BOUNDS AND OTHER NO ARBITRAGE CONDITIONS ON OPTIONS PRICES First we review the topics: Risk-free borrowing and lending and Short sales

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Title: BOUNDS AND OTHER NO ARBITRAGE CONDITIONS ON OPTIONS PRICES First we review the topics: Risk-free borrowing and lending and Short sales

1
BOUNDS AND OTHER NO ARBITRAGE CONDITIONS ON
OPTIONS PRICESFirst we review the
topicsRisk-free borrowing and lending and
Short sales
2
Risk-free lending and borrowing

Arbitrage A market situation in
which an investor can make a profit
with no equity and no risk.
Efficiency A market is said to be
efficient if prices are such that there
exist no arbitrage opportunities.
Alternatively,
a market is said to be inefficient if
prices present arbitrage opportunities
for investors in this market.

Risk-free lending and borrowing
PURE ARBITRAGE PROFIT
A PROFIT MADE
1.WITHOUT EQUITY INVESTMENT
and
WITHOUT ANY RISK
We will assume that
the options market is efficient.
This assumption implies that one cannot make
arbitrage profits in the options markets

4
We are now ready to analyze upper and lower
BOUNDS AND OTHER NO ARBITRAGE CONDITIONS ON
OPTIONS PRICES.The basic assumptions and
notations that underlie the analysis are
5
ASSUMPTIONS1. The market is frictionlessNo
transaction cost nor taxes exist. Trading are
executed instantly. There exists no restrictions
to short selling.2. Market prices are
synchronous across assets. If a strategy
requires the purchase or sale of several assets
in different markets, the prices in these markets
are simultaneous. Moreover,No bid-ask spread
existonly one market price.
6
ASSUMPTIONS3. Risk-free borrowing and lending
exists at the uniquerisk-free rate. Risk-free
borrowing and lending is done via selling short
and purchasing T-bills 4. There exist no
arbitrage opportunities in the options market.
7

NOTATIONS
t the current date.
St the current market price of the
underlying asset.
X the options exercise (strike) price. K in
the text book.
T the options expiration date.
T-t the time remaining to the options
expiration.
r the annual risk-free rate.
? the annual standard deviation of the
returns on the underlying asset.
D cash dividend per share.

NOTATIONS
Ct the current market premium of an
American call.
ct the current market premium of an
European call.
Pt the current market premium of an
American call.
pt the current market premium of an
European call.
In general, we express the options
premiums as the following functions
Ct , ct cSt , X, T-t, r, ?, D ,
Pt , pt pSt , X, T-t, r, ?, D .

FINAL REMARK
Many strategies described below use
lending or borrowing capital at the risk-
free rate. Mostly, the amount borrowed
or lent is the discounted value of the
options exercise price.
Namely, Xe-r(T-t).
The assumption here is that the holder of
the strategy can always buy T-bills (lend)
or sell short T-bills (borrow) for exactly
the amount of Xe-r(T-t). It follows that
upon terminating the strategy at the
options expiration time, the lender will
receive this amounts face value, namely,
a cash flow of X. If borrowed, the
borrower will pay this amounts face
value, namely, a cash flow of X.

RESULTS FOR CALLS
1. Call values at expiration
(3.1) page 102
CT cT Max 0, ST X .
Proof
At expiration the call is either
exercised, in which case CF ST X,
or you let the option expire
worthless, in which case, CF 0.

RESULTS FOR CALLS(P104)
2. Minimum call value
A call premium cannot be negative.
At any time t, prior to expiration,
Ct , ct ? 0.
Proof The current market price of a
call is the NPVMax 0, ST X ? 0.
3. Maximum Call value Ct ? St.
Proof The call is a right to buy the stock.
Investors will not pay for this right a price
that is higher than what the value the right to
buy gives them, I.e., the stock itself.

RESULTS FOR CALLS
4. Lower bound American call value
At any time t, prior to expiration,
Ct ? Max 0, St - X.
Proof Assume to the contrary that
Ct lt Max 0, St - X.
Then, buy the call and immediately exercise it
for an arbitrage profit of St X Ct gt 0.
Contradiction of the no arbitrage profits
assumption.

RESULTS FOR CALLS
11. The money value of calls
The higher the exercise price, the lower is the
value of a call.
Proof Let X1 lt X2 be the exercise prices for
two calls on the same underlying asset and the
same time to expiration. To show that c2 lt c1
assume, to the contrary, that c2 gt c1 or, c2 -
c1 gt 0. Then,
At expiration
Strategy ICF ST lt X1 X1ltST lt X2 ST gtX2
Sell c(X2 ) c2 0 0 -(ST X2)
Buy c(X1 ) -c1 0 ST X1 STX1
Total ? 0 ST X1 X2 - X1
In the absence of arbitrage the initial cash
flow cannot be positive.

RESULTS FOR CALLS
12. The money value of calls
Let X1 lt X2 then C1 - C2 ? X2 - X1
Proof Let X1 lt X2 be the exercise prices or
two American calls on the same underlying
asset and the same time to expiration.
Assume that C1 - C2 gt X2 - X1 or,
equivalently C1 - C2 (X2 - X1) gt 0. Then,
At expiration
Strategy ICF ST lt X1 X1ltST lt X2 ST gtX2
Sell C(X1 ) C1 0 -(ST X1) -(ST X1)
Buy C(X2 )-C2 0 0 STX2
Lend (X2 - X1) X2-X1i X2- X1i X2-
X1i
Total ? X2-X1i X2-STi i
i interest
Even if the sold call is exercised before
Expiration, the total value in hand is gt0.

RESULTS FOR CALLS
13. The money value of calls
Let X1ltX2 then c1-c2 ? (X2-X1)e-r(T-t)
Proof Let X1 lt X2 for two European calls
on the same underlying asset and the same
time to expiration. Assume that
c1-c2 gt (X2-X1)e-r(T-t) or,
c1-c2 -(X2-X1)e-r(T-t) gt0. Then,
At expiration
Strategy ICF ST lt X1 X1ltST lt X2 ST gtX2
Sell c(X1 ) c1 0 -(ST X1) -(ST X1)
Buy c(X2 )-c2 0 0 STX2
Lend -(X2-X1)e-r(T-t) X2-X1 X2-X1
X2-X1
Total ? X2-X1 X2- ST 0
In the absence of arbitrage, the initial
cash flow cannot be positive.

RESULTS FOR CALLS
5. Lower bound European call value
At any time t, prior to expiration,
ct ? Max 0, St - Xe-r(T-t).
Proof If, to the contrary,
ct lt Max 0, St - Xe-r(T-t), then,
0 lt St - Xe-r(T-t) - ct
At expiration
Strategy I.C.F ST lt X ST gt X
Sell stock short St -ST -ST
Buy call - ct 0 ST - X
Lend funds - Xe-r(T-t) X X
Total ? X ST 0
In the absence of arbitrage, the initial
cash flow cannot be positive.

RESULTS FOR CALLS
6. The market value of an American call is at
least as high as the market value of a European
call.
Ct ? ct ? Max 0, St - Xe-r(T-t).
Proof An American call may be exercised at any
time, t, prior to expiration, tltT, while the
European call holder may exercise it only at
expiration.

RESULTS FOR CALLS
7. The time value of calls
The longer the time to expiration, the higher is
the value of a call.
Proof Let T1 lt T2 for two calls on the same
underlying asset and the same exercise price. To
show that
c2 gt c1 assume, to the contrary, that c1 gt c2
or, c1 - c2 gt 0.
At expiration T1
Strategy I.C.F ST1 lt X ST1 gt X
Sell c(T1 ) c1 0 -(ST1 X)
Buy c(T2 ) - c2 c(TV) C(T2-T1)
Total ? c(TV) ?

RESULTS FOR CALLS
Case 1. The calls are American style.
The question is whether
C(T2-T1) - (ST1 X) ?
If gt 0 the proof is completed.
If lt 0, the open call can be exercised
immediately for (ST1 X) and the proof is
complete.
Case 2. The calls are European style.
In this case, result 5. Guarantees that the
cash flow above is gt0 because
ct1 ? St1 - Xe-r(T2-T1).

RESULTS FOR CALLS
8. Cash dividends and calls
Cash dividends and calls
It is not optimal to exercise an American call
prior to its expiration if the underlying stock
does not pay out any dividend during the life
of the option.
Proof If an American call holder wishes
to rid of the option at any time prior to its
expiration, the market premium is greater
than the intrinsic value because the time
value is always positive.
9. The American feature is worthless if the
underlying stock does not pay out any
dividend during the life of the call.
Mathematically Ct ct.
Proof Follows from result 8. above.

RESULTS FOR CALLS
10. Early exercise of Unprotected American calls
on a cash dividend paying stock. (Section 8.7)
Consider an American call on a cash
dividend paying stock. It may be
optimal to exercise this American call
an instant before the stock goes X
dividend.
Two condition must hold for the early
exercise to be optimal
First, the call must be in-the-money.
Second, the dividend/share, D, must
exceed the time value of the call at the
x dividend instant. To see this result
Consider

RESULTS FOR CALLS
FACTS
1. The share price drops by D/share
when the stock goes x-dividend.
2. The call value decreases when the price per
share falls.
3. The exchanges do not compensate call holders
for the loss of value that ensues the price drop
on the x-dividend date.

SCUMD
SXDIV
tA
tXDIV
tPAYMENT
Time line
4. SXDIV SCDIV - D.
23

Early exercise of
Unprotected American calls on a cash dividend
paying stock
The call holder goal is to maximize the
Cash flow from the call. Thus, at any
moment in time, exercising the call is
inferior to selling the call. This conclusion
may change, however, an instant before
the stock goes x dividend
Exercise Do not exercise
Cash flow SCD X cSXD, X, T-tXD-
Substitute SCD SXD D.
Cash flow SXD X D SXD X TV.
Conclusion Early exercise of American
calls may be optimal. If the call is in the
money and D gt TV, early exercise is
optimal.

Early exercise of
Unprotected American calls on a cash dividend
paying stock
This result means that an investor is
indifferent to exercising the call an
instant before the stock goes x
dividend if the x dividend stock
price SXD satisfies
SXD X D cSXD , X, tXD-t.
It can be shown that this implies
that the price price SXD exists if
D gt X1 e-r(T t).

RESULTS FOR CALLS
14. The money value of calls
Let X1 lt X2 lt X3 and
X2 ?X1 (1 - ?)X3 for 0 lt ? lt 1.
The premiums on the three calls must
satisfy c2 ? ?c1 (1 - ?)c3 At
Expiration
STRATEGY ICF ST lt X1 X1ltST lt X2 X2
ltST lt X3 ST gt X3
Buy ? calls X1 -?c1 0 ?(ST X1)
?(ST X1) ?(ST X1)
Sell one call X2 c2 0 0 -(ST
X2) -(ST X2)
Buy 1-? calls X3 -(1-?)c3 0 0 0
(1-?)(STX3)
Total c2-?c1-(1-?)c3 0 ?(STX1)
(1-?)(X3-ST) 0
All the cash flows at expiration are non
negative.
Hence, the Initial Cash Flow cannot be positive!
I.e.,
c2-?c1-(1-?)c3 ? 0.
Or, c2 ? ?c1 (1 - ?)c3.
Remark When ? 1/2 , the strategy is
called a Butterfly. In this case, 2X2 X1 X3
and the result asserts 2c2 ? c1 c3

RESULTS FOR CALLS
Example Let 80, 95 and 100 be the
exercise prices of three calls on the same
underlying asset and the same expiration.
Observe that 95 (.25)80 (.75)100.
Thus, in this case, ? ¼. Result 14.
Asserts that c(95) ? ¼c(80) ¾c(100). Or,
4c(95) ? c(80) 3c(100).
If the latter inequality does not hold, then
4c(95) gt c(80) 3c(100) or,
4c(95) - c(80) - 3c(100) gt 0
and arbitrage profit can be made by the
Strategy
Buy the 80 call
Sell four 95 calls
Buy three 100 calls.

RESULTS FOR CALLS
15. Volatility
The higher the price volatility of the
Underlying asset, the higher is the
Call value.
Proof The call holder never loses
more than the initial premium. The
upside gain, however, is unlimited.
Thus, higher volatility increases the
potential gain while the potential loss
remains Unchanged.

RESULTS FOR CALLS
16. The interest rate
The Higher the risk-free rate, the
Higher is the call value.
Proof The result follows from result 6

Ct ? ct ? Max 0, St - Xe-r(T-t).
With increasing risk-free rates, the difference
St - Xe-r(T-t) increases and the call value must
increase as well.
29

RESULTS FOR PUTS
17. Put values at expiration
PT pT Max 0, X - ST.
Proof
At expiration the put is either exercised, in
which case CF X - ST, or it is left to expire
worthless, in which case CF 0.

RESULTS FOR PUTS
18. Minimum put value
A put premium cannot be negative.
At any time t, prior to expiration
Pt , pt ? 0.
Proof The current market price of a put is the
NPVMax 0, X - ST ? 0.
19a.Maximum American Put value
At any time t lt T, Pt ? X.
Proof
The put is a right to sell the stock for X,
thus, the puts price cannot exceed the maximum
value it will create X, which occurs if S drops
to zero.

RESULTS FOR PUTS
19b.Maximum European Put value
Pt ? Xe-r(T-t).
Proof
The European put may be exercised only at
expiration. The maximum revenue it can create at
that time is X, ( in case S drops to zero), thus,
at any time point before expiration, the European
put cannot exceed the NPVX.

RESULTS FOR PUTS
20.Lower bound American put value
At any time t, prior to expiration,
Pt ? Max 0, X - St.
Proof Assume to the contrary that
Pt lt Max 0, X - St.
Then, buy the put and immediately
exercise it for an arbitrage profit of
X - St Pt gt 0. A contradiction of the
no arbitrage profits assumption.

RESULTS FOR PUTS(P116)
21. Lower bound European put value
At any time t, t lt T,
pt ? Max 0, Xe-r(T-t) - St.
Proof If, to the contrary,
pt lt Max 0, Xe-r(T-t) - St then,
0 lt Xe-r(T-t) - St - pt
At expiration
Strategy I.C.F ST lt X ST gt X
Buy stock -St ST ST
Buy put - pt X - ST 0
Borrow Xe-r(T-t) - X - X
Total ? 0 ST- X
In the absence of arbitrage, the initial
cash flow cannot be positive.

RESULTS FOR PUTS
22. The market value of an American put is at
least as high as the market value of a European
put.
Pt ? pt ? Max0, Xe-r(T-t) - St.
Proof An American put may be exercised at any
time, t, prior to expiration, tltT, while the
European put holder may exercise it only at
expiration.

RESULTS FOR PUTS
23. The time value of puts
The longer the time to expiration, the
higher is the value of an American put.
Proof Let T1 lt T2 for two American
puts on the same underlying asset and
the same exercise price.
To show that P2 gt P1 assume, to the
contrary, that P1 gt P2 or, P1 - P2 gt 0.
At expiration T1
Strategy I.C.F ST1 lt X ST1 lt X
Sell P(T1 ) P1 -(ST1 X) 0
Buy P(T2 ) -P2 P(T2-T1) P(TV)
Total ? ? P(TV)
You may exercise the put. The result
follows.

RESULTS FOR PUTS
24. American put is always priced higher than
its European counterpart.
Pt ? pt
Proof An American put may be
exercised at any time, t, prior to expiration,
tltT, while the European put holder may
exercise it only at expiration. Moreover, if
The price of the underlying asset fall below
a certain threshold price, it becomes
Optimal to exercise the American put
and earn X St. At that very same moment
the European put holder wants to
(optimally) Exercise the put but cannot
because it is a European put.
The next figure demonstrates the
relationship Between the American and
European puts premiums.

RESULTS FOR PUTS(P176,7)
24. American put is always priced higher than
its European counterpart. Pt ? pt

For Slt S the European put premium is less than
the puts intrinsic value. For Slt S the
Americanan put premium coincides with the puts
intrinsic value.
P/L
X
Xe-r(T-t)
P
p
S S X
S
38

RESULTS FOR PUTS
25. The money value of puts
The higher the exercise price, the higher is the
value of a put.
Proof Let X1 lt X2 for two puts on the same
underlying asset and the same time to expiration.
To show that p2 gt p1 assume, to the contrary,
that p2 lt p1 or, p1 - p2 gt 0.
Then,
At expiration
Strategy ICF ST lt X1 X1ltST lt X2 ST gtX2
Sell p(X1 ) p1 ST X1 0 0
Buy p(X2 ) -p2 X2 ST X2 - ST 0
Total ? X2 - X1 X2 - ST 0
In the absence of arbitrage, the initial cash
flow cannot be positive.