4 Implications of the Dominance Principle, ROP for Puts and Calls
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RATIONAL OPTION PRICING
375
Example of the Intrinsic Value Lower Bound for an American Call
Here we will take an American option’s price quote and examine whether it
satisﬁes CA(St ,,E)уMAX[St–E,0]. Google’s ofﬁcial closing stock price on
Nasdaq was approximately $575.62 on September 12, 2014. The GOOG\
14J18\500.0 (2014 Oct 500.00 Call) was trading at that time for an offer price
of $78.80.
Intrinsic value was therefore $575.62–$ 500.00=$75.62, which is below the
option price of $78.80. The difference of $78.80–$75.62=$3.18 consisted of
time premium per share.
The story does not end here, however, because the American feature, which
seems to enhance the value of an American call option relative to an otherwise
identical European call option, is illusory if there are no payouts to the
underlying over the life of the option. We will discuss this in section 11.4.6.
11.4.2 Lower Bound for a European Call Option on an Underlying
with no Dividends (LBEC)
In the case of European call options with no dividends, we can get a higher
lower bound than the intrinsic value bound. (This bound also holds for American call options where there are no dividends (see section 11.4.6)). In the
case of European options, early exercise is not possible so the arbitrage
opportunity we constructed in terms of intrinsic value does not work.
Ordinary intrinsic value is not an appropriate immediate value concept for
European options because, to wring the value out of a European option, you
have to wait until expiration. However, instead of using E in the deﬁnition
of intrinsic value, we can simply use the present value of E, PV(E)=e–r E.
This leads to a useful way to adjust intrinsic value for European call options.
Definition of Adjusted Intrinsic Value (AIV) for a European Call
Adjusted intrinsic value is intrinsic value with E replaced by e–r E,
AIVt=MAX[St–e–r E, 0]
A proven lower bound for a European (note the E in CE below) call option
on an underlying asset that has no payouts over the life of the option is adjusted
intrinsic value.
CE(St ,,E)уMAX[St–e–r E, 0]
(Lower Bound European Calls
(LBEC))
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OPTIONS
We will prove this using the dominance principle, not by constructing an
arbitrage strategy explicitly. To do so, we look at the right-hand side of (LBEC)
and work back from current costs to strategies. This technique is discussed in
section 11.5.4, but simply involves taking costs (and beneﬁts) and working
backwards to uncover the strategies that generate them.
The strategy represented by the current cost of St–e–r E is simply to borrow
–r
$e E and purchase the stock for $St. This will be called Strategy B.
Strategy A will be purchasing a European call option. Table 11.2 describes
the payoffs to each strategy in each relevant state of the world at expiration.
TABLE 11.2
Proving LBEC
State of the World at Expiration
ST уE
S T
Strategy A
Payoff to Long European Call Option
S T –E
0
Strategy B
i. Long the stock
ii. Borrow PV(E)
ST
–E
ST
–E
Total Payoff to Strategy B
S T –E
ST –E
Note that Strategy A has payoffs at least as great as the payoffs to Strategy
B in all states of the world. Note that the payoff to Strategy B is ST –E which
is negative if STStrategy B is St–e–r E.
n
CONCEPT CHECK 2
Show, by using the dominance principle, that the value of a European call
option also cannot be negative.
By the dominance principle, the current value of Strategy A must be greater
than or equal to the current value of Strategy B. Therefore, adjusted intrinsic
value is a lower bound for the value of a European call option with no payouts
over the life of the option, CE (St,,E) уMAX[St–e–rE, 0].
There are several important things to note about (LBEC). First, note that
this lower bound is higher than intrinsic value.
RATIONAL OPTION PRICING
n
377
CONCEPT CHECK 3
a. Show that adjusted intrinsic value for calls is greater than (unadjusted)
intrinsic value, except at maturity.
b. Show that adjusted intrinsic value for calls is equal to (unadjusted) intrinsic
value, at maturity.
Implications of the Lower Bound for European Calls
One of the implications is that the underlying asset itself can be interpreted
as a European call option. There are two interpretations:
Interpretation 1
The exercise price E=0. In this case, the underlying asset is economically
equivalent to a European call option with exercise price equal to zero and
with any maturity.
Interpretation 2
The exercise price E>0. In this case, the underlying asset is economically
equivalent to a European call option with any exercise price greater than zero
and inﬁnite maturity.
To prove interpretation 1, we compare the payoffs to holding any European
call option with E=0 and any maturity to that of holding the stock. There is
only one relevant state of the world since ST<0=E cannot occur for a limited
liability asset. This is described in Table 11.3.
TABLE 11.3
Interpretation 1 of the Underlying Asset as a
European Call Option
State of the World at Expiration
S T у0
Strategy A
Long European Call Option with E=0
ST –E =ST
Strategy B
Long the Stock
ST
It’s clear that the two strategies have exactly the same payoffs when E=0, so
they are economically equivalent and therefore have the same current costs,
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OPTIONS
St=CE(St ,,0). The value of any European call option with E=0 is equal to
the current underlying asset price St .
To prove interpretation 2, all we need to do is note that as the time to
maturity increases the PV(E)=e–r E approaches zero. Furthermore,
MAX[St–e–r E, 0]рCE(St,,E)рSt .
The ﬁrst inequality is (LBEC), and the second inequality follows because the
call option price cannot be greater than the underlying asset price. But as
increases without bound, MAX[St–e–r E, 0] approaches MAX[St , 0]=St , since
a limited liability asset always sells for at least a zero price.
Thus StрCE(St ,,E)рSt . This says CE(St ,,E)=St, as long as becomes
inﬁnitely large, no matter what the exercise price is. This is interpretation 2.
We next turn to the ROP bounds for American and European puts, once
again for assets with no payouts over the life of the option.
11.4.3 Lower Bound for an American Put Option on an Underlying
with no Dividends (LBAP)
The nominal lower bound for an American put option is intrinsic value for a
put IVt=MAX[E–St , 0]. We say nominal because there is a very real possibility
that the American put will be exercised prior to maturity. Once an option is
exercised it is dead and gone and no bounds can hold for its price.
PA(St ,,E)уMAX[E–St ,0]
(Lower Bound American
Puts (LBAP))
We can prove this by contradiction. If (LBAP) were not true, then PA(St,,E)<
MAX[E–St , 0] and an immediate arbitrage opportunity would arise. Note
that E>St and MAX[E–St , 0]=E–St , otherwise the assumption implies that
PA(St ,,E)<0 which is not possible (the put premium cannot be negative).
The arbitrage strategy would be to buy the put for PA(St ,,E), then acquire
the stock underlying the put by purchasing it. Then you would exercise the
put by delivering the stock in return for $E. This would leave you with the
positive amount E–St since E>St .
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The net cash ﬂow from this set of transactions is E–St–PA(St ,,E), which is
positive by the assumption that PA(St ,,E)no further implications to your trades—the put has been exercised, and you
disposed of the stock. Nothing is left to do except to walk away with your
arbitrage proﬁts.
Example of the Intrinsic Value Lower Bound for an American Put
Here we will take an American put option’s price quote and examine whether
it satisﬁes PA(St ,,E)уMAX[E–St , 0]. Google’s ofﬁcial closing stock price on
Nasdaq was approximately $575.62 on September 12, 2014. The GOOG\
14V18\500.0 (2014 Oct 500.00 Put) was trading at that time for an offer price
of $1.30.
Intrinsic value was MAX[E–St, 0]=0 since the stock price was above the put’s
exercise price. Therefore, the entire value of the option, $1.30 per share,
consisted of time premium.
FIGURE 11.1 Optimal Early Exercise for an American Put Option
(No Underlying Dividends) Along the Early Exercise
Boundary
Underlying Asset Price St ($)
50
49
48
47
46
45
44
C
Continuation Region
43
E – Sτʹ
42
41
40
E – St
39
38
E – S∼τ
37
36
S
Stopping Region
E – St
∼
τ
35
0
0.5
1.0
1.5
2.0
τʹ
t
2.5
3.0
3.5
4.0
Time
4.5
5.0
380
OPTIONS
The story of American puts doesn’t end here. Early exercise of American
put options is an advanced topic. However, we can give some idea of the
ﬂavor of this important topic. First, once you have read section 11.4.6 below,
compare it to this section.
Early exercise of American put options is not driven by dividends. For each
time t prior to maturity, there is a certain level called the early exercise boundary.
When the underlying asset price declines to and hits the early exercise
boundary for the first time, it is optimal to exercise the American put option.
Figure 11.1 illustrates. Note that ′ is the second time the early exercise boundary
is hit. Too late.
The optimal time to exercise the American put option is the random time
~
(not normal time to expiration , but the effective random time to early exercise)
and the optimal early exercise price is S~. At this critical stock price the value
of the American put comes into equality with intrinsic value, PA(S~,~,E)=
MAX[E–S~, 0]=E–S~, since the American put option has to be in-the-money
in order to justify exercising it.
11.4.4 Lower Bound for a European Put Option on an Underlying
with no Dividends (LBEP)
In the case of European put options, we can get a lower bound that is lower
than unadjusted intrinsic value for an American put. In this case, early exercise
is not possible, so the arbitrage opportunity we just constructed in terms of
intrinsic value again does not work.
However, instead of using E in the deﬁnition of intrinsic value we can use
the present value of E, PV(E)=e–r E. This is the corresponding way to adjust
intrinsic value for European puts.
Definition (Adjusted Intrinsic Value for a European Put Option)
Adjusted intrinsic value is ordinary intrinsic value with E replaced by e–r E,
AIVt=MAX[e–r E–St, 0]. We use the same notation as for calls. The context
dictates which form to use.
The lower bound for a European put option is then adjusted intrinsic value.
P E(St ,,E)уMAX[e–r E–St , 0]
(Lower Bound European Puts
(LBEP))
We will prove this using the dominance principle, not by constructing an
arbitrage strategy explicitly. To do so, we look at the right-hand side of (LBEP)
RATIONAL OPTION PRICING
TABLE 11.4
381
Lower Bound for a European Put Option
State of the World at Expiration
S T уE
ST
Strategy A
Payoff to Long European Put Option
0
E–ST
Strategy B
i. Short the Stock
ii. Lend PV(E)
–ST
E
–S T
E
Total Payoff to Strategy B
E–ST
E–ST
and work back from current costs to strategies. The strategy represented by
the current cost of e–r E–St is simply to lend $e–r E and short sell the stock,
thereby cutting current costs by St . This will be Strategy B. Strategy A will
be purchasing a European put option.
Table 11.4 describes the payoffs to each strategy in each relevant state of
the world at expiration.
Note that Strategy A has payoffs at least as great as the payoffs to Strategy
B in all states of the world. The payoff to Strategy B is E–ST<0 if ST уE,
while the payoff to Strategy A is 0. The current value of Strategy B is e–r E–St .
The value of a European put option also cannot be negative because its payoffs
dominate the zero-payoff strategy.
By the dominance principle the current value of Strategy A must be greater
than or equal to the current value of Strategy B. The maximum of adjusted
intrinsic value and zero is a lower bound for the value of the European put
option,
PE(St ,,E)уMAX[e–r E–St , 0]
(Lower Bound European Puts
(LBEP))
Note that adjusted intrinsic value for a European put option this time is lower
than unadjusted intrinsic value.
n
CONCEPT CHECK 4
a. Show that adjusted intrinsic value (AIV) for puts is lower than (unadjusted)
intrinsic value, except at maturity.
b. Show that adjusted intrinsic value for puts is equal to (unadjusted) intrinsic
value, at maturity.
382
OPTIONS
11.4.5 Lower Bound for a European Call Option on an Underlying
with Continuous Dividends (LBECD)
This material is similar to Chapter 4, sections 4.5 and 4.6, on the valuation
of forward contracts when the underlying carries a continuous dividend yield.
As we shall see after discussing put-call parity, European call and put options
are closely related to their corresponding forward contracts.
There are a number of ways of generating a lower bound in this scenario,
just as there were for forward contracts. The objective is to replicate Table
11.2 by adjusting for dividends. This is done in Table 11.5.
Let the stock price at time T, with dividends paid over [t,T] and re-invested
in the stock, be denoted by ST . Let S′T . be the terminal (time T) stock price
without the re-invested dividends. This is the terminal value of the capital
gains process, to use the terminology of Chapter 4, section 4.4. It is also the
future value of the current stock price St minus the PV(dividends paid out
over [t,T]).
Denote the current stock price, St , minus the PV(dividends paid out over
[t,T]) by S′t . We showed, in Chapter 4, Appendix 4.8, that the quick way to
calculate S′t is S′t =e–St . That is, multiply the current stock price St by the
discount factor e–.
The key thing to remember in this case is that the European call entitles
the exerciser to the terminal stock price at expiration, ST , minus the re-invested
dividends paid out over the life of the option [t,T]. That is, only to the adjusted
terminal stock price S′T ..
We start with a long position in a single European call option as our strategy
A. The payoff is in terms of the dividend-adjusted stock price so it doesn’t
pay to exercise unless S′T >E. (In terms of the normal stock price, it has to be
larger than the exercise price, E, plus the FV of the re-invested dividends for
this to occur.)
We now have to ﬁgure out how to partially replicate the call option’s payoffs.
Provided that we accomplish this with payoffs that are no larger than the payoffs
to the call option, the dominance principle will provide a lower bound to the
option’s current price.
Under strategy B, we purchase the current stock price, St, minus the PV
(dividends paid out over [t,T]) for the price S′t =e–St. That is, we purchase
fewer than one unit of underlying stock since e–<1.0. This is a form of tailing
the hedge, because we now can match up that part of the option’s payoff
corresponding to S′T . Note that S′t grows to S′T by deﬁnition. The other part
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RATIONAL OPTION PRICING
TABLE 11.5
Lower Bound for a European Call Option with
Dividends
State of the World at Expiration
ST′ уE
ST′
Strategy A
Payoff to a Long European Call Option
ST′ –E
0
Strategy B
i. Long e– units of the underlying stock
ii. Borrow PV(E)=e– r *E
ST′
–E
ST′
–E
Total Payoff to Strategy B
ST′ –E
ST′ –E
of strategy B is the usual borrowing of PV(E)=e–r *E, which will result in an
outﬂow of –E at time T.
We have matched up the option’s payoff in state of the world S′T уE. In
state S′T whereas the payoff to the option is 0.0. This is perfectly ﬁne. The dominance
principle tells us that the current price of strategy A, C E(St ,,E), must be greater
than or equal to the current price of strategy B, e–St–e–r E. It is also greater
than 0. Therefore, our lower bound in this case is,
CE(St,,E)уMAX[S′t –e–rE, 0]=MAX[e–St –e–rE, 0]
(LBECD)
Note that e–St–e–r E=St–PV(dividends payable over [t,T])–e–r E.
11.4.6 Lower Bound for an American Call Option on an Underlying
with Continuous Dividends (LBACD)
Once we enter the world of American options, the waters of ROP are muddied
because early exercise complicates the ROP discussion considerably. One has
to carefully distinguish between ROP results, which hold for all models, and
model-based results which hold only for the assumed processes, unless proven
otherwise.
However, the ﬁrst thing to note is that, even when early exercise is
possible, it is not always optimal to early exercise an American call option just
because it happens to be in-the-money. This is true whether or not the
underlying pays dividends. In the case of no dividends on the underlier, an
unambiguous exercise policy can be given, which is never to early exercise.
384
OPTIONS
Case 1: An American call on an underlier with no dividends
over the life of the option
In this case, it is never optimal to early exercise an American call option.
Therefore, while one can early exercise it due to the American feature, the
value of being able to do so is zero. It will consequently sell for the same
current market premium as an otherwise identical European option,
CA(St ,,E)=CE(St ,,E).
This result is not hard to prove using the tools we already have. But we
will defer its proof to End of Chapter Exercise 2. This result is a famous result
due to Robert Merton in 1973, and we will take the reader systematically
through all the steps.
Case 2: An American call on an underlier with continuous,
proportional dividends over the life of the option
In this case, it may be optimal to early exercise an American call. That depends
on the relative magnitude of the dividend to the loss of interest on the exercise
price, which will be incurred by exercising. Using the proportional dividend
model, the instantaneous dividend is St ⌬t over a small interval of time ⌬t.
Instantaneous interest, if the option were not exercised and the strike price
E were not lost, would be equal to er⌬tE over the same small interval of
time ⌬t.
Thus, a necessary condition for early exercise at time t is that St ⌬t>
r⌬t
e E, which says that the instantaneous dividend is greater than the loss of
instantaneous interest. The condition is roughly equivalent to St>rE, which
will hold at each time t prior to maturity if the stock price St is sufﬁciently
large.
Thus, early exercise of American call options is a dividend capture play,
since the option is written on the ex-dividend underlying asset price. Just as
it is for forward contracts. If the dividend one can capture by exercising
the option is relatively large enough, it can be optimal to early exercise.
Otherwise, not.
Any ROP result that we obtain for American calls on dividend-paying
underlying assets carries the qualiﬁer ‘only when the option is unexercised’. After
the option is exercised, it is extinguished, and no results can apply to an option
that has ceased to exist. Option pricing bounds only apply to alive options.
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More precisely, option pricing bounds apply where the underlying asset
price is in the continuation region (see Figure 11.1). When the underlying asset
price hits the early exercise boundary, its value is exactly equal to intrinsic
value. Beyond the early exercise boundary, in the stopping region exclusive of
the early exercise boundary, the option is no longer alive, if one has rationally
exercised it.
We now turn to show that the no-arbitrage lower bound in this American
scenario, for an American call on an underlier with proportional, continuous
dividends over the life of the option is the same as that for an American call
option on an underlying with no dividends given in section 11.4.1.
CA(St ,,E)уMAX[St–E, 0]
(Lower Bound American Calls
(LBACD)=(LBAC))
The implication of the dividend-adjusted lower bound, both for European
calls and European puts, discussed in sections 11.4.6 and 11.4.7 is that, if you
initiate a long position in a European call option at time t, then at expiration,
time T, you will receive only the adjusted terminal stock price S′T . As discussed, this is the stock price with dividends re-invested, ST , minus the future
value of the re-invested dividends paid out over the life of the option [t,T].
In the case of a European put, you only have to deliver S′T and make no
adjustment for the dividends paid over the period [t,T]. It is important to note
that it is the time difference T–t that creates the dividend adjustment issue.
And it is the European feature that precludes early exercise. These problems
go away when we turn to American options.
In the case of American options, early exercise is instantaneous so there is
no lag between the initiation date, t, of the long call or put position and the
potential exercise date T. That is, we can assume T=t if we want to do so.
When we do the arb discussed in End of Chapter Exercise 1, dividends will
not impact the value proposition, because we can buy the call option at time
t and then instantaneously exercise it, without losing any past dividends. Since
we can do this at any time t in a continuous-time, continuous-trading model,
the value of the American call must be no smaller than ordinary intrinsic value,
without any adjustment for dividends. That is, LBAC must prevail at all times
t′ in the maturity range of the option [t,T].
To connect up this lower bound with early exercise, it is the possibility of
attaining the lower bound in a world of dividends that drives early exercise.