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Properties ofStock Option Prices

Chapter 9

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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: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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FACTORS AFFECTING OPTIONS PRICES: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 }.

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FACTORS AFFECTING OPTIONS PRICES:

Factor European call

European put

American call

American put

St + - + -

K - + - +

T-t ? ? + +

+ + + +

r + - + -

D _ + - +

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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 ST > K

expires worthless ifST K

Cash Flow = Max{0, ST – K}

PUT is: exercised if ST < K

expires worthless if ST ≥ K

Cash Flow = Max{0, K – ST}

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

4. A Table Format

Every row is one part of the strategy.Every row is analyzed independently of theother rows.The total strategy is the vertical sum of

therows.The profit is the cash flow at expiration

plusthe initial cash flows of the strategy,disregarding the time value of money.

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5. 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 P/L atexpiration:

Long stock: –St + ST

Short stock: St - ST

Long call: -ct + Max{0, ST - K}

Short call: ct + Min{0, K - ST}

Long put: -pt + Max{0, K - ST}

Short put: pt + Min{0, ST - K}

Notice: the time value of money is ignored.

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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. Atthe option’s expiration, the lender receives K.If borrowed, the borrower will pay K, namely,a cash flow of – K.

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Bounds on options market prices

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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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.

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(Sec.9.3 p.209) 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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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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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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(Top p.210) 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.

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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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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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Eq(9.1)p.211Lower 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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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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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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Sec. 9.6 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 a European put may be exercise at expiration. If the price of the underlying asset fall belowsome price, it becomes optimal to exercisethe 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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The put-call parity.

European options: The premiums of European

calls 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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At expirationStrategy 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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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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The put-call parity for American options (Eq.(9.4) p.215)

The premiums on American options satisfy

the following inequalities:

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

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

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.

This is an arbitrage profit making strategy, which contradicts the supposition above.

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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.

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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.

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It can be optimal to exercise an American put on a non dividend paying stock early.

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.

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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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Early exercise: 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:

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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

SCUMD SXDIV

4. SXDIV = SCDIV - D.

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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.

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Conclusion:

Early exercise of American calls may be optimal:1. The call must be in the money And2. D > TV. In this case, the call should be

(optimally) exercised an instant before the stock goes x-dividend and the cash flow will be: SXD –K + D.

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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 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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Explanation

K Ke t)r(T

t)r(TKe- DK

*XDS

XDS

t)r(TXD KeS

DKSXD

XDC

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Eq.(9.7) p.219

The put-call parity. European options:Suppose that European puts and calls arewritten on a dividend paying stock. There

will be n dividend Payments in the amounts

Dj on

dates tj; j = 1,…,n, tn < T. rj = the risk-free

rate during tj – t; j=1,…,n,T. Then,

n

1j

t)(trj

t)(Trttt

jjT eDKeSpc

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j = 1,…,n At expirationStrategy I.C.F tj ST < K ST > K

Sell stock St -Dj - ST - ST

sell put pt ST - K 0

Buy call - ct 0 ST- K

Lend - Ke-rT(T-t) K KLend - Dje-rj(tj-t) DjTotal 0 0 0 0

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Eq(9.8) p.219

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

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A Risk-free rate with options:A Box spread: K1 < K2

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

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