzzle.
Journal of Finance, 46, 75-109.
Litzenberger, R., and Rolfo,
J.,
(1984), "Arbitrage Pricing, Transactions Costs and Taxation of Capital
Gains," Journal of Financial Economics, 13. pp. 337-351.
Merton, R., (1973), "An Intertemporal Capital Asset Pricing Model," Econometrica,
vol. 41.
pp.
867-887.
Schaefer.
S..
(
1982). "Taxes and Security
Market Equilibrium."
in
Financial
Economic Essavs
in
Hone >r
23
of Paul H. Cootner. Sharper, W., and Cootner, C,
Schwartz, E..
Hill, J.,
and Schneeweis,
eds.,
Prentice Hall. pp. 159-178.
E., (1986), Financial Futures,
Richard Irwin.
Stigum, M., (1983),
The Money Market, 2nd
edition,
Dow
Jones-Irwin.
Stigum, M., (1990),
The Money Market, 3rd
edition,
Dow
Jones-Irwin.
Vila,
Inc.
and Zariphopoulou. T., (1989), "Optimal Consumption and Portfolio Choice with
Borrowing Constraints," unpublished manuscript.
J.L.,
24
Table I
Frequency of riskless arbitrage bound violations
as a function of the level of holding costs
Table
I
reports the
number and percentage of observations which
violates the
no arbitrage
condition |x,|<s(t) for different levels of the holding cost. There are 1184 observations
Holding Cost
in all.
25
Table
OLS
dz
The
regression model
II
regression results from estimating the process
=
-pzdt
+ odb
with monthly observations.
is:
Zt
+
l
"
=
*!
P 2!
+
€
t
with 1125 observations.
The estimated value of p
is:
p
=
-.452
with a standard error of .0257.
The R squared
is:
R2 =
The estimated value of a
is
.216
taken to be the standard error of the residuals:
a
=
.253
Figure
1
$ Mispricing per $100 Face Value
co
J3
-L -
rb
ro
CO
Figure 2
Position Size
in
Face Value
(Thousands)
a*
o
c
D
Q.
CD
<5'
CO
C/5
r+
0)
CD
<
Q)
Q)'
C£
CO
N
en
O
o
CO
o
O
O
O
O
CO
O
O
cn
O
(J)
O
Figure 3
Certainty Equivalent
in
Dollars
Date Duef
-fr-*
Lib-26-67
,11
3
UBRARItS
TD60 00735bEl
E