Transcript Forwards and Futures

Properties of
Stock Option Prices
Chapter 8
1
ASSUMPTIONS:
1. The market is frictionless: No
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 exist; only one trading
price.
2
3.
Risk-free borrowing and lending exists
at the unique risk-free rate.
Risk-free borrowing is done by selling
T-bills short and risk-free lending
is done by purchasing T-bills.
4.
There exist no arbitrage opportunities
in the options market
3
NOTATIONS:
Ct = the market premium of an American call.
ct = the market premium of an European call.
Pt = the market premium of an American put.
pt = the market premium of an European put.
In general, we express the premiums as the
following functions:
Ct , ct = c{St , K, T-t, r, , D },
Pt , pt = p{St , K, T-t, r, , D }.
4
NOTATIONS:
t = the current date.
St= the market price of the underlying asset.
K = the option’s exercise (strike) price.
T = the option’s expiration date.
T-t = the time remaining to expiration.
r = the annual risk-free rate.
 = the annual standard deviation of the
returns on the underlying asset.
D= cash dividend per share.
q = The annual dividend payout ratio.
5
Options Risk-Return Tradeoffs
PROFIT PROFILE OF A STRATEGY
A graph of the profit/loss as a function of
all possible market values of the
underlying asset
We will begin with profit profiles at the
option’s expiration; I.e., an instant
before the option expires.
6
Options Risk-Return Tradeoffs At
Expiration
1. Only at expiry; T.
2. No time value; T-t = 0
CALL is: exercised if S > K
expires worthless if S  K
Cash Flow = Max{0, S – K}
PUT is: exercised if S < K
expires worthless if S  K
Cash Flow = Max{0, K – S}
7
3. All legs of the strategy remain open
till expiry.
4. A Table Format
Every row is one leg of the strategy.
Every row is analyzed separately.
The total strategy is the vertical sum of
the rows.The profit is the cash flow at
expiration plus the initial cash flows of
the strategy, disregarding the time
value of money.
8
6. A Graph of the profit/loss profile
The profit/loss from the strategy as a
function of all possible prices of the
underlying asset at expiration.
9
The algebraic expressions of profit/loss
at expiration:
Cash Flows:
Long stock:
ST – St
Short stock:
St - ST
Long call:
-c + MaX{0, ST -K}
Short call:
c + Min{0, K- ST }
Long put:
-p + MaX{0, K- ST}
Short put:
p + Min{0, ST -K}
10
Borrowing and Lending:
In many strategies with lending or borrowing
capital at the risk-free rate, the amount
borrowed or lent is the discounted value of
the option’s exercise price:
Ke-r(T-t).
The strategy’s holder can buy T-bills (lend) or
sell short T-bills (borrow) for this amount. It
follows that at the option’s expiration time,
the lender will receive this amount’s face
value, namely, a cash flow of K. If borrowed,
the borrower will pay this amount’s face
11
value, namely, a cash flow of – K.
RESULTS FOR CALLS:
1. Call values at expiration:
CT = cT = Max{ 0, ST – K }.
Proof:
At expiration the call is either
exercised, in which case CF = ST – K,
or it is left to expire worthless, in which
case, CF = 0.
12
RESULTS FOR CALLS:
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 NPV[Max{ 0, ST – K }]  0.
3. Maximum Call value: Ct  St.
Proof: The call is a right to buy the stock.
Investors will not pay for this right more
than the value that the right to buy gives
them, I.e., the stock itself.
13
RESULTS FOR CALLS:
4. Lower bound: American call value:
At any time t, prior to expiration,
Ct  Max{ 0, St - K}.
Proof:
Assume to the contrary that
Ct < Max{ 0, St - K}.
Then, buy the call and immediately exercise
it for an arbitrage profit of: St – K – Ct > 0; a
contradiction of the no arbitrage profits
assumption.
14
RESULTS FOR CALLS:
5. Lower bound: European call value:
At any t, t < T, ct  Max{ 0, St - Ke-r(T-t)}.
Proof: If, to the contrary,
ct < Max{ 0, St - Ke-r(T-t)}, then,
0 < St - Ke-r(T-t) - ct
At expiration
Strategy
I.C.F
ST < K
ST > K
Sell stock short
St
-ST
-ST
Buy call
- ct
0
ST - K
Lend funds
- Ke-r(T-t) K
K
Total
?
K – ST
0
15
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 - Ke-r(T-t)}.
Proof: An American call may be
exercised at any time, t, prior to
expiration, t<T, while the European call
holder may exercise it only at expiration.
16
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 < T2 for two calls on the same
underlying asset and the same exercise
price. To show that c2 > c1 assume that
c1 > c2 or, c1 - c2 > 0.
At expiration T1
Strategy I.C.F
ST1 < K
ST1 > K
Sell c(T1 )
c1
0
-(ST1 –K)
Buy c(T2 )
- c2
c
c
Total
?
c
?
17
RESULTS FOR CALLS:
8. Cash dividends and calls:
It is not optimal to exercise an American call
prior to its expiration if the underlying stock
does not pay 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.
18
RESULTS FOR CALLS:
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.
19
RESULTS FOR CALLS:
10. Early exercise of Unprotected American
calls on a cash dividend paying stock:
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:
20
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.
S
CUMD
t
A
4.
S
t
=S
XDIV
XDIV
S
XDIV
CDIV
t
- D.
PAYMENT
Time line
21
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 – K
c{SXD, K, T - tXD}
Substitute: SCD = SXD + D.
Cash flow: SXD –K + D SXD – K + TV.
22
Conclusion:
Early exercise of American calls
may be optimal:
If the call is in the money
and If
D > TV,
early exercise is optimal.
In this case, the call should be (optimally)
exercised an instant before the stock
goes x-dividend.
23
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 xdividend stock price S*XD satisfies:
S*XD –K + D = c{S*XD , K, T - tXD}.
It can be shown that this implies that the
Price, S*XD ,exists if:
D > K[1 – e-r(T – t)].
24
RESULTS FOR CALLS:
11. The money value of calls:
The higher the exercise price, the lower is
the value of a call.
Proof: Let K1 < K2 be the exercise prices for
two calls on the same underlying asset
and the same time to expiration. To show
that c2 < c1 assume, to the contrary, that
c2 > c1 or, c2 - c1 > 0. Then,
25
RESULTS FOR CALLS:
At expiration
Strategy
ICF ST < K1 K1<ST < K2 ST >K2
Sell c(K2 )
c2
0
0
-(ST –K2)
Buy c(K1 )
-c1 0
ST –K1
ST–K1
Total
?
0
ST –K1
K2 - K1
26
RESULTS FOR CALLS:
12. The money value of American calls:
Let K1 < K2 then C1 - C2  K2 - K1
Proof: Let K1 < K2 be the exercise prices for
two American calls on the same underlying
asset and the same time to expiration.
Assume that C1 - C2 > K2 - K1 or,
equivalently: C1 - C2 – (K2 - K1) > 0.
Then,
27
RESULTS FOR CALLS:
At expiration
Strategy ICF ST < K1
K1<ST < K2 ST
>K2
Sell C(K1 ) C1
0
-(ST –K1)
-(ST –K1)
Buy C(K2 )-C2
0
0
ST–K2
Lend – (K2 - K1) K2-K1+i K2- K1+i K2- K1+i
Total
?
K2-K1+i K2-ST+i
i
i = interest
Even if the sold call is exercised before
Expiration, the total value in hand is >0.
28
RESULTS FOR CALLS:
13. The money value of calls:
Let K1<K2 then c1-c2  (K2-K1)e-r(T-t)
Proof: Let K1 < K2 for two European calls
On the same underlying asset and the
same time to expiration.
Assume that c1-c2 > (K2-K1)e-r(T-t)
or, c1-c2 -(K2-K1)e-r(T-t) >0. Then,
29
RESULTS FOR CALLS:
At expiration
Strategy
ICF ST < K1 K1<ST < K2 ST >K2
Sell c(K1 )
c1
0
-(ST –K1) -(ST –K1)
Buy c(K2 )
-c2 0
0
ST–K2
Lend -(K2-K1)e-r(T-t) K2-K1
K2-K1
K2-K1
Total
?
K2-K1
K2- ST
0
30
RESULTS FOR CALLS:
14. The money value of calls:
Let K1 < K2 < K3 and
K2 = K1 + (1 - )K3 for any : 0 <  < 1.
The premiums on the three calls must
satisfy:
c2  c1 + (1 - )c3
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RESULTS FOR CALLS:
At Expiration
STRATEGY
ICF ST < K1 K1<ST < K2 K2 <ST < K3 ST > K3
Buy  calls K1 -c1
0
(ST – K1) (ST – K1) (ST – K1)
Sell one call K2
c2
0
0
-(ST – K2) -(ST – K2)
Buy 1- calls K3 -(1-)c3 0
0
0
(1-)(ST–K3)
Total
c2-c1-(1-)c3 0 (ST–K1) (1-)(K3-ST)
0
All the cash flows at expiration are non
negative. Hence, the Initial Cash Flow cannot
be positive! c2-c1-(1-)c3  0.
Or,
c2  c1 + (1 - )c3.
32
RESULTS FOR CALLS:
Example: Let $80, $95 and $100 be the three
exercise prices. $95 = (.25)$80 + (.75)$100.
Thus,
 = 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) > c(80) + 3c(100) or,
4c(95) - c(80) - 3c(100) > 0
and arbitrage profit can be made by the
Strategy: Buy the $80 call and sell four $95
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Calls and 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.
34
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 - Ke-r(T-t)}.
With increasing risk-free rates, the
difference St - Ke-r(T-t) increases and the call
value must increase as well.
35
RESULTS FOR PUTS:
17.
Put values at expiration:
PT = pT = Max{ 0, K - ST}.
Proof:
At expiration the put is either exercised, in
which case CF = K - ST, or it is left to expire
worthless, in which case CF = 0.
36
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
NPV[Max{ 0, K - ST}]  0.
37
RESULTS FOR PUTS:
19a.Maximum American Put value:
At any time t < T,
Pt  K.
Proof: The put is a right to sell the stock for
K, thus, the put’s price cannot exceed the
maximum value it will create: K, which occurs
if S drops to zero.
19b.Maximum European Put value:
Pt  Ke-r(T-t).
Proof: The maximum gain from a European
put is K, ( in case S drops to zero). Thus, at
any time point before expiration, the
European put cannot exceed the NPV{K}. 38
RESULTS FOR PUTS:
20.Lower bound: American put value:
At any time t, prior to expiration,
Pt  Max{ 0, K - St}.
Proof: Assume to the contrary that
Pt < Max{ 0, K - St}.
Then, buy the put and immediately
exercise it for an arbitrage profit of:
K - St – Pt > 0. A contradiction of the
no arbitrage profits assumption.
39
RESULTS FOR PUTS:
21. Lower bound: European put value:
At any time t, t < T, pt  Max{ 0, Ke-r(T-t) - St}.
Proof: If, to the contrary,
pt < Max{ 0, Ke-r(T-t) - St}
then,
Ke-r(T-t) - St - pt > 0.
At expiration
Strategy
I.C.F
ST < K
ST > K
Buy stock
-St
ST
ST
Buy put
- pt
K - ST
0
Borrow
Ke-r(T-t)
-K
-K
Total
?
0
ST - K 40
RESULTS FOR PUTS:
22. An American put is always priced higher than
an European put.
Pt  pt  Max{0, Ke-r(T-t) - St}.
Proof: An American put may be exercised at
any time, t, prior to expiration, t < T, while the
European put holder may exercise it only at
expiration.
If the price of the underlying asset fall below some
price, it becomes optimal to exercise the American
put. At that very same moment the European put
holder wants to (optimally) exercise the put but
cannot because it is a European put.
41
RESULTS FOR PUTS:
23. American put is always priced higher than its
European counterpart. Pt  pt
P/L
K
Ke-r(T-t)
For S< S** the European put
premium is less than the put’s
intrinsic value.
For S< S* the American put
premium coincides with the put’s
intrinsic value.
P
p
S*
S** K
S
42
RESULTS FOR PUTS:
24. The time value of puts:
The longer the time to expiration, the higher
is the value of an American put.
Proof: Let T1 < T2 for two American puts
on the same underlying asset and the same
exercise price. To show that P2 > P1 assume,
to the contrary, that P2 < P1 or, P1 - P2 > 0.
At expiration T1
Strategy
I.C.F
ST1 < K ST1 < K
Sell P(T1 )
P1
ST1 –K
0
Buy P(T2 )
-P2
P
P
Total
?
?
P
43
RESULTS FOR PUTS:
25. The money value of puts:
The higher the exercise price, the higher
is the value of a put.
Proof: Let K1 < K2 for two puts on the
same underlying asset and the same
time to expiration. To show that p2 > p1
assume, to the contrary, that p2 < p1
or, p1 - p2 > 0.
Then,
44
RESULTS FOR PUTS:
At expiration
Strategy ICF
ST < K1 K1<ST < K2
Sell p(K1 ) p1
ST –K1
0
Buy p(K2 ) -p2
K2 –ST
K2 - ST
Total
?
K2 - K1 K2 - ST
ST >K2
0
0
0
45
RESULTS FOR PUTS:
26. The money value of puts:
Let K1 < K2 then P2 - P1  K2 - K1
Proof: Let K1 < K2 be the exercise prices or
two American puts on the same underlying
asset and the same time to expiration.
Assume, contrary to the result’s assertion,
that
P2 - P1 > K2 - K1 or,
equivalently:
P2 - P1 – (K2 - K1) > 0.
Then,
46
RESULTS FOR PUTS:
At expiration
Strategy ICF
ST < K1 K1<ST < K2 ST >K2
Sell P(K2 ) P2
ST –K2
ST –K2)
0
Buy P(K1 )-P1
K1 – ST
0
0
Lend – (K2 - K1) K2-K1+i K2- K1+i K2K1+i
Total
?
i
ST-K1 +i K2- K1+i
i = interest
Even if the sold put is exercised before
expiration, the total value in hand is >0.47
RESULTS FOR PUTS:
27. The money value of puts:
Let K1<K2 then p2-p1  (K2-K1)e-r(T-t)
Proof: Let K1 < K2 for two European puts
on the same underlying asset and the same
time to expiration. Assume, contrary to the
result’s assertion, that
p2-p1 > (K2-K1)e-r(T-t) or,
p2- p1 -(K2-K1)e-r(T-t) >0. Then,
48
RESULTS FOR PUTS:
At expiration
Strategy ICF
ST < K1 K1<ST < K2 ST > K2
Sell p(K2 ) p2
ST –K2
ST –K2
0
Buy p(K1 ) -p1
K1 – ST
0
0
Lend -(K2-K1)e-r(T-t) K2-K1 K2-K1
K2-K1
Total
?
0
K2- ST
K2-K1
In the absence of arbitrage, the initial
cash flow cannot be positive.
49
RESULTS FOR PUTS:
28. Volatility:
The higher the price volatility of the
underlying asset, the higher is the put value.
Proof: The put holder never loses more than
the initial premium. The upside gain,
however, is increasing from zero to K. Thus,
higher volatility increases the potential gain
while the potential loss remains unchanged.
50
RESULTS FOR PUTS:
29. The interest rate:
The higher the risk-free rate, the lower is
the put value.
Proof: Follows from result 22:
Ct  ct  Max{ 0, St - Ke-r(T-t)}.
With increasing risk-free rates, the
difference Ke-r(T-t) - St decreases and the
put value decrease too.
51
RESULTS for PUTS and CALLS:
30. The put-call parity.
European options: The premiums of European
calls and puts written on the same non
dividend paying stock for the same expiration
and the same strike price must satisfy:
ct - pt = St - Ke-r(T-t).
The parity may be rewritten as:
Proof:
ct + Ke-r(T-t) = St + pt.
52
RESULTS for PUTS and CALLS:
At expiration
Strategy
I.C.F
ST < K ST > K
Buy stock
-St
ST
ST
Buy put
- pt
K - ST
0
Total
-(St+pt)
K
ST
At expiration
Strategy
I.C.F
ST < K ST > K
Buy call
- ct
0
ST-K
Lend
- Ke-r(T-t)
K
K
Total
-(ct+ Ke-r(T-t) ) K
ST
53
RESULTS for PUTS and CALLS:
31. Synthetic European options:
The put-call parity
ct + Ke-r(T-t = St + pt
can be rewritten as a synthetic call:
ct = pt + St - Ke-r(T-t),
or as a synthetic put:
pt = ct - St + Ke-r(T-t).
54
RESULTS for PUTS and CALLS:
32. The put-call parity. European options:
Suppose that European puts and calls are
written on a dividend paying stock and
suppose that there will be two dividend
payments during the life of the options:
D1 at t1 and D2 at t2.
The option’s premiums must satisfy the
following equation:
ct-pt = St-Ke-r(T-t) – D1e-r(t1-t) – D2 e-r(t2-t)
55
RESULTS for PUTS and CALLS:
33. Box spread:
At expiration
Strategy ICF ST < K1 K1<ST < K2 ST > K2
Buy p(K2) -p2 K2 - ST
K2 – ST
0
Sell p(K1) p1 ST - K1
0
0
Sell c(K2) c2
0
0
K2 - ST
Buy c(K1) -c1
0
ST - K1
ST - K1
Total
? K2-K1
K2-K1
K2-K1
Therefore, the initial investment is riskless.
c1 - c2 + p2 - p1 = (K2-K1)e-r(T-t)
56
RESULTS for PUTS and CALLS:
33. Box spread:
Again: An initial investment of c1 - c2 + p2 - p1
yields a sure cash flow of K2-K1. Thus,
arbitrage profit exists if the rate of return on
this investment is not equal to the T-bill
rate which matures on the date of the
options’ expiration.
c1 - c2 + p2 - p1 = (K2-K1)e-r(T-t)
1
K 2  K1
r
ln(
)
T  t c1  c 2  p 2  p1
57
RESULTS for PUTS and CALLS:
34. The put-call parity. American options:
The put-call parity for European options
asserts that:
ct - pt = St - Ke-r(T-t).
This result does not necessarily hold for
American options. The premiums on American
options satisfy the following inequalities:
St - K < Ct - Pt < St - Ke-r(T-t).
58
RESULTS for PUTS and CALLS:
Proof: Rewrite the inequality:
St - K < Ct - Pt < St - Ke-r(T-t).
The RHS of the inequality follows from the
parity for European options. The stock does
not pay dividend, thus, Ct = ct. For the
American puts, however, Pt > pt. Next
suppose that:
St - K > Ct - Pt
or,
St - K - Ct + Pt > 0.
It can be easily shown that this is an
arbitrage profit making strategy, which
contradicts the supposition above.
59
RESULTS for PUTS and CALLS:
When the options are written on a dividend
paying stock the RHS of the inequality
remains the same:
Ct - Pt < St - Ke-r(T-t).
Assuming two dividend payments, the LHS of
the inequality becomes:
St - K – D1e-r(t1-t) – D2 e-r(t2-t) < Ct - Pt
60