Properties of Stock Option Prices Chapter 9

1

Properties ofStock Option Prices

Chapter 9

2

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.

3

3. Risk-free borrowing and lending exists at the unique risk-free rate.

Risk-free borrowing is done by sellingT-bills short and risk-free lending is done by purchasing T-bills.

4. There exist no arbitrage opportunities in the options market

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

Factors affecting Options Prices(Secs. 9.1 and 9.2)

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 functions of the following variables:

Ct , ct = c{St , K, T-t, , r, D },

Pt , pt = p{St , K, T-t, , r, D }.

6

FACTORS AFFECTING OPTIONS PRICES AND THE DIRECTION OF THEIR

IMPACT:Facto

rEuropea

n callEuropea

n putAmerica

n callAmerica

n putSt + - + -K - + - +

T-t ? ? + + + + + +r + - + -D - + - +

7

Bounds on options market prices (Sec. 9.3)

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.

8

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

NPV[Max{ 0, ST – K }] 0.

9

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.

10

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.

11

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

NPV[Max{ 0, K - ST}] 0.

12

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.

13

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}.

14

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 simultaneously exercise it for an arbitrage profit of:

St – K – Ct > 0. a contradiction.

15

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, ct < St - Ke-r(T-t) 0 < St - Ke-r(T-t) - ct

P/L PROFILE OF

STRATEGY: ICFAT

EXPIRATIONST < K ST > K

SELL THE STOCK SHORT

St - ST - ST BUY CALL -ct 0 ST - K

LEND NPV OF K -Ke-r(t-t) K KTOTAL ? K - ST 0

16

American vs European Calls

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.

17

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 simultaneouslyexercise it for an arbitrage profit of: K - St – Pt > 0. A contradiction of the no arbitrage profits assumption.

18

American vs European Puts (Sec. 9.6)

Pt pt Max{0, Ke-r(T-t) - St}.

Proof: First, An American put may beexercised at any time, t; t < T. A European

putmay be Exercise only at T. If the price of

theunderlying asset fall below some price, itbecomes optimal to exercise the American put. At that very same moment the

Europeanput holder wants to exercise the put but cannot because it is European.

19

Second, the other side of the inequality:At any 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, pt < Ke-r(T-t) – St 0 < Ke-r(T-t) - St - pt

P/L PROFILE OF STRATEGY: ICF

AT EXPIRATION

ST < K ST > KBuy the stock - St ST ST Buy the put - pt K - ST 0

Borrow NPV OF K Ke-r(t-t) - K - KTOTAL ? 0 ST - K

20

American put is always priced higher than its European counterpart. Pt pt

S* S** K

P/L

K

S

Ke-r(T-t)

Pp

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.

21

The put-call parity (Sec. 9.4)European options:The premiums of European calls and putswritten on the same non dividend payingstock for the same expiration and the

samestrike price must satisfy:

ct - pt = St - Ke-r(T-t).

The parity may be rewritten as:

ct + Ke-r(T-t) = St + pt.

22

P/L PROFILE of STRATEGY

ICFAT EXPIRATION

ST < K ST > K

BUY STOCK

- St ST ST

BUY PUT - pt K - ST 0TOTAL - [St + pt] K STP/L

PROFILE ofSTRATEGY ICF

AT EXPIRATION

ST < K ST > K

BUY CALL - ct 0 ST - KLEND

NPV(K) - Ke– r(T – t) K KTOTAL - [ct + Ke– r(T – t)] K ST

23

Synthetic European options:

The put-call parityct + 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).

24

Synthetic Risk-free rate

The put-call parity ct + Ke-r(T-t) = St + pt

ttt cpS

KlntT

1r

For another synthetic risk-free rate we next analyze the Box Spread strategy:

25

Stock XYZ, TH SEP 21, 2007. All prices $/share.S=61.4

8calls puts

K OCT NOV JAN APR OCT NOV JAN APR40 22 23 .5650 12 .6355 8.13 11.5 1.25 3.6360 4.75 8.75 2.88 4.00 5.7565 2.50 3.98 5.75 8.63 6.00 6.63 8.38 10.0070 .94 3.88 9.25 11.2575 .31 5.13 13.38 16.7980 1.6390 .8195 .44

26

P/L of strategy ICF AT EXPIRATIONST < 75 ST > 75

Buy the STOCK -61.48 ST ST

Buy APR 75 PUT -16.79 75 - ST 0Sell APR 75 CALL 5.13 0 75 - ST

TOTAL -73.14 75 75P/L 1.86 1.86

This strategy guarantees its holder a sure profit of $1.86/share for an investment of $73.14/share in a 7 months period.For this strategy to work all the initial prices – the stock the put and the call must be available for the investor at the same instant.If the strategy is possible, it creates a RISK-FREE rate:

1. Stock and options markets

27

4.305%73.14

75ln7/12

1r

75 73.14e 127r

28

P/L of strategy ICF AT EXPIRATIONST < 75 ST > 75

sell the STOCK short

61.48 -ST -ST

short APR 75 PUT 16.79 -[75 – ST] 0long APR 75 CALL -5.13 0 -[75 – ST]

Buy T-bills -73.14 75.38 75.38TOTAL 0 0.38 0.38

0.38 0.38

Suppose that the yield on T-bills that mature on the option’s expiration is r = 5.17%. Then, to make arbitrage profit:

When the T-bills mature, the GOV pays you:

73.14e.0517[7/12] = 75.38.The above strategy guarantees you an ARBITRAGE PROFIT of 38 cents per share.

29

Another Synthetic Risk-free rateA Box spread (p. 235): K1 < K2.

P/L PROFILE ofSTRATEGY ICF

AT EXPIRATIONST < K1 K1<ST < K2 ST > K2

Buy p(K2) - p2

K2 - ST K2 - ST 0

Sell p(K1) p1 ST - K1 0 0Sell c(K2) c2 0 0 - [ST - K2]Buy c(K1) -c1 0 ST – K1 ST - K1

TOTAL ? K2 - K1 K2 - K1 K2 - K1

30

The initial cost of the box spread is:

c1 - c2 + p2 - p1

The certain income from the box spread at the options’ expiration, T, is:

K2 - K1

Thus:

c1 - c2 + p2 - p1 = (K2-K1)e-r(T-t)

31

Reiterating:For the Box spread strategy: An initial investment of c1 - c2 + p2 - p1

yields a sure income of K2-K1regardless of theunderlying asset’s market price. Thus, solving

c1 - c2 + p2 - p1 = (K2-K1)e-r(T-t)

for r, yields a risk-free rate:

1221

12

ppccKKln

tT1r

32

P/L of strategy ICF AT EXPIRATIONST < 55 55 < ST <

60ST > 60

Buy the JAN 55 CALL - 11.50 0 ST - 55 ST - 55Buy the JAN 60 PUT - 5.75 60 - ST 60 - ST 0

Sell the JAN 60 CALL 8.75 0 0 60 - ST

Sell the JAN 55 PUT 3.63 ST - 55 0 0TOTAL - 4.87 5.00 5.00 5.00

This strategy guarantees its holder a sure profit of $.13/share for an investment of $4.87/share in a 4 months period.For this strategy to work all the initial prices – the CALLS and the PUTS must be available for the investor at the same instant.If the strategy is possible, it creates a RISK-FREE rate:

2. Options markets only. A BOX SPREAD. (p. 235)

33

7.903%4.87

5ln4/12

1r

5 4.87e 124r

Of course, an annual risk-free rate of 7.903% is large and it indicates that one could NOT have been able to create this strategies with the prices given in the table.

34

SummaryWe have seen that there are strategies

thatyield synthetic Risk-free rates.1. The put-call parity yields a risk-free

rate that requires inputs from the options market and the stock market.

2. The Box spread yields a risk-free rate that requires inputs from the options market ONLY.

Of course, T-bill rates are risk-free.IN AN EFFICIENT ECONOMIC MARKETS ALLTHESE RATES MUST BE EQUAL.

35

SummaryIf the above rates are not equal

arbitrageprofit exists. You may use a strategy tocreate a positive, risk-free cash flow; i.e.,borrow at the lower risk-free rate, andinvest the proceeds in the strategy thatyields the higher risk-free rate.

The above is exactly what professionalarbitrageurs do, mainly, using the BoxSpread strategy.

36

ExampleSuppose that calculating the risk-free ratefrom a Box spread on slide 26 yields r = 3%.The options are for .5yrs and the 6-monthT-bill yield a risk-free rate of 5%.Arbitrage:Borrow money employing the reverse BoxSpread ( effectively borrowing at 3%) andinvest it in the 6-month T-bill. At expirationreceive 5% from the GOV and repay your

3% Debt for An ARBITRAGE PROFIT of 2%.

37

DIVIDEND FACTS:Firms announce their intent to pay

dividends on a specific future day – the X-dividend day. Any investor who holds shares before the stock goes – X-dividend will receive the dividend. The checks go out about one week after the X-dividend day.

Time linetAnnouncement tXDIV tPAYMENT

SCDIV SXDIV

4. SXDIV = SCDIV - D.

38

DIVIDEND 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.

Time linetAnnouncement tXDIV tPAYMENT

SCDIV SXDIV

4. SXDIV = SCDIV - D.

39

The dividend effectEarly exercise of Unprotected American calls

on a cash dividend paying stock:Consider an American call on a cashdividend 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:

40

The dividend effectThe 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 exerciseCash flow: SCD – K c{SXD, K, T - tXD}Substitute: SCD = SXD + D.Cash flow: SXD –K + D SXD – K + TV.

41

The dividend effectEarly exercise of American calls may be

Optimal if:1. The call is in the money

2. D > TV. In this case, the call should be (optimally)

exercised an instant before the stockgoes x-dividend and the cash flow will be:

SCD – K = SXD –K + D.

42

Early exercise of Unprotected American calls on a cash dividend paying stock:

The previous result means that an investor is

indifferent to exercising the call an instant before the stock goes x-dividend if thex- dividend 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)].

43

Early exercise of Unprotected American calls on a cash dividend

paying stock:

D > K[1 – e-r(T – t)].Example:r = .05T – t = .5yr.K = 30.

30[1 – e-.05(.5)] = $.74.Thus, if the dividend is greater than 74 cent per share, the possibility of early exercise exists.

44

Explanation

K Ke t)r(T

t)r(TKe- DK

*XDS

XDS

t)r(TXD KeS

DKSXD

XDC

-K+D> Ke-r(T-t)

D> K[1-e-r(T-t)]

45

Early exercise: Non dividend paying stock

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 itsexpiration, the market premium is greater than the intrinsic value because the time value is always positive.

46

Early exercise: Non dividend paying stock

The American feature is worthless if theunderlying stock does not pay out any dividend during the life of the call. Mathematically: Ct = ct.

Proof: Follows from the previous result.

47

American put on a non dividend paying stock

It may be optimal to exercise a put on a

non dividend paying stock prematurely.

Proof: There is still time to expiration and the stock price fell to 0. An American put holder will definitely exercise the put. It follows that early exercise of an American put may be optimal if the put is enough in-the money.

48

The put-call parity of European optionswith dividends:

Consider European puts and calls arewritten on a dividend paying stock.

The stock will pay dividend in the amounts Dj

on dates tj; j = 1,…,n, and tn < T. rj = the risk-free during tj – t; j=1,…,n.

Then:

n

1j

t)(trj

t)(Trttt

jjT eDKeSpc

49

P/L PROFILE

ofSTRATEGY

ICFAT EXPIRATIONtj ST < K ST < K

SELL STOCK SHORT

St -Dj - ST - ST

SELL PUT pt ST - K 0BUY CALL - ct 0 ST - K

LEND NPV(K)

- Ke-rT(T-t) K KLEND

NPV(DJ)- Dje-rj(tj-t) Dj

TOTAL 0 0 0 0

The put-call parity with dividends

50

The put-call “parity” for American options on a non dividend paying

stock:(p. 220)

At any time point, t, the premiums of American options

on a non dividend paying stock, must satisfy

the following inequalities:

St - K < Ct - Pt < St - Ke-r(T-t)

51

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: Ct = ct becausethe stock does not pay dividend. Moreover,For the puts Pt > pt. Next, on the LHS, suppose that:St - K > Ct - Pt i.e., St - K - Ct + Pt >

0.It can be easily shown that this is an arbitrage profit making strategy, and hence Cannot hold.

52

The put-call “parity” for American options on dividend paying stock:

Let NPV(D) denote the present value of the dividend payments during the life of the options.

Then:

St – NPV(D) – K < Ct - Pt < St - Ke-r(T-t)

n

1j

t)(trj

jjeD NPV(D)