CI
,Ai
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
DOUBLING STRATEGIES, LIMITED LIABILITY,
AND THE DEFINITION OF CONTINUOUS TIME
Mark Latham
WP 1131-80
June 1980
revised August 1981
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
DOUBLING STRATEGIES, LIMITED LIABILITY,
AND THE DEFINITION OF CONTINUOUS TIME
Mark Latham
WP 1131-80''
Support
from
the
Social
June 1980
revised August 1981
Sciences
and
Humanities
Canada and from M.I.T. is gratefully acknowledged.
C.
and
Merton
to
for
Council
of
My thanks to Robert
suggesting this problem and the approach to solving it,
Fischer
suggestions.
Research
Black,
Stewart
Myers,
and
Richard
Ruback
for
helpful
August 1981
DOUBLING STRATEGIES, LIMITED LIABILITY,
AND THE DEFINITION OF CONTINUOUS TIME
Mark Latham
ABSTRACT
The
used
assumptions
yield
arbitrage profits to
underlies
amends
these
preserving
assumptions
rule
to
Black-Scholes
the
out
result.
This
postulating limited liability for investors,
valid
continuous-time
corresponding
discrete-time
,
1
can
the
relationship,
as
the
^
to
while
doubling,
be
paper
achieved
by requiring that
be
approaches zero.
r-i
or
by
must
relationship
This
technique.
arbitrage
seem
One such model
strategy.
valuation
option
Black-Scholes
the
"bet-doubling"
a
models
finance
continuous-time
many
in
limit
time
of
by
any
the
increment
I.
Introduction
What
assumptions
and
mean,
we
do
what
continuous-time
of
we
should
finance
mean,
models?
by
I
standard
the
am
basic
referring
the
to
following assumptions:
1.
Continuous
—
time
Time
is
indexed
the
by
real
line
some
(or
subinterval thereof), any investor can trade at any point or points in
time, and markets clear at all times.
2.
—
Unrestricted borrowing at the market rate of interest
There is
no
limit on the amount that an investor can borrow, and the rate does not
increase with amount borrowed.
3.
Stochastic processes for stock prices that have ex ante uncertainty over
every subinterval of time, no matter how small.
4.
Price-takers
—
Each investor acts as if his actions have no effect
on
prices.
It
literally
may
they
'bet-doubling'
arbitrage
seem
quite
imply
a
strategy,
profits.
clear
what
Kreps
contradiction:
using one
Markets
cannot
assumptions
those
stock
and
clear
the
when
mean,
shows
[19793
riskless
there
is
that
arbitrage
opportunity, so the standard assumptions above contradict each other
are
not
"internally
consistent".
This
is
a
serious
a
yields
asset,
an
taken
but
—
matter,
contradictory assumptions tend to produce contradictory implications,
they
since
which
can
describe
possibly
not
consistency
internal
economy;
actual
an
is
Formal algebraic statement of assumptions and theorems can not
imperative.
only lessen the ambiguities of statement in English, but may also help
the
internal
proofs of
search for
rather,
taken in this paper;
Such
consistency.
approach
an
pursue the less ambitious goal of
I
in
not
is
amending
the assumptions to eliminate the arbitrage from doubling strategies.
The development of continuous-time techniques in
enlightening
many
results
could
that
not
derived
be
including Merton [1971] and Black and Scholes [1973].
the
claim
that
the
basic
assumptions
are
results of these models are not valid.
I
finance has led
discrete
in
time,
One might wonder
contradictory
implies
while
eliminating
those
that
do
not.
In
the
valid;
this paper amends the assumptions so as to preserve the results that
reasonable
if
that
believe the results are
to
seem
particular,
the
Black-Scholes (B-S) model for pricing options is shown to hold under the new
assumptions.
There
are
While
problem.
prohibit Kreps
'
approaches
possible
several
either
imposing
credit
arbitrage strategy,
I
to
limits
rectifying
or
discrete
doubling
the
trading
would
argue that these are unrealistic and
would not in general preserve the classical results.
Instead,
replace the
I
assumption of unlimited riskless borrowing by an institutional structure in
which
all
loans
have
a
limited
liability
investors
feature:
required to repay more than their total assets.
In
cannot
this setting the
be
Kreps
doubling strategy no longer produces arbitrage profits, but the B-S option
pricing
formula
problem
lies
in
still
holds.
defining
a
An
alternative
continuous-time
solution
economy
as
to
the
the
doubling
limit
of
a
sequence
of
discrete-time
economies,
as
trading
the
interval
approaches
Merton [1975] argues that the only valid continuous-time results
zero.
those that hold in the limit of discrete time.
The doubling strategy
this test, while the B-S formula is shown to hold in the limit.
are
fails
Finally,
the most sensible solution may be to combine this limiting definition
with
the limited-liability loan assumption.
Section II reviews the Black-Scholes economy and two derivations
Section III outlines the Kreps doubling strategy.
the B-S formula.
Section
IV proves the B-S formula in a world of limited-liability demand loans,
shows the flaw of doubling in such a world.
non-demand
using
section
section
In
V.
(limited-liability)
VI,
I
look
is
at
the
discrete-time economies with no limit on
formula in such
Section
VII
a
analyzed
and
rejected
continuous-time
liability;
a
that
the
B-S
formula
also
holds
proof of the
discrete-time economies with limited-liability loans.
the
in
in
limit
world is presented, and doubling is shown not to
shows
and
feasibility of doubling
The
loans
of
of
B-S
"work".
limit
of
Section VII gives
a
summary and conclusions.
II.
the Black-Scholes Economy
This
section
is
a
brief
review,
with
slight
arguments in Black and Scholes [1973] and Merton [1977].
the
simplest
highlight the
economy
possible
for
aspects of special
the
purposes
interest;
generalized to more complex economies.
no
of
the
main
will work
with
changes,
I
this
doubt
the
of
paper,
so
results
as
to
can
be
Let us assume:
1.
Capital markets operate (and clear) continuously.
2.
Capital markets are perfect
—
(a)
all investors are price-takers;
(b)
there are no taxes or transaction costs;
(c)
there are no restrictions on borrowing or selling short;
(d)
there are no indivisibilities.
3.
Investors prefer more to less.
4.
There is
riskless asset with
a
a
continuous yield
r
that
is
constant
through time.
5.
There
is
stock
a
paying
dividends,
no
whose
price
S
follows
the
geometric Brownian motion process
dS/S
where t is time,
Z
= ;jdt
+ adZ
is a standard Wiener process,
known
and p and 6 are
and constant through time.
6.
There is
a
European call option on the stock,
maturity date
T,
option
will
E,
and market price C.
Black and Scholes state that
the
with exercise price
depend
only on
"under
the
price
those assumptions,
of
the
variables that are taken to be known constants."
stock
(p.
and
641)
the value of
time
and
They did
on
not
prove this assertion, although their derivation of the B-S formula depends
on it.
Therefore,
also discuss
I
a
proof of the B-S result that does not
start by assuming it.
In the
buying
one
B-S derivation an instantaneously riskless hedge is formed by
call
selling
and
F^
atook.
of
aharesi
Using
stochastic
calculus, the rate of change in the hedge's value (in dollars per unit time)
is found to be
1
2
2
2^ ^ss^
—
with probability one
on investment,
(F -
no uncertainty.
F^S)r
,
^
*
This must equal the riskless return
giving the partial differential equation:
rFgS - rF . ^^FggS^ -
^
t
=
•
Solving this subject to the boundary conditions
F(S,t)
max[0,S-E]
=
F(0,t)
and
and a boundedness condition F/S ^
1
=
,
as S->oo
,
gives the B-S call option valuation formula:
F(S.t)
=
SN(d^) - Ee"''^^"^^N(d2)
where N(d) is the cumulative normal distribution function,
log(3/E) + (r+|<j^)(T-t)
For
d^
=
d2
=
,
d^
- a /T-t
future rei'erence,
and
.
let us denote
this
particular solution for F
(i.e.
the B-S formula) by G(S,t).
Merton [1977] shows that if
C
ever differs from G(S,t) then there will
This provides a proof of the formula
be an arbitrage opportunity.
assuming that the only relevant variables are S and
If C(0)
t.
without
G(S(0),0),
>
the arbitrage is achieved by selling one call, pocketing the difference C(0)
- G(S(0),0),
at
max[0,S-E]
and investing G(S(0),0) in such a way as to return
maturity
with
T
certainty,
paying
thus
off
the
call.
He
G(S(0),0) by continuously borrowing enough extra money to hold
shares
of
stock.
can
It
shown
be
that
this
portfolio is always worth G(S,t), even as S and
particular,
maturity it will be worth G(S,T)
at
G„(S(t),t)
strategy ensures
t
invests
that
his
change through time.
In
=
max[0,S-E],
which
must
equal the call's market value at maturity, C(T), no matter how wildly it may
So he will be able to pay off the
have been priced in the meantime.
that he sold at t
The
=
0.
Black-Scholes
option
pricing
fruitful in modern finance theory.
methodology has
these
skills,
extensive
and
the
analysis of bank
resting
structures
proven
remarkably
Among its various applications are
on
the
line
the
the measurement
valuation of different types of corporate liabilities,
market timing
call
commitments.
Black-Scholes
2/
foundation,
of
With
the
underlying assumptions deserve some careful attention.
Trouble in Paradise:
III.
the Problem of Doubling Strategies
The strong assumptions of the original Black-Scholes economy sheltered
the
analysis
rates,
from many possible
nonexistent
rates, and so forth.
variance
unknown
variance
stochastic
interest
real-world confusions:
rates,
jump
processes,
Then, armed with an understanding of what it takes
to
subsequent research has tackled these more
make the hedging argument work,
In this era of expanding frontiers, one
general problems with much success.
may be surprised to learn that back in the garden where it all began
there
the bet-doubling strategy outlined by
Kreps
is still a snake in the grass:
pp. 40-41]
[1979,
that
the
He
shows
a
from
which
we
shows
self-contradictory.
arbitrage
profits,
way
for
must
any
section
in
investor
conclude
to
are
above,
II,
make
limitless
market-clearing
that
is
Here is a version of his argument:
impossible.
Start
at
0.
=
t
investor
An
probability one, by the time t
this,
aasumptiooa
knowing
rate of return
x
and
<T,
;j,
>
r
r,
=
1,
going
is
make
to
$1
or
with
more,
with no investment of his own.
he first calculates a probability q
To
and
>
with probability at least q.
X
given in the Appendix.
He
then borrows and
being up
yields at least $1
until t
=
(A proof that such q and
There will be infinitely many possible
at t = 1/2,
4/
1/2 is at least q.
=
exist
choices.
buys enough stock so that by t
If his
=
previous losses and still be up
by
t
=
3/
)
asset
he borrows
and
3/4, with probability q, he will cover
any
a
dollar.
Again,
=
1/2,
if he wins he
shifts to
the riskless asset, while if he doesn't win he raises the stakes enough
net $1
is
investment
he invests the profits in the riskless
If he is not up a dollar or more at t
1.
x
at
invests enough money in the stock that the chance of
dollar or more at t
a
a
such that over any time interval of length less than or
equal to 1/2, the stock will yield a (continuously compounded) return of
least
do
7/8 with probability q,
once, so the chance of being up $1 by t
is certain to be up $1 by t =
1.
5/
and so on.
=
1
All he needs is to
- (1/2)" is
1
- (1-q)"
,
and
to
win
he
One's first impulse is to look for an incidence of cheating hidden
in
the above strategy, but it seems to break none of the rules of the game.
So
we are forced to conclude that, taken literally, the assumptions in
section
II are not mutually consistent.
One Solution:
IV.
Limited Liability Demand Loans
How should we amend the B-S assumptions?
Surely we do not want to
allow arbitrage profits to be achievable, by doubling or any other strategy.
What enables doubling to work in the B-S economy, and why would it not
in
practise?
unrestricted
"Continuous time yields the necessary opportunities to bet;
short
losses." [Kreps,
eliminate
points;
lacking."
the
work
p.
sales
41]
doubling
the
necessary
earlier
paper,
yields
In
an
problem
but, as Kreps [1979, p.
by
restricting
^31 admits,
resources
Harrison
trading
to
Kreps
and
to
cover
any
[1979]
discrete
time
"economic motivation for it
Even if we assume that no trading can occur at night when
is
the
organized markets are closed, the continuous opportunity for trading from 10
a.m.
to 4 p.m.
Think of
is enough to permit arbitrage by doubling.
a
gambler playing
on red and doubling if he loses).
a
doubling strategy at roulette (betting
$1
Aside from discrete time, one problem
he
will face is that the house will limit the size of his bet.
The parallel to
this in capital markets might be that if a doubler doubles too many
times,
his position would be so large that the "price-taker" assumption would
questionable.
This approach will not be pursued here.
gambler wants to finance his bets by borrowing.
In
But now suppose
practise,
be
our
prospective
lenders will ask what the money is to be used for, and will turn him down if
Doubling looks fine, even for lenders, except for that
he tells them.
chance of
one
string of losses long enough that the money to double again
a
is
not forthcoming.
This suggests another possible amendment:
borrowing.
Doubling would no longer 'work', but choosing some level for the
would
limit
credit
option prices.
leverage.
putting an upper limit on
be
arbitrary,
somewhat
If the constraint is binding,
Finally,
such
a
and
it
may
affect
equilibrium
investors may use options
we do not
limit would not be realistic:
for
find
that investors can borrow at r up to a ceiling on borrowing, but rather that
the interest rate charged depends on the term of the loan,
wealth,
and
what
he
plans
to
assumptions often prove useful,
limit that
is
do
I
with
the
money.
the
Although
investor's
unrealistic
shall propose an alternative to the credit
both more realistic and more appealing
in
its
analysis
and
results.
Append the following to the assumptions of section II:
7.
Each economic agent's wealth is always nonnegative and finite, so that
no contract can
require that an agent pay more than his total
assets
under any possible circumstances.
8.
All loans, short sales, and options are negotiated on a demand basis
—
either party may redeem the contract at any time without penalty.
In
the case of options
(American or European),
this
is
intended
that the option buyer can give the option back to the seller in
to
mean
return
10
for
sum equal to the market price of
a
no
similar option that has
a
default risk (for example, a covered call would have no default risk).
made
You may well ask how a contradictory set of assumptions can be
consistent
assumptions.
adding more
by
Well,
1
believe that
¥7
replaces an implicit assumption that negative wealth was allowed.
like
//7,
so that
it
Or it may
assumption
field had in mind an implicit
be that some researchers in this
fact
in
But in any case,
is merely being made explicit here.
the great appeal of this assumption is that negative wealth in fact has
no
realistic meaning.
You
may
also
whether
ask
limited-liability world
—
riskless
a
asset
wouldn't every loan have
a
exist
can
this
in
risk of default?
future
all else fails, the riskless asset can be defined as a contract for
delivery of $B cash that is secured by $B of cash sitting in
similarity to government debt should be apparent.)
sources of riskless assets.
a
Consider
a
But
If
vault.
a
there
are
(A
other
demand loan owed by an investor with
portfolio of stock and option positions, where the portfolio has
a
market
Since stock prices (and therefore put and
value greater than the loan owed.
call prices) have continuous sample paths with probability one, there is
no
risk of default on the loan, because if the portfolio value drops close
to
the amount owed, the loan can be called in.
instantaneously
called;
so
the
dropping
below
the
amount
proceeds
dominated
by
of
owed
riskless rate will be charged.
would lend to an investor with zero wealth
the
There is no chance of the value
his
investment),
investing directly in
because
whatever
before
the
loan
Also notice that
can
no-one
(unless he agreed to repay
such
the
a
all
would
be
planning
to
proposition
borrower was
be
11
invest in.
the new framework of assumptions
In
through
1
doubling
the
8,
strategy does not earn arbitrage profits, but the B-S option formula
holds.
First
economy,
it
I
will
prove
Black-Scholes
the
formula.
amended
this
In
will make a greater difference to the force and
still
form of
proof whether or not one starts by assuming that the call price is
the
known
a
function of only S and t.
unchanged.
proof
carries
over
almost
Unless the Black-Scholes partial differential equation
holds,
this
Given
assumption,
original
the
B-S
standard
any investor with positive wealth can make excess returns on the
stock,
of
portfolio
hedge
call
option,
riskless
and
More
asset.
specifically, denoting the call price function by F(S,t), if
rFgS - rF . l<y^F^^S^ . F^
=
b >
.
then the investor can buy n calls, sell short nF„ shares of stock, put
difference
the
in
asset,
riskless
chosen as large as he likes).
(unborrowed)
wealth,
make
and
$nb
In
other words,
this
is
an
The opposite investment strategy is followed if b
differential
equation
must
(n
can
be
This is in addition to any returns on his own
which he needed as collateral
with no default risk.
unit time
per
the
Also
hold.
the
<
in
order to short-sell
arbitrage
0.
opportunity.
So the B-S
boundary
partial
conditions
are
unchanged, and thus the B-S formula is valid in this economy.
If the call price is not assumed to be a function of only S and t, the
proof
is
price by C(t),
C(0)
buys
<
n
Denote
more difficult.
and an investor's
G(S(0),0),
calls,
and
sells
W(0)
>
short
0.
the
B-S
formula
by G(S(t),t),
(market-valued) wealth by W(t).
The call
nF2(S(0),0)
is
undervalued,
shares
of
stock,
so
the
and
the
call
Suppose
investor
puts
the
12
(including W(0))
remainder
continuously
position
in Merton
As
stock.
the
in
that
so
riskless
he
is
asset.
always
short
adjusts his
He
nF-(S(t),t)
can pay off the calls and still have money left over,
This is not
more undervalued?
a
but here there is a
problem for Merton, because his
can hold out until maturity, when he is sure he will be ahead.
case, if W(t)
for some
<
t
his position to maturity
This puts
I
a
—
even
investor
But in
before maturity, my investor will no longer
able to borrow (or sell short or sell calls),
of
investor
If a call can be undervalued, what is to prevent its becoming
hitch:
take.
shares
this is designed to ensure that the
[1977],
short
and thus could not
my
be
maintain
he would be out of the game.
limit on n,
the size of the position that he may
need to show that this limit is positive,
i.e.
safely
that there is
positive position that the investor could take and make excess profits
a
for
certain.
Lemma:
-E ^ G(S(t),t) - C(t) 4 E
for all t until maturity.
Proof:
(a)
S(t)-E ^ C(t) 4 S(t)
investors
could
collateral
for
limit,
sell
the
the
call,
call,
for all
buy
If
t.
the
C(t)
stock,
making arbitrage gains.
and
broke the upper limit,
put
If
it
up
the
broke
stock
the
lower
investors could sell the stock short, buy the call, and put up
call and $E as collateral for the short sale.
as
the
13
(b)
S(t)-E
<
G(S(t),t)
<:
S(t)
all
for
This
t.
is
a
well-known
property of the B-S formula.
(c)
F and C are trapped in
The lemma follows easily from (a) and (b).
moving interval of length
E,
and so can never be more than E apart.
If n < W(0)/{E-[G(S(0) ,0)-C(0)]
Theorem:
}
then W(t)
,
for all
>
t
until maturity.
The constraint on n implies that
Proof:
<
W(0) - En + n[G(3(0),0)-C(0)]
4 W(0) + n[G(S(0),0)-C(0)] - n[G(S(t) ,t)-C(t)
(by the Lemma)
]
^ {W(0) + n[G(S(0),0)-C(0)]}e'"^ - n[G(3(t) ,t)-C(t)
=
]
W(t)
(For the last line of the proof, recall that the investor starts with
skims off G(S(0) ,0)-C(0) immediately for every call he buys, is long
thenceforth,
portfolio,
and
short
ensures by continuous hedging that
in
stock
and
long
riskless
the
the
asset,
W(0),
n
remainder
always
calls
of his
has
value
-nG(S(t),t).)
The proof is similar for the case of an overvalued call, in which
limit on n would be W(0)/{E-[C(0)-G(S(0) ,0)
]
the
}.
Therefore any deviation from the B-S formula allows excess profits to
be made, and so the B-S formula must hold.
of position
an
investor
can
take,
the
Because of the limit on the size
above
is
more
in
the
spirit
of
a
14
dominance argument than an arbitrage argument.
If C(t)
>
G(S(t),t) then all
investors with positive wealth would be wanting to write the call;
So the call market
G(S(t),t) they would all want to buy the call.
only clear if C(t)
=
C(t)
if
could
G(S(t),t) for all t.
There remains the question of whether the Kreps doubling strategy can
produce
sure
gains
in
world of limited
a
doubler starts with wealth W(0)
>
liability.
Suppose
a
would-be
He may borrow as much as he wants, but
0.
if ever his net market position W(t)
becomes
the loan will be called,
will not be able to borrow any more, and so he will have lost
permanently.
The trouble is that no matter how small an amount he borrows at t
put
in
the
stock until
t
=
to
=
there is a positive probability that
1/2,
stock will drop enough to make W(t)
disaster doesn't strike him,
he
for some t
=
if he has
to
<
'double'
1/2.
And even if
the
that
several times in a
row
wealth
the stake will have risen so as to increase the risk of hitting zero
So arbitrage profits can not be made this way.
before the next double.
But that does not prove that there is no way to make excess profits by
The next section looks at some
some other type of doubling strategy.
of
these.
V.
Doubling With Non-Demand Loans
The
doubler
who
is
only
allowed
complain that this is an unfair handicap
is
such
that
if
only
he
weren't
to
—
borrow
using
demand
that the nature of his
interrupted
when
his
loans
may
strategy
wealth
drops
<
15
temporarily below zero, he would in the end pay off all his debts with
loans
and
section
I
conclude
introduction of such real-world
The
money left over.
of
lines
outline
an
permit
still
successful
structure
institutional
doubling
that
may yet
credit
All
fails.
with
effort
contracts as
doubling.
these
term
In
this
features,
but
wasted
not
is
some
though,
because the rules drawn up for this economy will be useful in sections
VI
and VII.
Replace assumption
8
of section IV by the following:
All kinds of financial contracts are allowed, provided that
8.
(a)
they satisfy the limited-liability assumption, #7;
(b)
they specify how each borrower will invest his assets (loans plus
wealth)
until
contract
the
expires,
problems in pricing contracts;
(c)
so
as
moral
avoid
to
own
hazard
and
all loans and lines of credit are of finite size.
The
idea
short-seller
or
is
that
option
if
there
writer)
will
is
a
not
chance
have
that
the
borrower
(or
make
the
enough wealth
to
prescribed payment, it is understood at the outset that he will only pay
much as he has;
and the contract is priced with that in mind.
Furthermore,
contracts may be written in which the borrower pledges an amount less
his
total
evolve
in
wealth.
a
This
relatively
institutional
unregulated
setting
seems
marketplace.
like
In
one
fact,
as
that
the
than
could
limited
16
explicit
feature
is
borrowing
through
corporations),
contracts
many
in
liability
and
is
implicit
that
in
observe
we
because
others
all
(e.g.
bankruptcy is allowed and slavery is not.
Notice that the proofs of the Black-Scholes formula given in section
IV apply here
also,
a
securities, not the nonexistsnije oC
Suppose
doubler
a
'demand'
They required the existence of
fortiori.
%^^W\l\aa-.
a®ftii^<»^'f\^
financed
his
stock
positions
by
term
maturity equal to the planned holding period for the stock.
loans
That
is,
=
he borrows on a promise to repay at t
=
1/2.
If he is not up $1
t =
1/2, he borrows on a promise to pay at t
=
3/4;
and so on.
can
not
t
be
put
out
of
the
by hitting
game
zero wealth
during
of
at
at
This way he
holding
a
period, but there is still a chance that at the end of any holding period he
will have to pay all his assets to his creditor.
In
such a case he
will
have zero wealth, and be unable to recoup his losses since no-one would then
lend to him.
Therefore riskless excess profits can not be made this way.
Suppose instead he negotiates
whole period from
t
=
to t
=
1
,
a
single line of credit of $B for
the
paying a fee of $f up front, and paying
the riskless rate on any amounts borrowed.
He then arranges his
with the aim of making $1 plus $f plus any interest paid.
'doubling'
Here the catch is
that he may suffer enough losses that the amount he needs to invest
would
After that he can not increase his
stock
involve borrowing more than $B.
holdings for more 'doubles', and so can not guarantee to be ahead by
t
=
1.
17
If a doubler could arrange an infinite line of credit, he could
arbitrage profits with Kreps
make
strategy while only borrowing finite amounts,
'
but he can set no sure upper limit in advance on his borrowing, and infinite
credit
lines
are
eliminated
assumption.
by
would
It
interesting
be
to
analyse what happens in the limit as B approaches infinity.
VI.
Limit of Discrete Time, No Limit on Liability
Quite
a
different approach to resolving the question of doubling is to
only allow as valid continuous-time economies those that have the property
of being the limit of
a
sequence of discrete-time economies, as the
interval h approaches zero.
lines.
This
preserving
turns
a
See Merton [1975] for a discussion along
eliminate
to
Black-Scholes
the
phenomenon as
out
result,
benefits
the
thus
doubling
of
identifying
singularity of the continuous-time extreme.
we do not assume that wealth is nonnegative.
a
trading
the
these
while
doubling
In this section
Although it is hard to imagine
meaning for negative wealth, we assume that investors have derived utility
of wealth functions defined on the entire real line.
We will set up a sequence of discrete-time economies with the aim
finding the price of
a
we want to price it at
European call in the continuous-time limit.
t
and it matures at t
=
from the set {1,2,4,8,16,...}
1.
,
=
T
>0.
Suppose
Where n is
drawn
define economy n by the assumptions:
Capital markets operate (and clear) only at the dicrete time points t
0,h,2h,3h,
.
. .
where h
=
of
T/n.
It
is
to be understood
in
what
=
follows
18
that for a given n the variable t takes on only these discrete values.
2.
Capital markets are perfect.
3.
Investors prefer more to less, and are risk-averse, throughout the range
of their utility functions.
4.
There is a riskless asset that returns $(1+rh) on an investment of
$1
for time interval h.
5.
There
is
a
stock paying
no
whose
dividends,
stock
price
Markov process such that given S(t), the mean of S(t+h) is
2
follows
S
a
S(t)[1+^h],
2
and the variance of S(t+h) is C [3(t)] h.
6.
There is
a
European call option on the stock,
with exercise price
E,
maturity date T, and market price C(t).
Merton
[forthcoming]
provides
assumptions
of
set
a
about
security
price dynamics in discrete time that are sufficient to make prices follow an
Its
process
in
the
continuous-time
Here
limit.
I
make
a
similar
but
stronger set of assumptions to give the continuous-limit process
dS/S
This
is
geometric
Black-Scholes,
and
= ;jdt
+
Brownian motion with
constant
drift
<?dZ
.
constant
rate
p.
simplicity, though the results do not require it.
^
variance
is
held
rate
O
2
,
constant
as
in
for
19
To
the
above
assumptions
six
economy
on
add
n,
the
following
cross-economy assumptions:
CI.
The parameters r,
[Notation:
C2.
Let e(t)
;a,
S(t) - E^_^{S(t)}.]
=
The distribution of e(t)
and
M <
OO
,
and T are the same in all economies.
E,
<J,
is
discrete,
and
both independent of n and t,
number of possible outcomes for e(t) is
there exist numbers m
< CX5
such that for each t
the
2
^^
m,
and e (t) ^ M for certain.
Merton invoices these to simplify the mathematics without imposing any
significant economic restrictions.
Choose some time
t
7/
with
Then e(t) may take on values d
in economy n.
%J
probabilities p.,
j
=
1
,2,3,
.
• .
.m.
For
a
fixed t and j, and increasing
n,
from
d. and p. can be thought of as functions of h, with h approaching zero
above.
Define the notation f(h)'>^h^ to mean that lim f(h)/h
is
a
nonzero
h-*0
real number.
Now assume:
C3.
For all t and
j,
d. and p. are sufficiently "well-behaved" functions of
q
f"i
that there exist numbers r. and q. such that d
J
J
.
J
^-^
h ^ and p
.
J
-^ h
•
^
•
20
C4.
For all t and
such that q
j
+ 2r
.
main
purpose
this
of
Merton shows that q
assumptions,
+ 2r
.
that
and
<
.
if
q
1
assume that r.
,
(The
1/2.
=
J
jump
out
rule
to
is
=
.
J
J
processes
in
limit.
the
is impossible as it would violate prior
1
.+
2r
.>
possible outcomes
some
for
1
j,
then those outcomes contribute nothing to the limiting continuous-time
stock price process.)
do not attempt to derive a formula for the price of the call option,
I
C(0), in any of the discrete-time economies.
it must converge
this sequence of prices is,
value,
G(S(0),0),
as
n
Rather,
—> oo
h-»0.
and
To
argue that
I
whatever
to the Black-Scholes
this,
do
assuming that it converges to some function of S and
t,
I
do
but
I
formula
start
not
by
assume that it
converges.
To simplify notation, let
o(h) stand for any function f(h) with the property lim f(h)/h
=
0;
h-»0
S(k)
=
stock price at
C(k)
=
option price at
N(k)
=
G2(S(k),k)
G(k)
=
G(S(k),k)
H(k)
=
value of hedge position at t
Suppose
C(0)
=
=
kh,
=
t
t
=
for k
=
0,1,2,...,n;
kh;
B-S hedge ratio at t
=
kh;
B-S formula call value at t
converges
to
a
=
=
kh;
and
than
G(S(0),0).
kh.
value
greater
Then
claim that in an economy with large enough n, arbitrage profits can be
by the following strategy:
8/
I
made
21
At t
and
pocket the difference
sell one call for C(0),
=
invest G(0)
in
composed of N(0)
stock portfolio
levered
a
C(0) - G(0),
stock and riskless borrowing of $[N(0)S(0)-G(0)
]
shares
of
to make up the difference.
So the portfolio's initial value H(0) will equal the B-S formula call
value
G(0), but will be less than the call's market price C(0).
Let D(t)
=
H(t) - G(t), the difference between the hedge value and the
So D(0)
B-S formula value.
that D(t)
by construction.
=
The next goal is to
show
for all t in the continuous-time limit.
=
The change in the hedge value from t
H(1) - H(0)
=
=
to t
=
h will be
N(0)[S(1)-S(0)] - rh[N(0)S(0)-H(0)].
At each stage (t=(k-1)h for k=2, 3,
.
. .
,n)
,
the hedge is revised so as to hold
N(k-1) shares of stock, financed by borrowing as necessary.
The change
in
the hedge value from t=(k-1)h to t=kh will be
H(k) -H(k-1)
Meanwhile
the
B-S
N(k-1)[S(k)-S(k-1)] - rh[N(k-1 )S(k-1 )-H(k-1)].
=
formula
also
value
call
changes.
Its
change
(1)
can
be
rewritten following the Taylor series argument in Merton [forthcoming]:
G(k)-G(k-1)
=
G2(k-1)[S(k)-S(k-1)]+ G^(k-1)h + |G22(k-1)[S(k)-S(k-1)]^ +o(h)
....(2)
Let u(k)
in
the
n-»oo,
sense
e(k)
g.
that
-
.y^.
whereas
This standardizes the driving random variable
.
the
outcomes of u(k)
the
Ej^_^{u(k)}
=
=
and Ej^_^{u^(k)}
S(k) - S(k-1)
=
outcomes
not,
do
=
1
of
e(k)
become
by assumption
for all n.
So
;jS(k-1)h + e(k)
= jLiS(k-1)h
+ oS(k-1)u(k)/h
.
C4
infinitesimal
above.
In
as
fact,
22
2
Then [S(k)-S(k-1
)]
=
2 2
2
(k-1 )u (k)h
aS
and equation (2) can be
+ o(h).
rewritten as
G(k) - G(k-1)
G^(k-1)[S(k)-S(k-1)] + G (k-1)h +
=
^^....(3)
^33(k-1)a^S^(k-1)u^(k-1)h + o(h)
- D(k-1)
Into D(k)
=
H(k)
- H(k-1)
-
[G(k)
- G(k-1)]
substitute
(1)
and (3) to give
D(k) -D(k-1)
=
N(k-1)[S(k)-S(k-1)] - rh[N(k-1 )S(k-1 )-H(k-1
) ]
G2(k-1)[S(k)-S(k-1)] -G^(k-1)h - -^^gCk-l )(7^S^(k-1 )u^(k)h + o(h)
recall
But
that
N(k-1)
=
G (k-1
)
,
and
that
Black-Scholes partial differential equation so that
(4)
satisfies
G
1
-
the
2 2
rG„S+G =rG - -^ S G„„.
Substituting these into (4) gives
D(k) -D(k-1)
=
rh[H(k-1)-G(k-1)] - ;^32(k-1)(J^S^(k-1)[u^(k)-1
]h + o(h)
....(5)
Defining y(k)
=
^^^(.k-^)a^S^(.ii-'\)[u^(.k)-^'\,
we have from the law of
n
numbers
large
martingales
for
that
lim
"*
h^y(k)
with
=
Consider the difference D(t) for some time t between
one.
D(t)
=D(0)
.
probability
k=l
and T:
f;tD(k)-D(k-1)]
k=1
where
K
=
nt/T
.
Substituting
from
(5)
taking
and
the
continuous-time
limit, this becomes
=
D(0) +
limV
""°°k =
K
K
K
D(t)
1
rh[H(k-1)-G(k-1)] -
limVy(k)h
""^k =
1
+
^limVoCh)
"^•"'kzl
23
K
limVrhDCk-D
+
k=1
t
/
rD(s)ds
.
3=
Solutions
D(0)
=
0,
to
this
integral
this means that D(t)
equation
satisfy
for all
=
D(t)
from
t
D(0)e
=
to T,
and
value H(t) will always equal the B-S formula call value G(t).
at t
=
T,
=
will
be
max[0,S(T)-E] so H(T)
the
hedge
finally
And
just
sufficient
to
max[0,S(T)-E] also -- the hedge
=
pay
off
on
the
call
were 'free'
=
—
at
arbitrage
is
achieved
by
buying
the
call
property
portfolio
maturity.
Thus
the
arbitrage.
By a similar argument, if the limit of C(0) as h -»
porfolio.
Since
but the Black-Scholes formula has the
profits that were pocketed at t
then
.
when the investor has to make good on the call he sold, he will
have to pay max[0,S(T)-E];
G(T)
rt
and
is less than G(0)
shorting
the
So it must be that the B-S formula gives the limit of the
hedge
call's
market price, as the trading interval approaches zero.
As
an
example of why doubling fails in the limit of discrete
imagine a doubler betting red on
.5),
a
time,
fair roulette wheel (probability of win
in a discrete-time world where negative wealth is allowed.
all his bets with riskless borrowing.
To make each bet a
=
He finances
fairly
priced
security, let us say that its outcome is independent of everything else, the
riskless rate is zero, and bets pay off at 2 to
to bet between t=0 and t=1,
1 .
He has n
opportunities
wants to be up $1 by t=1, and so he first bets
His net payoff as of t=1 will
$1
and keeps doubling until he wins once.
$1
with probability 1-(1/2)", or $(1-2") with probability (1/2)".
be
24
Now the parallel to Kreps
infinite,
so
that
payoff
the
'
doubling argument would be to next let
is
$1
with
probability
—
1
any
and
demand
But for any economy n, the doubling payoff only adds
for doubling is large.
to
be
disastrous
the
string of losses gets swept under the rug of zero probability,
noise
n
portfolio's
return,
and
so
would
not
undertaken
be
by
any
risk-averse investor [Rothschild and Stiglitz,
1970].
So in the sequence of
discrete-time
demand
for
economies
as
n
increases,
the
such
doubling
a
strategy is identically zero, and hence is zero in the limit.
Limit of Discrete Time, With Limit on Liability
VII.
Defining continuous time in terms of the limit of discrete time
is
very appealing in the way it seems to preserve results that make sense while
rejecting
those
requiring
wealth
imagine
realistic meaning for negative wealth.
methods
a
in
that
to
don't.
be
what may be
At
the
nonnegative
the
most
is
same
time,
desirable
the
other
because
it
approach
of
hard
to
is
This section combines both
reasonable way to resolve the
questions
addressed here.
To the six assumptions defining economy n in section VI,
append
the
following:
7.
Each economic agent's wealth is always nonnegative and finite,
no
contract can require that an agent pay more than his total
under any possible circumstances.
so that
assets
25
All kinds of financial contracts are allowed, provided that
8.
they specify how each borrower will
(a)
invest his
assets
until
the
contract expires;
(b)
all loans and lines of credit are of finite size;
(c)
transactions occur only at the time points
t
=
and
0,h,2h,...
The cross-economy assumptions CI, C2, C3, and C4 are also retained.
In this
setting,
a
riskless asset is much harder to come by.
discrete time interval, any loan on
a
portfolio with finite pledged
In
wealth
Note
and some stock-price risk will have a positive probability of default.
that
in
option-stock
an
hedge
portfolio
cases,
the contracts will
have
be priced
to
cannot
be
any short sale
of
stock-price
Furthermore,
completely hedged away in discrete time.
stock or writing of options will
the
default risk also.
risk
all
In
take account of the default
these
risk.
Define the default premium to be the difference between the prices today
a
given promise to pay with and without the default risk.
a
of
The price without
default risk can be imagined as the limit of the price as the pledged wealth
(collateral) approaches infinity or, better still, no-default securities can
be invented.
The price for a short sale is not a problem
the stock price if there is no default risk.
to
borrow
on
such
terms,
the
it should equal
A call writer can
not to default by putting up the stock as collateral.
riskless if backed by enough cash;
—
guarantee
And a loan can
although no private investor would
government
existence of riskless government debt.
might,
so
let
us
assume
be
want
the
26
The pursuit of
a
doubling strategy in this world would suffer from all
the handicaps of sections V and VI above, so it will not generate
arbitrage
showing
The remainder of this section is devoted to
profits here either.
that in spite of default problems in the discrete-time economies, the market
price of
limit.
call will approach the B-S formula value in the
a
continuous-time
of
My plan is to adapt the discrete portfolio adjustment argument
and to show that in a time interval of length h,
section VI,
default
each
probability is o(h), each default premium must be o(h), so that as h —>
effect
cumulative
of
these
probabilities
Just as in section IV,
disappears.
there will be
t
to
=
t
limit on the size
a
position that an investor may take to make money on
since all
from
premia
and
the
a
in
T
of
but
mispriced call,
investors would be wanting to transact on the call
=
same
the
direction, dominance would prevent the market from clearing until the price
is right.
Let
us
take
care
some
in
defining
discrete-hedging investor is going to invest in.
government
debt
(the
stock
is
not
a
our
When he buys stock, calls,
or bonds, let him do so no a no-default-risk basis:
or
sort of contracts
what
just
by buying covered calls
problem).
sells
When he
short,
writes calls, or borrows, there will be no way to avoid some default risk on
his part,
means
of
but
the
let the
following
risk be confined to one time unit per
arrangement.
Each
of
his
contract
liabilities
is
to
resettled after each time unit h (in the style of futures contracts).
is
nothing unusual
stock)
~
in
this
for
these are to be repaid next period.
option with some distant maturity date T,
I
There
of
have him write
an
fact will mean for him
to
sales
But when
in
be
(borrowings
short
borrowing and
by
I
contract to pay, at time h from now, the market price that that option
will
27
then command if it has no default risk from then on (e.g.
of
a
covered call).
options on
portfolio
any
that
price
This is the next best thing in discrete time to
continuous-time demand-basis assumption #8 on page
Consider
the market
stock,
and
of
short
and/or
9,
long
section IV.
positions
in
a
with positive pledged wealth
bonds,
In order for default to occur on any of the short positions,
~h
stock,
(positive
market value), where all short positions are to be resettled each
must move by an amount
the
period.
the stock price
in a time interval of length h.
By
assumption
C4 the probability of such an outcome is o(h).
Now the default premium can be thought of as the market value of
loan guarantee;
o(h),
the guarantor will only have to pay off with
in an amount that is bounded in magnitude.
of such a liability were larger than o(h).
such claims would give,
as
payoff of probability zero
h -> 0,
—
a
arbitrage.
probability
Suppose the market
value
Then rolling over a sequence
summed value greater than
a
of
zero for
So the default premium must
a
be
o(h).
The hedge strategy to be followed is the same as that in section
except that there is
sold,
a
VI,
limit on the number of calls that can be bought or
for the same reason as in section IV.
Here,
the change in the
hedge
value from time k-1 to time k, conditional on default not occurring, is
H(k) - H(k-1)
where
the
o(h)
=
N(k-1)[S(k)-S(k-1)] - rh[N(k-1 )S(k-1 )-H(k-1
term
includes any default premium.
With D(t)
before, we again get
K
D(t)
=
^lim2]rhD(k-1)
k=1
t
=
\
s=0
)]
rD(s)ds
+ o(h)
defined
as
28
conditional on there being no default between time
probability of such
a
default
bounded
is
which approaches zero as h
2_]o(h),
time t.
But
the
by something of the
above
—>0.
and
So we have
D(t)
form
D(0)e
=
with
k=1
probability one, the call position can be closed at maturity using only
from
proceeds
completes
the
the
portfolio,
hedge
dominance
argument
and
for
riskless
gains
are
Black-Scholes
the
the
This
made.
formula
this
in
setting.
Conclusions
VIII.
Kreps
The
doubling
strategy
raised
doubts
about
internal
the
consistency of the standard assumptions in continuous-time finance
Two ways of resolving the doubling problem were presented:
to
define
a
economy
continuous-time
as
the
limit
of
a
the
fails to
produce arbitrage
argument still obtains.
It
gains,
is
all three cases,
while
the
the doubling
of
These two
methods were combined in the last section, to form what seems to be
In
other
sequence
discrete-time economies, as the trading interval approaches zero.
reasonable model of the world.
make
one was to
explicit the natural requirement that wealth is never negative;
was
models.
a
more
strategy
Black-Scholes option
hoped that the amended assumptions
pricing
(any
the three sets) will provide a general partition of results that make
of
sense
from those that do not.
Furthermore, an explicit limited-liability requirement may prove to be
a
useful tool for applications well beyond those of the present paper,
example
p.
1122].
in
such
general
equilibrium
existence
proofs
as
in
Green
for
[1973,
=
29
FOOTNOTES
1.
Subscripts to functions denote partial derivatives, e.g.
2.
See
Black
Scholes
and
[19733,
Merton
F„
Henriksson
[1981],
=
dP/dS
Marten
and
[1981], and Hawkins [1982].
3.
Kreps'
statement that
"there
is
some positive probability q that over
any time interval the second security [stock] will strictly outperform
the
first
a
r
<
[riskless asset]"
then
any q
for
[1979,
40]
p.
there exists
>
a
not quite accurate.
is
time
interval T large
that the probability of the stock beating the riskless asset
than q.
If
enough
is
less
But this is not a problem for his doubler, who can put an upper
bound on the lengths of his holding periods.
4.
To calculate the amount that our "bet-doubler" should borrow and invest
in the stock each time,
let
n
=
the number of consecutive losses he has suffered so far;
W
=
his net winnings up to that time (may be negative);
T
=
(1/2)
=
and
the length of his next holding period.
To be up $1 with probability at least q at the end of the next
period, he borrows and invests $
—-W
1
e
=
.
Then if y(T)
>
x,
holding
his
net
- e
position will be
1
,,
W .
e
5.
A
simpler
,T
- W
- e
,
rT
yT
^^
rT,
)
.
>
,,
W .
1
a
strategy,
- W
,
-^f-^ie
e
bet-doubling
without reference to
- e
which
xT
- e
rT,
=
)
,
1
.
- e
can
be
defined
specific stochastic process,
is
and
analyzed
found by making
30
sell
short
a
enough
large
amount
holding-period return is less than
exceeds
q =
r;
fails
this
but
(but not quite) to set q
It almost works
use of short sales.
if
whenever
r,
the
and
the
to go
median
median
of
long when
equals
r.
1/2,
=
to
stock
the
median
the
Instead,
set
go long when the upper quartile exceeds r, and in the case when
1/4,
it does not exceed r go short an amount calculated with reference to the
lower quartile of the stock holding-period return.
This works unless
both quartiles equal r, a possibility which can be safely ignored.
6.
This sequence (powers of 2) is chosen for
capital
markets
open
are
in
n
economy n,
so that any time t for which
will
have
the
property
that
capital markets are open at that time in all economies of higher n.
7.
(a)
Merton is not explicit about
ra
and M being independent of n and
t,
but clearly intends it that way.
(b)
In
the following discussion,
if any e(t)
has less than m
possible
outcomes, imagine augmenting the number of outcomes until it equals m by
adding outcomes of zero probability.
8.
The argument that follows is based on proofs in Merton [1977],
and
[forthcoming].
The
preceding assumptions correspond to those
Merton [forthcoming] as follows:
guarantee Merton
his E.4.
9.
's
in
My assumption 5 is more than enough to
assumptions E.I,
E.2,
E.3,
E.5,
and E.6.
My C3
is
My C4 is his "Type I" partition.
Remember that the notation o(h)
statement
[1978],
o(h) + o(h)
=
o(h)
refers to
means
that
a
the
class of functions.
class
is
closed
The
under
31
addition.
10.
See the argument in Merton [forthcoming] leading to equation (32),
)
32
APPENDIX
Theorem
Given the stock price process
:
dS/S
^dt + adZ
=
probability
and the (constant) riskless rate of return r, there exist a
and
q >
length
rate of return
a
T
stock
the
1/2,
<:
x
such that over
r
>
will
yield
rate
a
any time
of
interval
least
at
(All rates of return are meant as
probability at least q.
of
with
x
continuously
compounded.
Proof
Let y denote the (random variable) rate of return on the stock
:
Then S(T)
in time T.
the
S(0),
=
S(0)e^^
=
2
-
p
<j
variance
=
/2
(T
X
=
a
:^log[s(T)/S(0)]
=
S(T)
of
implies
that
log S(0) + aT and variance
=
For
.
given
log S(T)
=
2
<T
Therefore y is distributed normally with mean
.
is
where
T,
a
=
and
/T.
Case
Choose
and y
distribution
lognormal
distributed normally with mean
a
,
and q
=
—
a > r
Then clearly for any T (including T ^ 1/2)
1/2.
probability that y ^
1
x
is
1/2 (^q), because
x
is the mean of y's
the
normal
distribution.
Case 2
Choose
X
=
r
+
0.001
and
q
=
—
a
Prob{y ^
.^
x
r
for T = 1/2}.
condition on q holds by the definition of q.
For
(Clearly q
>
T = 1/2
the
0.)
For
(Appendix cont'd)
33
smaller T, it is intuitively obvious that since
and
(a)
the mean of y is unchanged,
(b)
the variance of y is increased,
(c)
X
is greater than the mean of y,
it must be
true that
Pr{y ^ x}
increases,
and
so is
> q.
Here is
formal proof:
q = Pr{ y(1/2)
>^
x
}
^p^^. y(1/2)-a ..^jLjLJL}
Note that both have the standard normal distribution.
=
?r{( ^^V~^
<
Pr{
-
Pr{ y(T)
y^V"^
>,
'>.
>x
X
^^-^^-^
^S^
)
.
}
}
for all T,
for any T
including T
<
1/2
,
<
since
1/2
x-a
^^^
;
^
>
x-a
^,
a
34
REFERENCES
Black, F., and M. Scholes [1973],
"The Pricing of Options and Corporate
Liabilities", Journal of Political Economy
Green,
[1973],
R.
J.
"Temporary
General
,
Equilibrium
in
Trading Model With Spot and Futures Transactions",
vol.
Harrison,
41,
J.,
pp.
and
D.
Kreps
Securities
vol. 20, pp.
381-408.
G.
[1982],
D.
R.
D.,
and
Markets",
Journal
C.
Investment Performance.
,
Merton
II.
vol.
D.
III,
[1979],
no.
10,
"On
[1981],
Sequential
Econometrica
,
of
Arbitrage
Economic
in
Theory
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Agreements",
1.
Market
Timing
and
Statistical Procedures for Evaluating
Forecasting Skills", Journal of Business
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and
Analysis of Revolving Credit
"An
R.
"Martingales
[1979],
Journal of Financial Economics
Henriksson,
a
1103-1123.
Multiperiod
Hawkins,
637-659.
vol. 81, pp.
,
vol. 54, October.
"Three Essays on Capital Markets"
(especially essay
"Continuous Time Models"), Institute for Mathematical Studies
in the Social Sciences
University.
(IMSSS), Technical Report No. 261, Stanford
35
Merton,
R.
[1971],
C.
Continuous-Time
and
Model",
Journal
Economic
vol.
Time",
of
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Rules
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in
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a
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Finance
of
from
Perspective
the
of
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,
659-674.
10,
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C.
[1977],
R.
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[1975],
C.
R.
Continuous
Merton,
Consumption
373-413.
pp.
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"Optimum
"On
Modigliani-Miller
the
Pricing
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Contingent
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Journal
of
Claims
Financial
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the
Economics
,
vol. 5, pp. 241-249.
[1978], Seminar notes for "On the Mathematics and Economic
Merton, R. C.
Assumptions of Continuous-Time Models", unpublished manuscript.
Merton,
I.
R.
[1981],
C.
An
R.
,
vol. 54, pp.
[forthcoming],
C.
Assumptions
Essays
Market
Timing
Investment
and
Performance.
Equilibrium Theory of Value for Market Forecasts",
of Business
Merton,
"On
in
of
363-406.
"On
Continuous-Time
Honor
of
Journal
Paul
the
Mathematics
Models",
in
Cootner
,
and
Economic
Financial Economics:
W. F.
Sharpe
(ed.),
Prentice-Hall, Englewood Cliffs, New Jersey.
Rothschild,
M.,
and
J.
E.
Stiglitz
[1970],
Definition", Journal of Economic Theory
,
"Increasing
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vol.
225-243.
2,
pp.
I.
A
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