rrP0^
J
"U''-
/'
AA^.mi.'^^
'
.'--Trcg
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
NONNEGATIVE WEALTH, ABSENCE OF ARBITRAGE,
AND
FEASIBLE CONSUMPTION PLANS
Philip
H.
Dybvig and Chi-fu Huang
March 1986
Revised, December 1987
W.P./-
1976-88
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
QY^
NONNEGATIVE WEALTH, ABSENCE OF ARBITRAGE,
AND
FEASIBLE CONSUMPTION PLANS
Philip
H.
Dybvig and Chi-fu Huang
March 1986
Revised, December 1987
W. ?.-: 1976-88
NONNEGATIVE WEALTH, ABSENCE OF ARBITRAGE,
AND
FEASIBLE CONSUMPTION PLANS*
Philip H. Dybvigf
and Chi-fu Huang|
March 1986
Revised, December 1987
Abstract
A
restriction to nonnegative wealth
financial
models that have
is
sufficient to preclude all arbitrage opportunities in
risk neutral probabilities that are valid for all
simple strategies.
Imposing nonnegative wealth does not constrain agents from making the choice they would
make under
the standard integrability condition.
This conclusion does not depend on
whether the markets are complete.
'We are grateful for useful convereatione with Truman Bewley, Offer Kella, Dave Kreps, Jon Ingersoll, Bob Merton, Paul
Milgrom, and Steve Ross We arc also grateful for support for Dybvig under the Sloan Research Fellowship program and
for Huang under the Batterymarch Fellowship program. Any opinions are our own and not necessarily those of the Sloan
Foundation or of Batterymarch Financial Management. We are responsible for all errors.
tYale University
{Massachusetts Institute of Technology
1
1.
i
FEB
8 1988
Table of Contents
1
Introduction
2 Nonnegative
1
Wealth and Absence of Arbitrage
3 Nonnegative Wealth
4 Concluding remarks
and
Feeisible
Consumption Streams
2
13
17
1.
Introduction
While much of the intuition
and portfolio choice can be exhibited
in option pricing
in discrete
time models, continuous time models using Ito calculus have been dominant in these areas of
finance.^
One
reeison is that
it is
generally easier to derive a closed-form solution to a differential
equation than to a difference equation.
Early development of continuous-time finance using Ito
calculus tended to be intuitively based and
(see, for
sufficiency of the natural first order conditions
example, Merton [1971] and Black and Scholes
as
and Rubinstein
[1979]),
was rigorously
The
[1973]).
was consistency with hmiting versions
was reassuring,
analysis
assumed
intuitive appeal of the results
of discrete-time results (as in Cox, Ross,
but at that time no attempt was made to maike sure that the mathematiced
correct.
Harrison and Kreps [1979] set out to give the continuous time analysis a rigorous foundation.
They showed that
can be obteiined using
this teisk is not straightforward, since arbitrage profits
seemingly reasonable strategies called doubling strategies (after the strategy of doubling one's
bet at roulette).
would take
Having continuous trading allows one to do
many
infinitely
in
any
finite
time interval what
turns at the roulette wheel. Presence of the doubling strategies strikes
at the core of the continuous time model, rendering
it
Having
vacuous.
eu^bitrage opportunities
precludes having a solution to the optimal investment problem (for strictly monotone preferences)
and, of course, invalidates option pricing theory based on the assumption that there
is
no arbitrage
opportunity. Harrison and Kreps removed arbitrage possibilities by restricting trading strategies
to simple trading strategies that allow trade only at finitely mainy times chosen in advance. This
restriction allowed
[1976].
them
to use
and formalize the risk-neutral pricing approach of Cox and Ross
Cox and Ross argued that
probabiUties to give
all
assets the
in
the absence of arbitrage, one could always reassign the
same expected
the miirtingale approach because of
its
returns. Harrison
and Kreps called
this
approach
relation to martingale theory.
Using simple strategies salvaged some of the pricing
results,
but logically invalidated the results
pertaining to particular strategies for hedging or optimal portfolio choice, except for degenerate
cases that
happened to
lie
in the restricted set.
model, the hedging strategy used to duplicate a
For example, in the Black and Scholes [1973]
cadi
option
is
not simple. As a result,
we were
in
the unhappy position of having a whole set of results that haA been rendered logically inconsistent.
If
we
restrict ourselves to using simple strategies,
are not available, while
What we need
is
if
portfolio choice
is
unrestricted there
a condition that rules out
rich variety of useful strategies.
hedging strategies and optimal portfoUo choices
all
is
arbitrage.
arbitrage possibilities without also ruling out the
One approach proposed by Harrison and
Pliska [1981]
is
to impose
Arguably, numerical solutions of binomial models such as those of Cox, Ross, and Rubinstein [1979] are now more prevalent
models built by practitioners. The binomial model provides more intuition over a single period,
but the solution is a black box when comparative statics can be performed only numerically.
*
in applications, especially in
an integrability condition on the trading strategies. The set of strategies satisfying the restriction
is
hard to justify economically. The best interpretation so far
strategies with p
Huang
>
is
that within a class of U' trading
these are the limits in U' of simple trading strategies (see,
1,
e.g.,
Duffie «md
[1985]).
Harrison and Kreps [1979] suggested a more economically interpretable condition. They conjectured that arbitrage
weedth at
is
ruled out
trading strategies are restricted to those having nonnegative
Note that
points in time.
all
if
it
is
trivial
that any lower
bound on wealth
rules out
the doubling strategy, since wealth does go arbitrzurily negative with a strictly positive probability
What
given a doubling strategy.
all
more subtle
is
is
to
show that a lower bound on wealth
Dybvig
trading strategies that generates a free lunch.
[1980] confirmed Harrison
conjecture in the special fixed-coefficient Bleick and Scholes [1973] model.
explored this possibility more heuristically.
Heath and Jarrow
constraints,
[1987] have analyzed
Still
Latham
rules out
and Kreps'
[1984] has also
within the fixed-coefficient Black-Scholes model,
margin requirements that are similar to nonnegative wealth
and they show that the margin requirements do not
rule out the hedging strategies
that duplicate put and call options.
The purpose
that
is
of this paper
is
to provide a general analysis of the nonnegative wealth condition
not restricted to the fixed-coefficient Black-Scholes example, and to demonstrate the relation
between the nonnegative wecilth constraint and the integrabihty condition.
nonnegative wealth constraint precludes arbitrage opportunities quite generally.
for agents
whose choice space
the set of feasible choices
in the
is
the
is
We show
We
also
that the
show that
the set of consumption plans satisfying the integrabihty condition,
same up
to closure under the
paper require completeness of the market, although
in
two
restrictions.
None
of the results
one case we give a simple alternative
proof under completeness.
The
paper
rest of this
is
organized as follows.
Section 2 contains our assumptions and the
proof that the nonnegative wealth constraint precludes arbitrage. Results are in Section
concluding remarks are in Section
2.
Some
4.
Nonnegative Wealth and Absence of Arbitrage
model
All uncertainty in the
denoted by
u;
=
probabihty space
where J
is
of agents.
G
{w{t)\,t
(f2,
is
[0,1]}.
generated by an A^ -dimensional standard Brownian motion,
Formally,
T,P), where each
a;
6
all
fl is
random
variables are defined on the complete
a complete description of the state of the nature,
a sigma-algebra of distinguishable events, and where
Let
F =
{7t',t
G
[0>l]}
family of sub-sigma-fields of
for
3.
an agent
who
observes
u;
7,
P is
the conunon probabihty belief
^^ '^^ filtration generated by w. (A filtration
and
is
the formal description of
and no other
useful information.)
economy are endowed with the same information,
how information
We assume
specified by F,
that
is
an increasing
arrives over time
all
agents in the
and therefore the sample paths
of
completely specify distinguishable events (7i
tv
motion starts from zero almost
t
e
random
relations involving
all
We
[0,1].
denote by
.,M. Security zero
Security zero
r(f).
r{t)
is
it
will be, for
For j
We want
—
the
same
example,
1,2,...,
P
conditional on
7t,
for all
and by
E[-]
as .E[-|7o]).
M+
1
long-lived securities, indexed by j
=
if r is
M,
that security zero
is
its
price process
is
worth one
is
=
given by B{t)
and never
at time zero
expf/^
bounded below from zero uniformly across
For our
r(fl)ds}.
states
and time, as
never negative.
security j can be locally risky (as
it
will
be except
in degenerate cases).
to allow our analysis to be general enough to allow consumption payments and dividends
lumpy
to be
is
[w.l.o.g.)
assume that B{t)
will
P
not necessarily riskless over longer time periods, because the interest rate
is
has any distributions, and therefore
we
events in Iq have probability
a locally riskless bond that pays interest at the instantaneous rate
may be random. We assume
analysis
is, all
expectation operator under
^[-l^t] the
consider a frictionless securities market with
0,1, 2,..
that
trivial,
variables are to hold almost surely with respect to
the unconditional expectation (which
We
almost
is
Note that since a standard Brownian
T).
All the stochastic processes to appear will be adapted to F, and unless otherwise
zero or one.
noted,
surely, Jo
=
(at
a point in time), continuous, or a connbination of the two. Also, while we require
consumption to be nonnegative, we allow negative dividends
cial assets
that are not of limited
consumption pattern
liability.
(i.e.,
assessments), to allow finan-
The simplest and most
intuitive
way
to represent a
by a process with nonnegative and nondecreasing sample functions, repre-
is
senting accumulated consumption over time. Similarly,
we use processes with bounded
vaxiation
sample functions to represent accumulated dividends.
Therefore,
we
will represent the
using the ex-dividend price at time
t,
t,
investment opportunities presented by the risky security j
denoted by Sj{t), and cumulative dividends from time
time
denoted by Dj{t). To define properly the integrals characterizing the wealth process, we need
lumpy dividends and
to choose carefully the accounting conventions for
time.
The convention we choose
is
for the
cumulative dividend process or price at
and Sj{t) are assumed
to exist.
The lump
and A5j(t)
=
f.
dividend or stock price change at
to be right continuous,
-
Sj{t-)
this interpretation for
t
=
0,
is
and
t
their left
We
=
we
=
will use vector
thus ADj{t)
follow the convention that Dj{0-)
all
dividends occur only in lumps, then Dj{t)
Dj{0)
is
=
payment
and that 5,(0-)
=
at
f
=
t
is
0,
both Dj{t)
Dj{t)
the change in price as the stock goes ex-dividend.
of no real use to us to have a dividend
that Py(O-)
to be included in the
bmits Dj{t-) and Sj{t-) are eissumed
by security j
continuous, the dividend rate D'At) exists for almost
specieil case, if
t
In technical terms, this says that for each j,
net dividend paid out at time
Sj{t)
price changes at a point in
and Dj{t)
=
0.
If
Dj{t)
is its integral.
-
Dj{t-),
To maintain
is
absolutely
As another
a random step function. Because
we
it is
eissume, without loss of genereJity,
5,(0), Vj.
notation for the price and dividend processes for the risky assets:
5
=
,Sm) and D^ = [Di, D2,
{Si,S2,-
to denote the trace of <ra^
|<t|'
Because we
,
.
when
will not consider
any taxes or other frictions
from holding one share of any security
gains.
We
deviation
will
ff(t).
assume that
this total
vector
is
for a period
change
our model, the chcinge
in
in
wealth
depends only on the sum of income and capital
and a
in value includes a local drift f (t)
standard
local
In Ito differential notation,
dS{t)
where dS{t)
will use
a matrix or a vector.
is
cr
where ^ denotes the transpose. Also, we
Df^^),
,
.
.
+
dD{t)
the local capital gain and dD[t)
random process and a
is
an
Mx
=
i{t)dt
is
the local income. Note that
+
a{t)dw{t),
(2.1)
an Af -dimensional
f is
matrix random process. (In other words, a embodies
A^
the local variances and covariances.) For (2.1) to be well-defined
we assume
that
-1
<
\dD{t)\dt
00,
(2.2)
/„
1
\i{t)\dt
/
<
oc,
(2.3)
'0
and
1
2,
\a{t)\^dt
L
<
00.
(2.4)
'0
(See Liptser and Shiryayev [1977, Chapter
4].
are implicitly P-a.s.) For sake of generality,
Recall our convention that statements such as these
we
are intentionally avoiding the
common
convention
of stating changes in asset values in terms of return per dollar invested, thus allowing risk
expected change
be nonzero even when the value
in value to
include contracts having unlimited
The consumption
We
set
admits
fairly arbitrary
model cumulative consumption, that
statements we
AC(t)
=
C{t)
made about
is,
C(O-) =
will take
0, i.e.,
is
C{t)
0,
is
C=
=
for
t
<
1
and therefore we can make
lump
the dividend at
not necessarily zero.)
and C(l)
is
structure on the consumption set, but for
we
obtain
in this section for this
G
[0,1]} to
of
consumption at time
is
assumed to be
Our convention
is
now we
will leave
it
set will also
[0,(].
parallel statements to the
0,
t is
we
given by
will allow
consistent with continuous
the terminal consumption. In Section
consumption
{C{t)]t
the total consumption over the time interval
consumption, lumpy consumption, or any combination of the two. For example,
at the end, C{t)
allows us to
nonnegative patterns of consumption over time.
dividends. For example, the
C(0)
Our convention
and right-continuous process
— C{t-). (However, although
consumption at
zero.
liability.
will use a nonnegative, nondecreeising,
By convention, we
is
and
if all
3,
consumption
we
will
is
put more
at this level of generality.
The
be immediately valid
any subset,
for
results
by nature of the results. For example, sufficient conditions to preclude arbitrage opportunities are
still
sufficient
when
choice set
is
further constrained.
The
is
objects of choice
eire
portfolio strategies.
the vector of share holdings in the
processes that are Umits of
D
convention for
left
M+
1
A
portfolio strategy (a,^^)
We
securities.
In other words, 6{t)
t.
is
6[t)'^
dS{t) where both 9 and
predictable processes,
time
t
to time
t.
we
=
at
and
C
jump
different
from our
who
+
a{t)B{t)
are content to
are always zero.
-
S{t)
S{t-). This convention
can have jumps. Because the portfolio strategies are
e{t)^{S{t-)
be the portfolio held from an instant before
+ A5(t) + AD(0) - AC(0,
+ AD{t)) - AC(0,
0{t)^{S{t)
t,
less
(2.5)
payment
for
consumption
at
assume that dividends and consumption are never lumpy, S D,
are continuous (and therefore have the
AD, and AS
is
for defining the values of integrals
Ill]
the value of the portfoho plus the dividend received at
For readers
t.
,^m)
is
f
= a(OB(0 +
is
S
will interpret {a[t),9[t)) to
The wealth
W{t)
which
•
the risky portfoUo position across the
consistent with the convention used by Jacod [1979, ch.
/
•
•
be predictable random
9 to
continuous processes with right limits. This
dividend payment D{t) - D{t-) and corresponding stock price
of the form
(a, ^1,^2,
and S, because we want {a[t),6{t)) to be the portfolio position held across
the dividend payment at time
is
a and
take
=
same value
In this case, (2.5)
at t—
,
and
t,
identical to the
is
M-),
and the changes
budget constraint
AC,
Merton
in
(1971).
To ensure that the wealth process
/
\a{t)
is
well-defined,
we
will
assume that
B lt)r{t) + 9 {if g{t)\dt<
00
(2.6)
and
\9{t)^(7[t)\Ut
f
(We
will refer to these conditions after
<
00.
we write down the wealth process
the space of trading strategies satisfying these two conditions.
Royden
(1968, p. 114)),
H
is
(2.7)
By
in (2.8).)
Let
H
denote
the Minkowski Inequality (see
a linear space. Note that (2.2), (2.3), and (2.4) imply that
H
contains
the simple strategies^ considered by Heirrison and Kreps [1979].
all
Now we want to define
the budget constraint.
To avoid cumbersome
notation,
we again
sacrifice
complete generzJity by eissuming that there are no intermediate cash inflows. In discrete time, we
would start with some given amount of wealth
one period to the end of the next period
is
at
time zero. The change of wealth from the end of
equal to net gains plus net income, less any consumption.
Also, to preclude borrowing without repayment, terminal wealth cannot be negative.
'A
strategy
is
said to be simple
Formally, {a, #)
is
a simple trading strategy
*ni Vyni
n =
0,.
.
.
,S -
1
and
j
if it
is
bounded and changes
if
its vjilues
there exist time points
= 1,...,M,
at a finite
= to<'i <
such that i„ and yj„ are
number
=
of nonstochastic time points.
and bounded random variables
measurable with respect to 7(„ and a{t) = i„ and
<
t/^
1
The continuous time budget
initial
wealth
Wq >
level
constraint
is
given by analogous conditions.
Recall from (2.5) that the wesilth at time
0.
value of the portfolio plus the value of the dividends at
t
>
0,
t,
t,
We
start with a given
denoted by W{t),
minus the consumption
at
t.
is
the
For
all
the change in wealth over the interval dt must equeil to the change in value of investments
inclusive of income, less
=
(M'{t)
Because C((>-)
=
^^(0
0,
consumption withdrawal:
a{t)B{t)r{t)dt
= ^^(0-)+
is
is
+
a{s)B{s)r{8)ds+ f ${8)^ {^{s)ds
/
t
>
0.
(2.8)
is
<T{8)dw{8))
-
C{t).
(2.9)
Jo
a natural budget constraint and
Besides (2.8), an agent
+ a{t)dw{t))-dC{t)
e{ty{^{t)dt
the integral form of (2.8)
^0
Relation (2.8)
+
constrained by
initial
is
often referred to as the self-Bn&ncing constraint.
wealth, and nonnegative fined wealth (or else the
agent could borrow without repayment):
We
will say that the
define
W{t) by
Wq
if
Note that
(2.5).
(2.10)
W^(1)>0.
(2.11)
C=
consumption plan
{a, 9) with initial wealth
W{0-) = Wo,
{C{t);t
e H, and
{a,0)
€
[0,1]} is financed
(2.10),
(2.8),
consumption
initial
wealth
by {a, 6) €
Kreps
plain
we
C
Wq =
0.
is
C(l) <
More
formally, an arbitrage opportunity
and Huang
and
[1985]
total
consumption C(l) >
showed that a
is
value
is
computed using the
alent to each other
is
if
consequence of our
An
sirbitrage opportunity
all
sufficient condition for
(Two
e
is
absence of arbitrage
in
the existence of an equivalent
assets are priced by expected present value,
rolled-over spot rate.
a
H given an
plan C financed
{oc,6]
a consumption
is
with a strictly positive probabihty.
the L^ limit of consumption patterns given by simple strategies
probability reeissignment under which
when we
oc.
nonzero and nonnegative and financed by some
H such that Wq =
[1981]
is,
give a definition for an arbitrage opportunity.
which
satisfied
Therefore, every consumption plan financed by a feasible
trading strategy has a finite cumulative consumption, that
this point
and (2.11) are
(2.8) (or equivalently (2.9)) is well-defined as a
conditions (2.6) and (2.7) defining H.
At
by the portfolio strategy
where present
probability measures are said to be equiv-
they have the same sets of probability zero.) This probabihty reassignment
referred to by Harrison aind Kreps as an equivalent mastingaJe measure, which
for the artificial risk neutraJ probabilities of
probabihty reassignment
Cox and Ross
[1976].
We
will
is
another term
always assume such a
exists.
AssiixnptioD 2.1. There exists a probability measure
S(i)
B[t)
_ ^.
Q
equivalent to
/^^^(O+^I^.
6
P
such that (Vf < s)
(2.12)
Equivalence of
which we
will call
P
r?,
and
Q
implies that the
strictly positive.
is
1
=
f
Radon-Nikodym
Because
Q
the expectation of
is finite
r)
and
is
equal to
r,{t)
This process
Remark
a martingale under
is
2.1. Since
P
surely with respect to
our convention that
and
P
all
Q
P
with
=
fj{0)
with respect to P,
dT,{u;)dP{oj)
m,
=
i.e.,
Q
a probability,
is
dQ(u)^ f
derivative of
1.
We
can therefore define a process
E[r,\7t].
=
1.
have the same probability zero
statements that are true almost
sets,
must be true almost surely with respect
Q, and vice versa. Therefore,
to
statements are almost surely-P means equivalently that
all
statements are
almost surely-Q.
We
will first record
Proposition
2.1.
some implications
There exists an
of
Assumption
N -dimensional
2.1.
standard Brownian motion
tv'
under
Q
such
that
5(0
(In other words, if
we take the
cumulative dividends
is
Proof.
the product measure of
Because
{r){t);t
as numeraire,
we have
local price of taJcing on the risk in
^{t)
is
bond
riskless
that each stock price plus
a driftless ltd integral under Q.) Furthermore, dw'[t)
where k consistently gives the
wiiere v
^'-''^
^I!wr^'^='^'^^l!m'^'^'^-
B{t)
G
[0, l]}
iV-dimensional process {p[t)]t
is
€
-
P
r{t)S{t)
^
dw, that
u-a.e.,
-(T{t)K{t)
and the Lebesgue measure on
—
K[t)dt,
is,
[0, 1].
a martingale adapted to the Brownian fields F, there exists an
[0, 1]}
with
r\p{t)\ut
<oo
such that
=
dw{t)
(2.14)
Jo
ri{t)
=
r?(0)
+ / p{sydw{s)
Jo
t
G
[0, 1]
P
-
a.s.;
see Clark [1970]. Since
positive, Ito's
T][t) is strictly
dlnr){t)
lemma
= K{tydw{t)
implies that
- -\K{t)\Ut,
where
"">^iEquivalently,
,,(0
Girsanov's
Theorem then
=
exp
jy"
K(5)^du;(fi)
-
\K{s)\'ds^
\
.
I'
implies that
w/(f) =
- f K{s)ds
w{t)
Jo
is
an
A'^
-dimensional standard Brownian motion under Q.
Next, Ito's
By
definition of
lemma and
Q
the definition of w* imply that
in (2.12), the left side of (2.15)
is
a martingale under Q, and the integrals on the
right side are well-defined by our regularity conditions
Ito integral
is
a martingale only
if it
and the equivalence of
P
and Q. But an
has zero drift (Lipster and Shiryayev [1977]),^ and therefore
f(0-r(05(0 +
<7(0'c(t)=0,
i^-a.c,
and consequently
B{t)
Our second
result
^l!wr^'^='^'^^IoW)'-'^'^-
shows how expectations under the new measure E' price the consumption
plans that can be purchased
when
there
is
a nonnegative wealth constraint.
however, two technical lemmeis are included for completeness. The
context of the result that a local martingale that
Lemma
2.1.
is
bounded below
first is
is
Before
we proceed,
a specialization to our
a supermartingale.
Let the process y{t) be such that
dy{t)
=
4>{t)dz{t)
(2.16)
< oo
(2.17)
witii
/•I
Jo
'The
converse
is
false:
an ltd integral with zero
drift is a local
martingale, but
may
not be a martingale.
&nd
y{t) is uniformly
That
To prove
Proof.
for
t
any
<
<
t
s,
<
E{y{t)]
oo Vt
G
[0, l]
and
E[y{8)\It]
<
y{t) Vt
€
[0, 1].
supermartingale under P.
y{t) is a
is,
bounded below. Then
that y{t)
E[y{s)\7t]
is
<
we have
a supermartingale
!/(0-
^^
'^^^^
to
show that
prove here that E[t/(1)]
<
£^[y(t)]
y(0).
<
oo Vt
The proof
€
[O, l]
and
for arbitrary
s is similar.
Putting
Jo
let
r„
the
time that $(t)
first
By
stopping time.
= inf({l}U{t€[0,l]:$(t)>n}),
greater than or equal to n, or
is
J^ds <
1
a.s.
In the (stochastic)
Liptser and Shiryayev [1977].
—
lemma
>
1 a.s.,
always smaller than n.
T,i is
a
'
'
time interval [0,T„] y{t)
is
a martingale under P; see
Thus
^[y(T„)]
As T„
is
oo,
'
>
$(f)
the hypothesis that
Jo
'0
we know Tn —
1 if
we know y(r„)
—
=
Vn.
y(0)
y(l) a.s. Since y{t)
is
(2.18)
bounded below uniformly across
t,
Fatou's
implies that
E{y{l)]
<
=
where the equality follows from
expectation
is
(2.18).
lim ^[y(r„)]
n—OO
y(o),
Hence y(l)
heis
a finite expectation under
P
and the
smaller than y(0).
I
It will
be convenient to work with a normalized and re-invested wealth process
B{t) as numeraire eind treats past consumption as
asset. Formally,
we
where having the lower Umit 0-
The
following
had been reinvested
which uses
in the locally riskless
define
W(t)
zero in the integral,
if it
v(t)
/"'
1
(in place of 0) indicates to
include the effect of any
jump
in
C
at
i.e.
Lemma tells
by risky investments and
us that v
is
is
a driftless Ito integral under
independent of the
"split"
the bond.
9
Q
and that v
is
determined solely
between consumption and the investment
in
Lemma
Let
2.2.
according to
(2.5).
C
be 6nanced by {a,
Then, for
all
We
H
6
Wq and
with &n initieJ wealth
let
W
{t)
be deBned
t,
v{t)
Proof.
6)
= ^0 +
£ e{sy^^dw'{s).
(2.21)
have that
Mt) = <i[Z^)^''^^'^
5(0'
B{t)J
(by definition of
xj)
':mt)~mm)''''^'^
B{t)
B[iY
B{t)
'
•
^
'
lemma)
(by Ito's
{a{t)B{t)r{t)dt
+ e{ty
B[t)
a{t)B{t)
+
e{ty{S{t)
(f (t)dt
+
- dC{t))
(j{t)dw{t))
+ ADjt)) - AC(t)
dC{t)
(by definition of B{t), (2.5) and (2.8))
1
--.^T
e{t)' {{^{t)
- r{t)S{t))dt + a{t)dw{t))
B{t)
(by cancellation and deletion of terms like
AD[t)dt that integrate to
zero)
= ^(0^|||d-'(0
(by Proposition 2.1). Because t;(0-)
(2.22)
= W{0-)/B{0-) = Wq,
(2.21) follows directly
from
(2.22).
I
Lemma
2.1
and
2.2 tells us the local behavior of v under the
2.2 together
Lemma
Proof. Because
consumption
Therefore, the result
tion
under
if the
Q
is
2.2.
is
is
immediate from Lemmas
2.1
and
C
is
Lemmas
a supermartingaie under Q.
is
also nonnegative.
2.2.
Given the nonnegative wealth constraint W{t) >
of the total discounted value of consumption
consumption plsm
0, v is
nonnegative, nonnegative wealth implies that v
(asset pricing)
(2.8-2.11).
a supermartingale.
Under the Donnegative wealth constraint W{t) >
2.3.
Proposition
imply that "globally" v
budget constraints
Rnanced by {a,0) with
is
initial
no larger than
the expecta-
initial wealth.
Formally,
weedtb Wq,
^•[£;^dC(01<W^o.
10
0,
(2.23)
Proof.
We
have that
E*
< E'Hi)]
C si^^^'"
(by definition of v and nonnegativity of W(l)/B(l))
<
by
Lemma
2.3, definition of v,
and
=
t;(Q-)
Wo,
(2.10).
I
The
fact that (2.23) contains
an inequality
is
due to the possibility of "suicidal strategies" that
throw away money, such as running a doubling strategy
For practical economic purposes,
all
in reverse (Harrison
and Pliska
[1981]).
strategies having a strict inequality in (2.23) can be ignored
(provided that preferences are strictly increasing). This
the justification for referring to Propo-
is
sition 2.2 as a pricing result (in spite of the inequality).
The
following theorem
is
the main result of this section.
wealth precludes arbitrage even
An
implication of this theorem
process that
Theorem
is
shows that a lower bound on
absence of any other restrictions on the trading strategies.
in the
is
It
that an arbitrage opportunity must have a corresponding wealth
unbounded from below.
2.1.
(absence of arbitrage) Given the nonnegative wealth constraint W[t)
>
0,
no
arbitrage opportunities exist.
Proof.
Assume
to the contrary that there
is ein
arbitrage opportunity
C
financed by [a, 6)
G H.
Proposition 2.2 shows that
E*
because the
initial cost of
not identically zero.
By
an arbitrage opportunity
the assumption that B{t)
^'
<o,
L 4^<^<"
is
>
Next note that C(l)
zero.
0,
we know
[£ wf"^'^]
(2.24)
^
is
nonnegative and
that
"•
a contradiction to (2.24).
The
traditional definition of arbitrage
model. Implicit
in the
notion of arbitrage
is
we
are using
is
not the only possible choice for our
the idea that
it
can be undertaken at arbitrary
In models like ours in which the set of feasible consumption streams
the nonnegative wealth constraint), the availability of a net trade
and the
scale of the net trade.
is
not a linear space (because of
may depend on
Because the nonnegative wealth constraint
11
scale.
is
the starting point
scale independent,
our sort of ao-bitrage opportunities could
may be
other net trades that could be
(if
available) be undertaken at any scale.
made
at arbitrary scale starting
from some strategy (such
as holding the bond) that one would want to label as arbitrage opportunities.
we would have
case,
arbitrage would
still
a different definition of arbitrage.
work
However, there
that were the
If
Fortunately, our proof of the absence of
in these cases, since the pricing relation implies that increasing scale
would eventually violate the budget constraint.
To
relate Proposition 2.2
and Huang
and Theorem
[1985] have
and
Duflfie
The
integrability condition
some p E
is
Before
and Theorem
we proceed, we record
Letnraa 2.4. Let
2.1.
Furthermore, pricing as
in
(2.25)
Proposition 2.2
some p €
6 satisfy the integrability condition (2.25) for
an U'-martingale under Q, that
E'
"'^'
is, it is
dw*{s)
BW
is
f\{s)
[l,oo).
(€[0,1]
a martingale under
with equality.
dw' [s)
Q
<
Tien
(2.26)
with
oo
(2.27)
B{s)
Jo
for all
oo,
a technical lemma.
L
is
ffl^.n<
This condition can be used instead of the non-negative wealth constraint
[l,oo).
in Proposition 2.2
Harrison and PUska [1981]
used an integrability condition to rule out doubhng strategies.
.-,(/;
for
2.1 to previous literature,
t.
Proof. See Jacod [1979].
Theorem
some p G
That
is,
2.2. Let
[l,oo).
be financed by
(a,tf)
the discounted value of consumption
=
0.
Furthermore, there
The proof essentially
the process
t;
is
H
€
satisfying the integrability condition (2.25) for
Then
an equality ifW{l)
Proof.
C
is less
is
no
than the
initial wealth.
arbitrage.
follows the proofs of Proposition 2.2
and Theorem
an Z^-martingale under Q. Thus
W{1)
E'
B{
•1
hLwM='''12
The inequality becomes
2.1.
By Lemma
2.3,
By
(2.11)
and
proof of no arbitrage
is
Theorem
identical to that of
theorem instead of Proposition
3.
2.1,
except that
£issertion
we
use the
is
obvious.
first
The
pait of this
2.2.
Normegative Wealth and Feasible Consumption Streams
In Section 2,
condition
we showed that a nonnegative wealth
is
condition. For p
G
[l,oo),
we
consumption plans
set of feasible
an equiveilent martingale measure.
is
we show that
essentially the seune
wM
in a
under either
C
such that
1^1
1
I
economically motivated while
is
denote the space of consumption plans
will use L''{Q) to
•1
<
oo.
(3.1)
consider separately the special case with complete markets and the general case in which
may be
incomplete.
because the result
much
is
economically. In this section,
difficult to interpret
is
wide variety of situations, the
markets
there
interesting because the nonnegative wealth constraint
the integrebility condition
We
constraint can substitute for an integrability
when
a proof of the absence of arbitrage
in
This result
is
The second
positivity of B[l), the first assertion follows.
for
We
provide a separate proof for the special case of complete markets
complete markets
slightly stronger
is
simpler than the general proof. Here
Lemma
Tie measure
3.1.
unique and
let
(2.25) for the
C E
U'{Q)
same p
Q
for
is
unique
if
some p €
that finances
C
than the general result and the proof
a characterization of
is
and only
Tien tiere
(l,oo).
with W^(l)
if o[t) is
=
and an
when markets
are complete.
Suppose
nonsingul&r u-a.e.
exists {a, 6)
E
H
Q
is
with 9 satisfying
initial cost
(3.2)
The wealth process
for this strategy
is
W{t)/B{t)^E'^l'~-^dC{s)\Tt
Proof.
to P.
K
is
A
measure
From
Q
is
completely determined by
the proof of Proposition 2.1
uniquely determined
if
and only
if
K{t)
Next
let
C
e f^{Q)
[1979]), there exists
for
some p G
we can
a[t)
is
its
Random-Nikodym
see that
fj
nonsingular
in
turn
is
i>-a.e. In
derivative
determined by
r]
k.
with respect
From
(2.14),
such event,
= -a{t)-'{m-r{t)S{t)).
(l,oo).
By
the meurtingale representation theorem (see Jacod
an A'^-dimensionaJ process ^(t) with
E'
(3.3)
.
C\m?dt
Jo
13
<
oo
such that
C ^"^c =
Now
''
IC
iio^^">]
- i'*(«)^^"-w
define
It is easily verified
Now
set
that 9 satisfies the integrability condition.
W(l) =
Lemnnas
0.
2.2
and
2.3 imply that
^(0
= ''[l!w)'''^'^^\
B{t)
That
is,
wealth at any time
t is
equal to the expected present value (using
defining a(t) by inverting (2.6),
initial
it is
for discounting).
C
then straightforweird to verify that {a, 9) finances
Now,
with an
wealth
''
and a
B
final
—
wealth W[l)
^From Lemma
3.1
we
''
^
II 4)'^'"
0.
learn that any
C
€
I/'{Q),
when
strategy satisfying the p-integrability condition
I
< p <
oo,
is
finainced
by some trading
there exists a unique equivalent martingale
measure. In such event, we say that the markets are complete. Here
is
our equivalence theorem for
complete markets.
Theorem
3.1.
(equivalence of nonnegative wealth and the integrability condition
kets are complete) Fix p
L^iQ)
Proof.
is
the
We
€
(l,oo)
and
initial
This can be seen from
and nondecreasing, the left-hand side of
in
0)
and the integrability condition
(2.25).
claim that any strategy satisfying the integrability condition satisfies the nonneg-
ative wealth constraint.
plan
wealth Wq. The set of feasible consumption plans in
same under nonnegative wealth (W{t) >
first
when the mar-
(3.2)
(3.2).
Since a consumption plan
is
nonnegative
must be nonnegative. Therefore any consumption
L^{Q) financed by a strategy satisfying a p-integrability condition must be available with
C
E I^iQ) be financed by
a nonnegative wealth constraint. Conversely,
let
nonnegative wealth constraint with an
wealth Wq. Suppose
initial
Wo<Wo =
Lemma
3.1
finances
C
E'
shows that there exists a strategy
with an
initial
wealth
Wq tmd
i'
first
{a, 9)
6
14
satisfying a
that
4^^'"
{oi,9) satisfying the p-integrability
a zero
H
final
wealth.
condition that
Using the same risky investment
strategy 6 and a position in the locally riskless
consumption plan
C
with a
This strategy certainly
Then Lemma
finances
C
Remark
3.1
with
+ Wq — Wq
p-integrabihty condition and finances
shows that there
W {\) =
a(t)
can finance the given
wealth equal to
satisfies the
C
Next,
if
Wq =
IVq-
exists a strategy satisfying the p-integrability condition that
0.
When p =
3.1.
final
bond equal to
Lemma
1,
3.1 fails
because there exist consumption plans
are not financed by strategies satisfying the integrability condition.
It is still
in
true that
L^{Q) that
all
feasible
strategies satisfying the integrabiUty condition also satisfy the nonnegative wealth constraint,
and
Theorem
3.2.
the two consumption sets are the same up to closure, as will be demonstrated in
Therefore, the consumption set for nonnegative wealth
a way that
may
Theorem
solve
some
3.1 says that
is
larger than the set
under integrabiUty in
closure problems.
when markets
are complete, any consumption plan in
p E (1,00) financed by a strategy satisfying the nonnegative wealth constraint
strategy satisfying the p-integrability condition that possibly throws away
final
wealth.
some
I^(Q)
is
for
some
financed by a
strictly positive
Conversely, a strategy satisfying the p-integrability condition must also satisfy the
nonnegative constraint.
The second equivalence theorem
In this case
we have been unable
to
deals with the case where the markets
show that the two
be the same - we just don't know). Our result
sets are the
may
not be complete.
same (although they may always
that the two sets are the seime up to closure,
is
with the consumption set under nonnegative wealth no smaller than the consumption set under
the integrabiUty condition. Therefore, as in the case p
that the integrabiUty condition
is
=
1 for
at least as attractive, since
complete mcirkets, we can only say
it
has some potential of solving (or
partially solving) a closure problem.
Theorem
may
in
3.2.
fequivaience of nonnegative weaJtb and the integrabiUty condition
not be complete) Fix p G [l,c») and
I^{Q)
(2.25),
Proof.
is
initiid
wealth Wq. The set of feasible consumption plans
no smaller under nonnegative wealth (W{t) > 0) than under the
and up
to closure the
The proof that any
when markets
integrability condition
two sets are the same.
strategy satisfying a p-integrabiUty condition must also satisfy the
nonnegative wealth constraint
is
the
same
as in
15
Theorem
3.1.
So we need to prove that any
consumption plan
L^{Q) financed by a strategy satisfying the nonnegative wealth constraint
in
in the closure of the set of
is
consumption plans financed by strategies satisfying the p-integrability
condition.
If
we can show
that v(l)
is
essentially uniformly smaJler across states for the nonnegative
wealth strategy than for some strategy satisfying the integrabihty condition, we will be done.
We
can simply follow the same consumption withdrawal strategy
both cases and the strategy
in
satisfying the integrabihty condition will have nonnegative residual wealth at the end. (Recall from
Lemma
Let
2.2 that v
X
(2.25) with
depends only on
6
zmd not the mix between
£in
a.)
W [l]
wealth Wq, imposing the budget constraint including
initial
easily verified that JV
is
an
affine subsp2w;e of
U'{Q,T,Q)
>
t;(l)
in the
or
W{t) >
>
(which
for
norm topology and
<
t
is
1.
show that
0(1)
G cl{X — L^{Q,T,Q)), where
cl
indicates the closure and
denotes the positive orthant of Z/(n, J,P). This follows because
the set of feasible v(l) under integrability once
we allow
{X -
L^^{n,
7 ,Q))
is
It is
therefore
Fix some 0(1) generated by a strategy satisfying the nonnegative wealth constraint.
suffices to
is
and
be the space of v(l) generated by strategies satisfying the p-integrability condition
(2.11^, but without imposing any further requirement that
convex.
C
It
L^{n,T,P)
f] L''+{n,
for the free disposal of residual
I ,Q)
wealth
at the end.
To show
that 0(1)
6 cl[X — L^{Q,T ,Q)), we
use a proof by contradiction: suppose not. In
our proof by contradiction we are supposing that 0(1)
Because cl[X — L^{Cl, T,Q))
is
is
not an element of cl{X
— L^(n, J,Q)).
closed and convex, and because a set containing a single point
compact and convex, a standard separation theorem (Rudin
there exists a linear functional A
€ ^'(n, 7,Q) (where l/p+
Theorem
[1973],
1/g
=
1)
3.4(b)) tells us that
such that
E'[Xv{l)]>E'[X{v{l)-S)]
for all v(l)
e
X
and
8
e L^(n,7,Q).
Clearly, A
i/+(n, 7,(5)), and X
^
subspace,
must be the same constant
£'*[At;(l)]
{oT else (3.4) could not
>
have a
K
(3.4)
(or else (3.4) could not hold for all 6
strict inequality). Also, since
for all v(l)
almost-sure limit of elements of X, and therefore £?'[AO(l)]
<
G X. As we show
A"
is
X
is
an
shortly, 0(1)
G
affine
is
the
by Fatou's lemma, contradictiong
(3.4).
We
have yet to prove that 0(1)
is
the almost-sure Umit of elements of
strategy that corresponds to 0(1). Define
;(0
_
^ //-'i^wMfll!,
Jo
and
T„
= inf({l}U{<GlO,l]:S(t)>n}).
16
X. Let
be the risky
=
Define v„(w,t)
same
the
v[u/,t
A
Tn{ijj)).
(The symbol A indicates the minimum.) In other words,
t;„
as v until the cumulative variance of r, in units of the locally riskless bond, equals n,
constant thereafter. Each Vn corresponds to the same portfolio strategy as
t)(l) until
is
and
the integral
defining
E
reaches n, and corresponds to holding only the locally riskless bond afterwards. But this
integral
is
the
same
as the integral in the integrability condition (2.25),
for the strategy underlying v„ (since the integral
But
II(l)
<
oo
is
is
bounded uniformly by
implied by (2.7) and the fact that B[t)
«„(!) converges to v(l) almost surely. But each i'n(l)
that 0(1)
is
the almost-sure Umit of elements of
which
is
is
n).
therefore satisfied
is
Therefore Vn(l) G X.
uniformly bounded below. Therefore,
an element of X. Therefore we have shown
X, which
is all
we had
left
to show.
I
The conclusion
sumption
sets
consumption
of
Theorem
3.2 is
weaker than that of Theorem
p >
1 is
set
under nonnegative wealth
that while
X- L\{n,7,Q)
4.
It
says that the con-
under nonnegative wealth and integrabiUty are the same up to closure and that the
is
no smaller. The case p
treatment because the closure problem now exists for
for
3.1.
is
X
is
closed and free disposal
all p.
In
=
1
no longer requires
special
terms of the proof, the problem
— L^(n, 7 ,Q)
is
closed,
we do not know
that
closed.
Concluding remarks
In the
dynamic choice problems considered above, the commodity spaces are
martingale measure. This
is
unnatural.
We
specified
under the
demonstrated that solution sets under the nonnegative
wealth constraint and under the integrability condition coincide. But we did not deal with whether
the solution sets are empty.
for an extensive
We
refer readers to
treatment of these two
The nonnegative wealth
constraint
Cox and Huang
[1987a, 1987b]
and Pages
[1987]
issues.
is
more
intuitively understandable than the integrability
condition. They, however, are functionally indistinguishable for a leirge class of economic problems!
17
References
1.
F. Black
and M. Scholes, The pricing of options and corporate
Economy
2. J.
Clark,
liabilities,
Journal of Political
81, 1973.
The
representation of functionals of Brownian motion by stochastic integrals, AnnaJs
of Mathematical Statistics 41, 1970, 1282-1295.
3. J.
Cox and
C. Huang,
A
variational
problem arising
in finemcial
economics,
MIT mimeo,
last
revised November, 1987a.
4. J.
Cox and C. Huang, Optimal consumption and
diffusion process, 1987b, to appear in Journal of
5. J.
Cox, and
S.
Ross,
A
Survey of Some
New
portfolio policies
when
asset prices follow a
Economic Theory.
Results in Financial Option Pricing Theory,
Journai of Finance 21, 1976, 383-402
6. J.
Cox,
cial
7.
S.
Ross, and
Economics
7,
M. Rubinstein, Option
Pricing:
A
Simplified Approaich, Journal of Finan-
1979, 229-263.
D. Duffie and C. Huang, Implementing Arrow-Debreu equilibria by continuous trading of few
long-lived securities, Econometrica, 1985.
8. P.
Dybvig,
A
positive wealth constraint precludes arbitrage in the Black-Scholes model, Prince-
ton mimeo, 1980.
9.
M. Harrison and D. Kreps, Martingales and multiperiod
securities markets, JourrnJ of Eco-
nomic Theory 20, 1979, pp. 381-408.
10.
M. Harrison and
Martingales and stochastic integrals in the theory of continuous
S. Pliska,
trading. Stochastic Processes
11.
and Their Applications 11, 1981, pp. 215-260.
D. Heath and R. Jarrow, Arbitrage, Continuous Trading, and Margin Requirements, Journal
of Finance, forthcoming 1987.
12. C.
Huang, Information structure and viable price systems. Journal of Mathematical Fkonomics
14, 1985.
13. J.
Jacod, Calcul Stochastique
et
Problems de Martingales, Lecture Notes
in
Mathematics 714,
Springer- Verlag, 1979.
14.
M. Lathem, DoubUng
strategies
and the nonnegative wealth
constreiint,
mimeo. University of
California at Berkeley, 1984.
15. R. Liptser
New
16. R.
and A. Shiryayev,
Statistics of Random Processes:
General Theory, Springer- Verlag,
York, 1977.
Merton,
Optimum consumption and
Economic Theory
3, 1971,
portfolio rules in a continuous time model, Journal of
373^13.
18
17.
H. Pages, Optimal portfolio policies
18.
H. Royden, Resd Analysis, 1968,
19.
W. Rudin,
when markets
New
Functional Analysis, 1973,
are incomplete,
MIT mimeo,
York: Macmillan Publishing Co.
New
York: McGraw-Hill.
19
1987.
k5 3 h
\
85
Date Due
FEB 04
1990
Lib-26-r)7
MIT LIBRARIES
3
TDflD
DDS 13D T3E