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Properties ofStock Option Prices

Chapter 8

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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.

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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

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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 thefollowing functions:

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

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

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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.

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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.

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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}

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3. All legs of the strategy remain open till expiry.

4. A Table FormatEvery 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

ofthe strategy, disregarding the timevalue of money.

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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.

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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}

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Borrowing and Lending:In many strategies with lending or

borrowingcapital at the risk-free rate, the amountborrowed 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 value, namely, a cash flow of – K.

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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.

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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.

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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.

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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 - KLend funds - Ke-r(T-t) K KTotal ? K – ST 0

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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.

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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 ?

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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 lifeof the option.

Proof: If an American call holder wishes torid 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.

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RESULTS FOR CALLS:9. 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 result 8. above.

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RESULTS FOR CALLS:10. Early 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:

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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.

Time linetA tXDIV tPAYMENT

SCUMD SXDIV

4. SXDIV = SCDIV - D.

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

The call holder goal is to maximize the Cashflow 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 xdividend:

Exercise Do not exerciseCash flow: SCD – K c{SXD, K, T - tXD}

Substitute: SCD = SXD + D.

Cash flow: SXD –K + D SXD – K + TV.

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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.

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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 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)].

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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,

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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

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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,

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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 = interestEven if the sold call is exercised before Expiration, the total value in hand is >0.

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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,

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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

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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.

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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 Calls and buy three $100 calls.

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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.

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RESULTS FOR CALLS:

16. The interest rate:The Higher the risk-free rate, the higher

is thecall 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.

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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.

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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.

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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}.

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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.

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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 expirationStrategy I.C.F ST < K ST > K

Buy stock -St ST ST

Buy put - pt K - ST 0

Borrow Ke-r(T-t) - K - KTotal ? 0 ST - K

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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.

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RESULTS FOR PUTS:23. 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.

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RESULTS FOR PUTS:

24. The time value of puts:The longer the time to expiration, the

higheris 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

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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,

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RESULTS FOR PUTS:At expiration

Strategy ICF ST < K1 K1<ST < K2 ST

>K2

Sell p(K1 ) p1 ST –K1 0 0

Buy p(K2 ) -p2 K2 –ST K2 - ST 0

Total ? K2 - K1 K2 - ST 0

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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,

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RESULTS FOR PUTS:

At expirationStrategy 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 K2- K1+i

Total ? i ST-K1 +i K2- K1+i

i = interestEven if the sold put is exercised before expiration, the total value in hand is >0.

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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,

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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.

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RESULTS FOR PUTS:28. Volatility:The higher the price volatility of theunderlying 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.

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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.

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RESULTS for PUTS and CALLS:30. The put-call parity.European options: The premiums of

Europeancalls and puts written on the same non dividend paying stock for the same

expirationand the same strike price must satisfy:

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

The parity may be rewritten as:

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

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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 expirationStrategy I.C.F ST < K ST > K

Buy call - ct 0 ST-K

Lend - Ke-r(T-t) K KTotal -(ct+ Ke-r(T-t) ) K ST

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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).

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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)

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RESULTS for PUTS and CALLS:33. Box spread: At expirationStrategy 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)

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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 onthis 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)

)ppcc

KKln(

tT

1r

1221

12

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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).

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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.

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RESULTS for PUTS and CALLS:When the options are written on a dividendpaying 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