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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
DOUBLING STRATEGIES, LIMITED LIABILITY,
AND THE DEFINITION OF CONTINUOUS TIME
Mark Latham
June 1980
WP 1131-80
1
I
^
lb revised November 1981
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
DOUBLING STRATEGIES, LIMITED LIABILITY,
AND THE DEFINITION OF CONTINUOUS TIME
Mark Latham
June 1980
WP 1131-80.
|
98"
lb revised November 1981
Support from the Social Sciences and Humanities Research Council of Canada
and from M.I.T. is gratefully acknowledged.
My thanks to Robert C. Merton
for suggesting this problem and the approach to solving it, and to Fischer
Black, Robert Jarrow, Stewart Myers, and Richard Ruback for helpful suggest ions
The principal difference between this and the August 1981 revision is the
addition of a convergence proof on page 20.
to the middle paragraph of page 23,
footnote.
Consequent adjustments were made
to footnote 8,
and to this title page
August 1981
DOUBLING STRATEGIES, LIMITED LIABILITY,
AND THE DEFINITION OF CONTINUOUS TIME
Mark Latham
ABSTRACT
The
assumptions
yield
many
in
arbitrage profits to
underlies
amends
used
these
preserving
assumptions
"bet-doubling"
a
Black-Scholes
the
option
to
Black-Scholes
the
continuous-time
rule
out
result.
continuous-time
corresponding
relationship
discrete-time
approaches zero.
models
strategy.
valuation
This
or
by
can
seem
This
doubling,
be
paper
while
achieved
by requiring that
must
be
the
relationship,
as
the
to
One such model
technique.
arbitrage
postulating limited liability for investors,
valid
finance
limit
time
of
by
any
the
increment
I.
Introduction
What
assumptions
mean,
we
do
and
continuous-time
of
should
what
finance
we
mean,
by
models?
I
standard
the
basic
referring
am
the
to
following assumptions:
1.
Continuous
—
time
Time
is
indexed
by
the
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.
H.
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
contradiction:
using
Markets
one
cannot
assumptions
those
stock
Kreps
and
clear
the
when
mean,
shows
[1979]
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
for
search
proofs
internal
of
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
enlightening
many
results
that
could
techniques in finance has led
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
discrete
in
time,
One might wonder
contradictory
implies
while
eliminating
those
that
not.
do
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
'
possible
several
imposing
either
approaches
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,
I
replace the
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
while the B-S formula is shown to hold in the limit.
this test,
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.
using
(limited-liability)
non -demand
section
section
In
V.
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
changes,
arguments in Black and Scholes [1973] and Merton [1977].
the
simplest
economy
possible
for
highlight the aspects of special
the
purposes
interest;
generalized to more complex economies.
no
of
I
this
doubt
the
of
the
main
will work
with
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 les3.
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
is time,
t
Z
= jjdt
+ odZ
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
although their derivation of the B-S formula depends
prove this assertion,
Therefore,
on it.
also discuss
I
a
proof of the B-S result that does not
start by assuming it.
In
buying
the B-S derivation an instantaneously riskless hedge is formed by
one
call
sailing
and
shares
F.
stock.
of
Using
stochastic
calculus, the rate of change in the hedge's value (in dollars per unit time)
is found to be
K
1
—
with probability one
F
ss
s
2
+ F
t
This must equal the riskless return
no uncertainty.
F„S)r
(F -
on investment,
2
giving the partial differential equation:
,
^T^S 2
rF S - rF +
s
F
=
.
fc
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)
=
SNCd^
- Ee"
r(T_t)
N(d
2
)
where N(d) is the cumulative normal distribution function,
2
log(3/E) + (r+|o- )(T-t)
d
=
,
and
1
d
For
2
=
-
d
0"
v/f^t
.
1
future reference,
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
be an arbitrage opportunity.
a
proof of the formula
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)
and investing G(S(0),0)
- G(S(0),0),
at
maturity
G(S(0),0)
shares
with
T
certainty,
in
paying
thus
by continuously borrowing enough
stock.
of
can
It
shown
be
that
off
this
maturity it will be worth G(S,T)
at
the
call.
He
extra money to hold
portfolio is always worth G(S,t), even as S and
particular,
max[0,S-E]
such a way as to return
strategy
t
invests
G,,(S(t),t)
ensures
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
have been priced in the meantime.
that he sold at t
The
=
So he will be able to pay off the
0.
Black-Scholes
pricing methodology has
option
fruitful in modern finance theory.
valuation of different
market
these
timing
skills,
extensive
call
the
on
the
line
the
the measurement
liabilities,
analysis of bank
resting
structures
remarkably
Among its various applications are
types of corporate
and
proven
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
nonexistent
rates, and so forth.
possible
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. UO-41]
[1979,
shows
that
the
way
self-contradictory.
He
shows
a
profits,
from
which
we
arbitrage
impossible.
Start
any
for
must
section
in
investor
conclude
to
are
above,
II,
make
limitless
market-clearing
that
is
Here is a version of his argument:
at
t
=
investor
An
0.
probability one, by the time
this,
assumptions
knowing p,
rate of return
x
o",
>
r
and
t
=
make
to
$1
a
probability
with
more,
or
with no investment of his own.
first calculates
he
r,
1,
going
is
q
To
and
>
x
with probability at least q.
given in the Appendix.
He
then borrows and
=
proof that such q and
t = 1/2,
t
4/
1/2 is at least q.
=
exist
choices.
buys enough stock so that by t
If his
=
previous losses and still be up
$1
by t
=
3/
)
asset
he borrows
and
3/4, with probability q, he will cover
any
a
Again,
dollar.
=
1/2,
if he wins he shifts to
the riskless asset, while if he doesn't win he raises the stakes enough
net
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
yields at least $1 at
t
(A
There will be infinitely many possible
being up a dollar or more at
until
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
- (1/2) n is
1
- (1-q) n
is
,
to
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
II are not
mutually consistent.
One Solution:
IV.
section
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
the
p.
sales
41]
doubling
lacking."
the
necessary
earlier
paper,
yields
In
an
problem
but, as Kreps [1979, p.
points;
work
by
restricting
43] admits,
resources
Harrison
trading
to
Kreps
and
to
any
cover
[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. is enough to permit arbitrage by doubling.
Think of
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.
credit
Doubling would no longer 'work', but choosing some level for the
would
limit
option prices.
leverage.
putting an upper limit on
be
somewhat
arbitrary,
If the constraint is binding,
Finally,
such
a
and
it
may affect
equilibrium
investors may use options
limit would not be realistic:
we do not
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.
the
case of options
(American or European),
this
is
intended to
that the option buyer can give the option back to the seller in
In
mean
return
10
for
sum equal
a
to
default risk (for example,
a
covered call would have no default risk).
made
be
You may well ask how a contradictory set of assumptions can
consistent
adding more assumptions.
by
Well,
1
believe that
field had
be that some researchers in this
so that
it
in mind
is merely being made explicit
an
Or it may
assumption
implicit
here.
fact
in
#7
replaces an implicit assumption that negative wealth was allowed.
like #7,
no
similar option that has
the market price of a
But in any case,
the great appeal of this assumption is that negative wealth in fact has
no
realistic meaning.
You
also
may
limited-liability world
all else fails,
whether
ask
—
riskless
a
asset
wouldn't every loan have
a
exist
can
risk of default?
delivery of $B cash that is secured by $B of cash sitting in
sources of riskless assets.
a
Consider
a
But
If
future
the riskless asset can be defined as a contract for
similarity to government debt should be apparent.)
this
in
vault.
a
are
there
(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
hi3
investment),
investing directly
in
because
whatever
before
the
loan
Also notice that
can
no-one
(unless he agreed to repay
such
the
a
proposition
borrower was
be
would
all
be
planning to
11
invest in.
the new framework of assumptions
In
through
1
8,
doubling
the
strategy does not earn arbitrage profits, but the B-S option formula
holds.
First
economy,
it
I
will
prove
Black-Scholes
the
formula.
In
will make a greater difference to the force and
this
still
amended
form of
proof whether or not one starts by assuming that the call price is
the
known
a
function of only S and t.
Given
this
assumption,
original
the
B-S
carries
proof
Unless the Black-Scholes partial differential
unchanged.
over
almost
equation
holds,
any investor with positive wealth can make excess returns on the
of
portfolio
hedge
stock,
call
option,
riskless
and
standard
More
asset.
specifically, denoting the call price function by F(S,t), if
rF S - rF +
s
i^S
2
+ F
5 b
then the investor can buy n calls, sell short nF
difference
the
in
riskless
asset,
chosen as large as he likes).
(unborrowed)
wealth,
>
make
and
shares of stock, put
s
$nb
In
other words,
must
time
(n
can
the
be
This is in addition to any returns on his own
this
is
an
The opposite investment strategy is followed if b
equation
unit
per
which he needed as collateral
with no default risk.
differential
,
fc
Also
hold.
the
<
in
order
to
arbitrage
0.
short-sell
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,
proof
is
more difficult.
price by C(t),
and an investor's
C(0) < G(S(0),0),
buys
n
calls,
Denote
and
sells
W(0)
>
short
0.
the
B-S
formula
by G(S(t),t),
(market-valued) wealth by W(t).
The call
nF„(S(0),0)
is
undervalued,
shares
of
stock,
so
the
and
the
the
call
Suppose
investor
puts
the
12
(including W(0))
remainder
continuously
position
As
stock.
in
Merton
the
in
he
that
so
[1977],
asset.
riskless
always
is
short
This is not
more undervalued?
a
but here there is a
what is to prevent its becoming
But in
for some t before maturity, my investor will no longer
<
his position to maturity
—
be
he would be out of the game.
This puts a limit on n,
I
my
maintain
able to borrow (or sell short or sell calls), and thus could not
take.
even
investor
problem for Merton, because his
can hold out until maturity, when he is sure he will be ahead.
case, if W(t)
of
investor
this is designed to ensure that the
If a call can be undervalued,
short
shares
nF (S(t),t)
can pay off the calls and still have money left over,
hitch:
adjusts his
He
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) ^ E
for all t until maturity.
Proof:
(a)
S(t)-E ^ C(t) ^ S(t)
investors
could
collateral
for
limit,
sell
the
the
call,
call,
for all
buy
If
t.
the
stock,
C(t)
and
making arbitrage gains.
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)
for
all
This
t.
is
a
well-known
property of the B-S formula.
(o) The lemma follows easily from (a)
moving interval of length
E,
and (b).
and so can never be more than E apart.
If n < W(0)/{E-[G(S(0),0)-C(0)
Theorem:
F and C are trapped in
] }
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)]
« W(0)
n[G(S(0),0)-C(0)] - n[G(S(t) ,t)-C(t)
« (W(0) + n[G(S(0) t 0)-C(0)]}e
=
rt
(by the Lemma)
]
- n[G(3(t) ,t)-C(t)
W(t)
(For the last line of the proof, recall that the investor starts with
W(0),
skims off G(S(0),0)-C(0) immediately for every call he buys, is long n calls
thenceforth,
portfolio,
and
short
ensures by continuous hedging that
in
stock and
long
riskless
the
the
asset,
remainder
always
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;
G(S(t),t)
they would all want to buy the call.
only clear if C(t)
=
C(t)
if
So the call market
could
G(S(t),t) for all t.
There remains the question of whether the Kreps doubling strategy can
produce
sure
gains
in
world
a
doubler starts with wealth W(0)
of
>
limited
liability.
Suppose
would-be
a
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
=
there
1/2,
stock will drop enough to make W(t)
disaster doesn't strike him,
is
a
if he has
to
t
<
'double'
1/2.
And even if
several times in
the stake will have risen so as to increase the risk of hitting zero
before the next double.
to
=
positive probability that
for some
=
he
the
that
a
row
wealth
So arbitrage profits can not be made this way.
But that does not prove that there is no way to make excess profits by
some other type of doubling strategy.
The next section looks at some
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
some
money left
term
loans
lines
and
section
of
outline
I
conclude
The
over.
introduction of such real-world contracts
credit
an
permit
yet
institutional
doubling
that
may
still
successful
structure
fails.
All
with
effort
doubling.
these
is
not
as
In
this
features,
but
wasted
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,
(b)
they specify how each borrower will invest his assets (loans plus
wealth)
until
contract
the
expires,
problems in pricing contracts;
(c)
//7;
so
as
avoid
to
hazard
moral
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
enough
the
borrower
(or
make
the
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.
total
evolve
in
wealth.
a
This
relatively
institutional
unregulated
setting
seems
marketplace.
like
In
one
fact,
as
Furthermore,
contracts may be written in which the borrower pledges an amount less
his
own
that
the
than
could
limited
16
explicit
liability
feature
is
borrowing
through
corporations),
contracts
many
in
and
is
implicit
that
others
all
in
observe
we
(e.g.
because
bankruptcy is allowed and slavery is not.
Notice that the proofs of the Black-Scholes formula given in section
apply here also,
IV
They required the existence of
fortiori.
a
'demand'
securities, not the nonexistence of nondemand securities.
Suppose
maturity equal
t
:
t
=
can
doubler
a
to
his
stock
planned holding period
the
he borrows on
financed
promise to repay at
positions
for
the
term
by
is not up
=
1/2.
If he
1/2,
he borrows on a promise to pay at t
=
3/^;
and so on.
not
be
put
out
of
game
the
period, but there is still
hitting
by
That
stock.
t
a
zero wealth
loans
of
is,
at
$1
at
This way he
during
a
holding
chance that at the end of any holding period he
a
will have to pay all his assets to his creditor.
In
such
a
will
case he
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
paying
the riskless rate on any amounts borrowed.
with the aim of making
that
he may suffer
$1
a
fee of $f up front,
He then arranges his
plus $f plus any interest paid.
the
paying
and
'doubling'
Here the catch is
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
be
interesting
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
interval h approaches zero.
lines.
This
preserving
turns
out
See Merton [1975] for a discussion along
these
doubling
while
eliminate
to
result,
Black-Scholes
the
benefits
the
thus
of
identifying
phenomenon as a singularity of the continuous-time extreme.
we do not assume that wealth is nonnegative.
a
trading
sequence of discrete-time economies, as the
the
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 European call in the continuous-time limit.
we want to price it at t
and it matures at t
=
from the set {1,2,4,8,16,...}
1.
,
=
T >0.
Suppose
Where n is
.
.
.
where h
=
drawn
define economy n by the assumptions:
Capital markets operate (and clear) only at the dicrete time points
0,h,2h,3h,
of
T/n.
It
is
to be understood
in
what
t
=
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 dividends,
stock price
whose
follows
S
Markov process such that given S(t), the mean of S(t+h) is
a
S(t)[1+jih],
2
2
and the variance of S(t+h) is a [3(t)] h.
6.
There is
a
European call option on the stock,
maturity date
Merton
T,
with exercise price
E,
and market price C(t).
[forthcoming]
provides
assumptions
of
set
a
security
about
price dynamics in discrete time that are sufficient to make prices follow an
It6
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
+ adZ
Brownian motion with
constant
drift
.
constant
rate
/J.
simplicity, though the results do not require it.
jj
variance
is
held
rate
2
o*
,
constant
as
in
for
19
To
the
above
six
assumptions
economy
on
add
n,
the
following
cross-economy assumptions:
C1.
The parameters r,
Let e(t)
[Notation:
/j,
=
a,
and T are the same in all economies.
E,
S(t) - E
(S(t)
} .
]
1
C2.
The distribution of e(t)
and M
<
CO
,
both
is
discrete,
and
independent of n and t,
there exist numbers m
such that
each t
for
<
CO
the
2
number of possible outcomes for e(t) is ^ m, and e (t) ^ M for certain.
Merton invokes these to simplify the mathematics without imposing any
significant economic restrictions.
Choose some time t in economy n.
probabilities p.,
j
=
1,2,3, ... ,m.
7/
Then e(t) may take on values d. with
For a fixed t and j, and increasing
d. and p. can be thought of as functions of h, with h approaching zero
n,
from
above.
Define the notation
f(h)~h 3
to mean
that lim f(h)/h
3
is
a
nonzero
h-*0
real number.
Now assume:
C3.
For all t and
j,
d. and p. are sufficiently "well-behaved" functions of
that there exist numbers r. and q. such that d
J
J
.
J
~h
J
and d
j
~h
J
*
:
.
;
;
20
CM.
For all
and
t
such that q. + 2r
j
J
purpose
main
this
of
Merton shows that
assumptions,
q
is
+ 2r
.
that
and
rule
to
<
.
if
q
assume that r.
1,
(The
1/2.
=
J
processes
jump
out
limit.
the
in
is impossible as it would violate prior
1
+ 2r
.
=
.
J
.
>
for
1
possible outcomes
some
j,
then those outcomes contribute nothing to the limiting continuous-time
stock price process.)
do not attempt to derive
I
formula for the price of the call option,
a
Rather,
C(0), in any of the discrete-time economies.
it must converge
this sequence of prices is,
value, G(S(0),0), as
—*
n
oo
and h
->•
To simplify notation,
0.
stand for any function f(h) with the property
S(k)
=
stock price at
C(k)
=
option price at
N(k)
=
G
G(k)
=
G(S(k),k)
H(k)
=
value of hedge position at
(S(k),k)
=
=
kh
=
t
=
,
for k
lc(0) - G(S(0),0)|
>
£
in
;
kh
t
=
kh
B-S formula call value at
for
=
0,1,2,...,n;
=
B-S hedge ratio at
that
such
>
formula
let
——
lim
h-»0
t
=
t
=
kh
any
N
economy n.
and
;
kh
Then there exists
Suppose C(0) does not converge to G(S(0),0).
£
whatever
argue that
the Black-Scholes
to
o(h)
t
I
there
an
is
For that
£,
n
>
N
some
such
that
consider the following
two sequences:
:-
{all n such that C(0) - G(S(0),0)
>
£.
in economy n}
Sequence 2 :-
{all n such that C(0) - G(S(0),0)
<
£
in economy n}
Sequence
1
At least one of these two squences must be infinite.
infinite.
Suppose sequence
Then applying the following argument to economies with
from sequence
large enough n
1
n
shows that arbitrage profits can be made in an economy
1
is
drawn
with
]
21
At t
invest G(0)
and
pocket the difference
sell one call for C(0),
=
in
of N(0)
composed
portfolio
stock
levered
a
stock and riskless borrowing of $[N(0)S(0)-G(0)
]
C(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
value
call
formula
also
changes.
Its
(1)
)].
change
can
be
rewritten following the Taylor series argument in Merton [forthcoming]:
G(k)-G(k-1)
=
(k-1 )[S(k)-S(k-1
G (k-1)[S(k)-S(k-1)]+ G (k-1)h + ^G
ss
s
t
2
)
+o(h)
....(2)
Let u(k)
in
the
n-»co,
E
k_1
3ense
=
e(k)
TV7i?
;/h
—
—=r,
This standardizes the driving random variable
•
ot>(.k-i
that
outcomes
the
(u(k)}
=
whereas
of
the
u(k)
and E _ (u 2 (k)}
k 1
S(k) - S(k-1)
outcomes
not,
do
=
1
by
of e(k)
become
assumption
for all n.
So
=
/iS(k-1)h + e(k)
=
/jS(k-1)h + dS(k-1)u(k)/h
C4
infinitesimal
above.
In
as
fact,
22
Then
[S(k)-S(k-1
2
a S (k-1
=
)]
and equation
(k)h + o(h),
)u
can be
(2)
rewritten as
G(k) - G(k-1)
=
G
s
(k-1)[S(k)-S(k-1)] + G (k-1 )h +
^G
Into D(k)
-
2
2 2
(k-1)o- S (k-1)u (k-1)h + o(h)
ss
D(k-D
- H(k-1)
H(k)
=
-
9/
- G(k-1)]
[G(k)
....(3)
substitute
(1)
and (3) to give
D(k) - D(k-1)
G
s
N(k-1)[S(k)-S(k-1)] - rh[N(k-1)S(k-1)-H(k-1)] -
=
(k-1)[S(k)-S(k-1)] - G (k-1)h t
But
recall
N(k-1)
that
=
2 2
^ ss (k-1 )o S
G (k-1
)
,
(k-1 )u
that
and
Black-Scholes partial differential equation so that
2
(k)h + o(h)
(4)
satisfies
G
rG S+G =rG -
irS
the
G
Substituting these into (4) gives
D(k) - D(k-1)
=
rh[H(k-1)-G(k-1)] - ^G
3S
2 2
2
(k-1)o- S (k-1)[u (k)-1]h + o(h)
(5)
Defining y(k)
=
^G
ss
(k-1
)cj
2 2
2
S (k-1 )[u (k)-1
],
we have from the law of
n
large
numbers
martingales
for
lim hV'y(k)
that
n-»oo
=D(0)
+
£ [D(k) ^
k=
where
1
Consider the difference D(t) for some time
D(t)
K
=
nt/T
.
Substituting
from
probability
',
k=
one.
with
=
/—
t
between
and T:
(k-1 }]
1
(5)
taking
and
the
continuous-time
limit, this becomes
K
D(t)
=
D(0) +
K
limV
n
"°°k =
1
rh[H(k-1)-G(k-1)] -
K
limVy(k)h
n
^~k =
1
+
lim5o(h)
n
—
k=
1
'
25
K
limV
n -»<».*—
,00
+
=
k=
rhD(k-l)
1
t
rD (s)ds
/
s=0
Solutions
D(0)
=
this
to
integral
this means that
0,
D(t)
equation
for
=
satisfy
D(t)
from
to
all
t
=
D(0)e
and
T,
value H(t) will always equal the B-S formula call value G(t).
at
t
=
T,
=
will
be
the
hedge
when the investor has to make good on the call he sold,
he will
max[0,S(T)-E]
just
so
sufficient
but the Black-Scholes formula has the
H(T)
=
to
pay
arbitrage
portfolio.
So
is
C(0)
t
achieved
must
max[0 S(T )-E
,
off
the
on
also
]
call
—
were 'free'
=
—
at
property
the hedge
portfolio
maturity.
Thus
the
arbitrage.
if sequence 2 were infinite instead of sequence
By a similar argument,
then
Since
finally
profits that were pocketed at
1,
.
And
have to pay max[0,S(T)-E];
G(T)
rt
by
buying
the
call
converge to G(S(0),0):
and
the
shorting
B-S
hedge
the
formula is
the
limit of the call's market price, as the trading interval approaches zero.
an
As
example of why doubling fails in the
imagine a doubler betting red on
.5),
all
in a
his
a
limit of discrete
time,
fair roulette wheel (probability of win
discrete-time world where negative wealth is allowed.
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
to bet between t=0 and t=1,
2 to
1
.
He
has
n
opportunities
wants to be up $1 by t=1, and so he first
$1
and keeps doubling until he wins once.
$1
with probability 1-(1/2)
n
,
or $(1-2
n
)
His net payoff as of t=1 will
with probability (1/2)
n
.
bets
be
24
Now the parallel to Kreps
infinite,
so
that
payoff
the
'
doubling argument would be to next let
is
$1
with probability
—
1
disastrous
the
string of losses gets swept under the rug of zero probability,
for doubling is large.
noise
any
to
n be
and
demand
But for any economy n, the doubling payoff only adds
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
that
requiring
wealth
to
imagine
methods
a
don't.
be
At
the
nonnegative
is
same
time,
desirable
realistic meaning for negative wealth.
in
what
may be
the
most
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
under any possible circumstances.
total
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 C1, 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
and some stock-price risk will have a positive probability of default.
that
in
option-stock
an
hedge
portfolio
cases,
the contracts will
stock-price
Furthermore,
completely hedged away in discrete time.
stock or writing of options will
the
have default risk
be priced
to
risk
Note
cannot
be
any short sale
of
also.
all
In
take account of the default
these
risk.
Define the default premium to be the difference between the prices today
a
a
given promise to pay with and without the default risk.
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.
A call
not to default by putting up the stock as collateral.
riskless if backed by enough cash;
to
borrow
on
such
terms,
the
—
it should equal
writer can
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
doubling strategy in this world would suffer from all
a
the handicaps of sections V and VI above, so it will not generate
arbitrage
The remainder of this section is devoted to
profits here either.
showing
that in spite of default problems in the discrete-time economies, the market
price of
call will approach the B-S formula value in the
a
continuous-time
My plan is to adapt the discrete portfolio adjustment argument
limit.
section VI,
and
to show that
in
a
time interval of length h,
each
probability is o(h), each default premium must be o(h), so that as
cumulative
effect
of
these
probabilities
Just as in section
disappears.
IV,
premia
and
there will be
a
position that an investor may take to make money on
since all
from
default
h
—*
the
to
=
t
t
limit on the size
a
mispriced call,
investors would be wanting to transact on the call
in
of
the
=
T
of
but
same
direction, dominance would prevent the market from clearing until the price
is right.
Let
us
take
care
some
in
defining
what
just
discrete-hedging investor is going to invest in.
or bonds,
or
debt
(the
stock
is
not
a
of contracts
our
When he buys stock, calls,
let him do so no a no-default-risk basis:
government
sort
problem).
by buying covered calls
When
he
sells
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 contract
arrangement.
Each
of
his
liabilities
is
to
resettled after each time unit h (in the style of futures contracts).
is
nothing unusual
stock)
—
in
this
for
borrowing and
these are to be repaid next period.
option with some distant maturity date T,
I
short
in
I
be
There
(borrowings
of
have him write
an
sales
But when
by
fact will mean
for him
contract to pay, at time h from now, the market price that that option
to
will
27
then command if it has no default risk from then on (e.g.
of
a
continuous-time demand-basis assumption #8 on page
options on
portfolio
any
stock,
that
price
This is the next best thing in discrete time to
covered call).
Consider
the market
and
of
short
bonds,
and/or
9,
long
section IV.
positions
in
with positive pledged wealth
In order for default to occur on any of the short positions,
~h
stock,
a
(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
CM the probability of such an outcome is o(h).
Now the default premium can be thought of a3 the market value of
o(h),
probability
the guarantor will only have to pay off with
loan guarantee;
in an amount that is bounded
of such a liability were larger than o(h).
3uch claims would give,
as
h-»0,
—
payoff of probability zero
Suppose the market
in magnitude.
a
Then rolling over
arbitrage.
value
sequence
a
value greater than
summed
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-D] - rh[N(k-1 )S(k-1 )-H(k-1
term
includes
any default premium.
With
before, we again get
K
D(t)
=
n
limY]rhD(k-1)
-°°kTl
t
=
f rD(s)ds
J
s =0
)]
D(t)
+ o(h)
defined
as
28
conditional on there being no default between time
probability of such
a
default
bounded
is
and
time t.
But
the
by something of the
above
form
K
which approaches zero as h
2_,o(h),
—»0.
So we have
D(t)
D(0)e
=
with
k=1
probability one, the call position can be closed at maturity using only
from
proceeds
completes
the
the
hedge
dominance
portfolio,
argument
and
riskless
are
Black-Scholes
the
for
gains
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:
one was to
explicit the natural requirement that wealth is never negative;
was
define
to
a
economy
continuous-time
as
the
limit
of
models.
a
make
other
the
sequence
discrete-time economies, as the trading interval approaches zero.
These two
more
methods were combined in the last section, to form what seems to be a
reasonable model of the world.
fails
to
produce
In
arbitrage gains,
argument still obtains.
It
the three sets) will provide
is
a
all
three cases,
while
the
the doubling
of
strategy
pricing
Black-Scholes option
hoped that the amended assumptions
(any
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
Merton
[19733,
F
Henriksson
[1981],
=
3F/dS
Merton
and
[1981], and Hawkins [1982].
3.
statement that "there is some positive probability q that over
Kreps*
any time interval the second security [stock] will strictly outperform
the
a
[riskless asset]"
first
< r
then for any q
[1979,
there exists
>
a
not quite accurate.
is
**0]
p.
time
enough
interval T large
that the probability of the stock beating the riskless asset
If
is
less
But this is not a problem for his doubler, who can put an upper
than q.
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,
n = the
let
number of consecutive losses he has suffered so far;
W
=
his net winnings up to that time (may be negative);
T
=
(1/2)
n+
and
the length of his next holding period.
=
holding
To be up $1 with probability at least q at the end of the next
—
- W
e
- e
1
period, he borrows and invests
$
=•
Then if y(T)
.
>
x,
his
net
position will be
W +
—xT= 1
e
5.
A
simpler
W
- e
,
=-(e
yT
J
rT
bet-doubling
- e
rT.
)
.
>
,.
W +
—xT- 1
e
strategy,
which
W
.
xT
=-(e
- e
rT
can
be
- e
rT.
.
=
)
defined
without reference to a specific stochastic process,
is
1
and
.
analyzed
found by making
30
U3
of short sales.
e
short
sell
a
large
enough
nolding-period return
exceeds
q
=
1/4,
r
if
and
to
median
the
stock holding-period
both quartiles equal r,
6.
than r,
the
go
median
of
long when
equals
r.
median
the
Instead,
a
return.
set
This works unless
possibility which can be safely ignored.
This sequence (powers of 2) is chosen for n so that any time
capital
stock
the
go short an amount calculated with reference to the
quarcile of the
lower
less
fails
whenever
amount
to
1/2,
=
q
go long when the upper quartile exceeds r, and in the case when
does not exceed
it
is
this
but
r;
almost works (but not quite) to set
It
markets
are
open
in
economy
n,
will
have
the
for which
t
property
that
capital markets are open at cnat time in all economies of higher n.
7.
Merton is not explicit about m and M being independent of
(a)
and
n
t,
but clearly intends it that way.
(o)
En
the
following discussion,
if any e(t)
possible
has less than m
outcomes, imagine augmenting the number of outcomes until it equals m by
adding outcomes of zero probability.
d.
I
thank Robert Jarrow for emphasizing to me the importance of
proving
rather than just assuming convergence and
proving
that C(0)
converges,
that it must converge to G(3(0),0).
on Merton
[1977
J,
[197dJ,
The argument that follows is
and [forthcoming].
The preceding assumptions
correspond to those in Merton [forthcoming] as follows:
l:;
more
E.5,
than
and E.6.
enough
to
guarantee
My Cj is his E.4.
Merton'
s
based
assumptions
My assumption 5
E.1,
E.2,
My C4 is his "Type I" partition.
E. i,
31
9.
Remember
statement
that
o(h)
the
+
notation o(h)
o(h)
=
o(h)
refers to
means
that
a
the
class of functions.
class
is
closed
addition.
10.
See the argument in Merton [forthcoming] leading to equation (j2).
The
under
)
32
APPENDIX
Theorem
Given the stock price process
:
dS/S
+ adZ
= jjdt
and the (constant) riskless rate of return r, there exist
q
and
>
length
rate of return
a
T 4
stock
the
1/2,
probability at least
x
r
>
will
such that over
yield
rate
a
any
of
probability
a
time
at
interval
least
(All rates of return are meant as
q.
of
with
x
continuously
compounded.
Proof:
Let y denote the (random variable) rate of return on the stock
time T.
in
S(0),
the
Then S(T)
lognormal
=
S(0)e
=
/j
-
variance
2
o-
/2
=
o"
2
x
of
=
£log[s(T)/S(0)]
S(T)
implies
that
log S(0) + aT and variance
=
For
.
given
log S(T)
=
2
cr
is
where
T,
=
a
and
/T.
and q
a
=
and y
Therefore y is distributed normally with mean
.
Case
Choose
,
distribution
distributed normally with mean
a
yT
=
>y
—
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
r
+
0.001
and
q
=
—
a
Prob{y >
4
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,
must be true that Pr{y
it
>y x}
increases,
and
so
is
> q.
Here
is
formal proof:
q
=
Pr{ y(1/2)
-
Pr(
x
>,
y(1/2)-a
}
X^a
:
}
>x
Note that both have the standard normal distribution,
=
Pr{l y
v.
<
Pr{
}
^vt"f ;»^-=-^
o-yr
^7=7
Pr{ y(T)
}
for all T,
}
for any T
including T
<
1/2
;
/
>,
>,
x
jLZ-£
}
.
<
1/2
,
since
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,
J.
[19733,
R.
"Temporary
General
vol.
81,
Equilibrium
in
,
Trading Model With Spot and Futures Transactions",
vol.
Harrison,
11,
J.,
and
D.
Kreps
Securities
vol.
381-408.
Hawkins,
G.
pp.
[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
,
Agreements",
1.
Market
Timing
and
Statistical Procedures for Evaluating
Forecasting Skills", Journal of Business
Kreps,
and
Analysis of Revolving Credit
"An
R.
"Martingales
[1979],
Journal of Financial Economics
Henriksson,
a
1103-1123.
pp.
Multiperiod
20,
637-659.
pp.
,
vol. 54, October.
"Three Essays on Capital Markets"
(especially essay
"Continuous Time Models"), Institute for Mathematical Studies
in the Social
University.
Sciences (IMSS3), Technical Report No. 261, Stanford
35
Merton,
Continuous-Time
Model",
Journal
vol.
Portfolio
and
Theory
Economic
of
Rules
,
in
a
vol.
3,
of
from
Finance
Perspective
the
of
Journal of Financial and Quantitative Analysis
Time",
,
659-674.
10,
pp.
C.
[1977],
R.
"Theory
[1975],
C.
R.
Continuous
Merton,
Consumption
373-413.
pp.
Merton,
"Optimum
[1971],
C.
R.
"On
the
Pricing
Theorem",
Modigliani-Miller
Contingent
of
Journal
of
Claims
Financial
the
and
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
"On
Market Timing
Investment
and
Performance.
Equilibrium Theory of Value for Market Forecasts",
Journal
of Busines s, vol. 54, pp. 363-406.
Merton,
R.
C
[forthcoming],
Assumptions
Essays
in
of
"On
the
Mathematics
Continuous-Time
Models",
Paul
Cootner
Honor
of
in
,
and
Economic
Financial Economics:
W.
F.
Sharpe
(ed.),
Prentice-Hall, Englewood Cliffs, New Jersey.
Rothschild,
M.,
and
J.
E.
Stiglitz
[1970],
"Increasing
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Risk:
225-243.
I.
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