Copyright © 2001 by Harcourt, Inc. All rights reserved.1 Chapter 3: Principles of Option Pricing Order and simplification are the first steps toward mastery

Copyright © 2001 by Harcourt, Inc. All rights reserved.

1

Chapter 3: Principles of Option Pricing

Order and simplification are the first steps toward mastery Order and simplification are the first steps toward mastery of a subject - the actual enemy is the unknown.of a subject - the actual enemy is the unknown.

Thomas MannThomas Mann

The Magic MountainThe Magic Mountain

19241924


2

Important Concepts in Chapter 3

Role of arbitrage in pricing optionsRole of arbitrage in pricing options Minimum value, maximum value, value at expiration and Minimum value, maximum value, value at expiration and

lower bound of an option pricelower bound of an option price Effect of exercise price, time to expiration, risk-free rate Effect of exercise price, time to expiration, risk-free rate

and volatility on an option priceand volatility on an option price Difference between prices of European and American Difference between prices of European and American

optionsoptions Put-call parityPut-call parity


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Basic Notation and Terminology

SymbolsSymbols SS00 (stock price) (stock price)

X (exercise price)X (exercise price) T (time to expiration = (days until expiration)/365)T (time to expiration = (days until expiration)/365) r (see below)r (see below) SSTT (stock price at expiration) (stock price at expiration)

C(SC(S00,T,X), P(S,T,X), P(S00,T,X),T,X)


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Basic Notation and Terminology (continued)

Computation of risk-free rateComputation of risk-free rate Date: May 14. Option expiration: May 21Date: May 14. Option expiration: May 21 T-bill bid discount = 4.45, ask discount = 4.37T-bill bid discount = 4.45, ask discount = 4.37

Average T-bill discount = (4.45+4.37)/2 = 4.41Average T-bill discount = (4.45+4.37)/2 = 4.41 T-bill price = 100 - 4.41(7/360) = 99.91425T-bill price = 100 - 4.41(7/360) = 99.91425 T-bill yield = (100/99.91425)T-bill yield = (100/99.91425)(365/7)(365/7) - 1 = .0457 - 1 = .0457 So 4.57 % is risk-free rate for options expiring May 21So 4.57 % is risk-free rate for options expiring May 21 Other risk-free rates: 4.56 (June 18), 4.63 (July 16)Other risk-free rates: 4.56 (June 18), 4.63 (July 16)

See See Table 3.1, p. 74Table 3.1, p. 74 for prices of America Online options for prices of America Online options


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Principles of Call Option Pricing The Minimum Value of a CallThe Minimum Value of a Call

C(SC(S00,T,X) ,T,X) 0 (for any call)

For American calls:For American calls: CCaa(S(S00,T,X) ,T,X) Max(0,S Max(0,S00 - X) - X)

Concept of intrinsic value: Max(0,SConcept of intrinsic value: Max(0,S00 - X) - X) Proof of intrinsic value rule for AOL callsProof of intrinsic value rule for AOL calls

Concept of time valueConcept of time value See See TableTable 3.2, p. 763.2, p. 76 for time values of AOL calls for time values of AOL calls

See See Figure 3.1, p. 77Figure 3.1, p. 77 for minimum values of calls for minimum values of calls


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Principles of Call Option Pricing (continued) The Maximum Value of a CallThe Maximum Value of a Call

C(SC(S00,T,X) ,T,X) S0

Intuition See Figure 3.2, p. 78, which adds this to Figure 3.1

The Value of a Call at ExpirationThe Value of a Call at Expiration C(SC(STT,0,X) = Max(0,S,0,X) = Max(0,STT - X) - X)

Proof/intuitionProof/intuition For American and European optionsFor American and European options See See Figure 3.3, p. 80Figure 3.3, p. 80


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Principles of Call Option Pricing (continued)

The Effect of Time to ExpirationThe Effect of Time to Expiration Two American calls differing only by time to expiration, TTwo American calls differing only by time to expiration, T11

and Tand T22 where T where T11 < T < T22..

CCaa(S(S00,T,T22,X) ,X) C Caa(S(S00,T,T11,X),X) Proof/intuitionProof/intuition

Deep in- and out-of-the-moneyDeep in- and out-of-the-money Time value maximized when at-the-moneyTime value maximized when at-the-money Concept of time value decayConcept of time value decay See See Figure 3.4, p. 81Figure 3.4, p. 81 and and Table 3.2, p. 82Table 3.2, p. 82 Cannot be proven (yet) for European callsCannot be proven (yet) for European calls


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The Effect of Exercise PriceThe Effect of Exercise Price The Effect on Option ValueThe Effect on Option Value

Two European calls differing only by strikes of XTwo European calls differing only by strikes of X1 1

and Xand X22. Which is greater, C. Which is greater, Cee(S(S00,T,X,T,X11) or C) or Cee(S(S00,T,X,T,X22)?)? Construct portfolios A and B. See Construct portfolios A and B. See Table 3.3, p. 82Table 3.3, p. 82.. Portfolio A has non-negative payoff; therefore,Portfolio A has non-negative payoff; therefore,

• CCee(S(S00,T,X,T,X11) ) C Cee(S(S00,T,X,T,X22))

• Intuition: show what happens if not trueIntuition: show what happens if not true Prices of AOL options conformPrices of AOL options conform


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The Effect of Exercise Price (continued)The Effect of Exercise Price (continued) Limits on the Difference in PremiumsLimits on the Difference in Premiums

Again, note Again, note Table 3.3, p. 82Table 3.3, p. 82. We must have. We must have

• (X(X22 - X - X11)(1+r))(1+r)-T-T C Cee(S(S00,T,X,T,X11) - C) - Cee(S(S00,T,X,T,X22))

• XX22 - X - X1 1 CCee(S(S00,T,X,T,X11) - C) - Cee(S(S00,T,X,T,X22))

• XX22 - X - X1 1 CCaa(S(S00,T,X,T,X11) - C) - Caa(S(S00,T,X,T,X22))

• ImplicationsImplications See TSee Table 3.4, p. 85able 3.4, p. 85.. Prices of AOL options Prices of AOL options

conformconform


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The Lower Bound of a European CallThe Lower Bound of a European Call Construct portfolios A and B. See Construct portfolios A and B. See Table 3.5, p. 86Table 3.5, p. 86.. B dominates A. This implies that (after rearranging)B dominates A. This implies that (after rearranging)

CCee(S(S00,T,X) ,T,X) Max(0,S Max(0,S00 - X(1+r) - X(1+r)-T-T)) This is the lower bound for a European callThis is the lower bound for a European call See See Figure 3.5, p. 86Figure 3.5, p. 86 for the price curve for for the price curve for

European callsEuropean calls Dividend adjustment: subtract present value of Dividend adjustment: subtract present value of

dividends from S; adjusted stock price is S´dividends from S; adjusted stock price is S´


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American Call Versus European CallAmerican Call Versus European Call CCaa(S(S00,T,X) ,T,X) C Cee(S(S00,T,X),T,X)

But SBut S00 - X(1+r) - X(1+r)-T-T > S > S00 - X prior to expiration so - X prior to expiration so

CCaa(S(S00,T,X) ,T,X) Max(0,S Max(0,S00 - X(1+r) - X(1+r)-T-T)) Look at Look at Table 3.6, p. 88Table 3.6, p. 88 for lower bounds of AOL for lower bounds of AOL

callscalls If there are no dividends on the stock, an American call If there are no dividends on the stock, an American call

will never be exercised early. It will always be better to will never be exercised early. It will always be better to sell the call in the market.sell the call in the market. IntuitionIntuition


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The Early Exercise of American Calls on Dividend-Paying The Early Exercise of American Calls on Dividend-Paying StocksStocks If a stock pays a dividend, it is possible that an If a stock pays a dividend, it is possible that an

American call will be exercised as close as possible to American call will be exercised as close as possible to the ex-dividend date.the ex-dividend date.

IntuitionIntuition The Effect of Interest RatesThe Effect of Interest Rates The Effect of VolatilityThe Effect of Volatility


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Principles of Put Option Pricing The Minimum Value of a PutThe Minimum Value of a Put

P(SP(S00,T,X) ,T,X) 0 (for any put)

For American puts:For American puts: PPaa(S(S00,T,X) ,T,X) Max(0,X - S Max(0,X - S00))

Concept of intrinsic value: Max(0,X - SConcept of intrinsic value: Max(0,X - S00)) Proof of intrinsic value rule for AOL putsProof of intrinsic value rule for AOL puts

See See Figure 3.6, p. 92Figure 3.6, p. 92 for minimum values of puts for minimum values of puts Concept of time valueConcept of time value

See See TableTable 3.7, p. 933.7, p. 93 for time values of AOL puts for time values of AOL puts


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Principles of Put Option Pricing (continued)

The Maximum Value of a PutThe Maximum Value of a Put PPee(S(S00,T,X) ,T,X) X(1+r)-T

Pa(S0,T,X) X

Intuition See Figure 3.7, p. 94, which adds this to Figure 3.6

The Value of a Put at ExpirationThe Value of a Put at Expiration P(SP(STT,0,X) = Max(0,X - S,0,X) = Max(0,X - STT))

Proof/intuitionProof/intuition For American and European optionsFor American and European options See See Figure 3.8, p. 95Figure 3.8, p. 95


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The Effect of Time to ExpirationThe Effect of Time to Expiration Two American puts differing only by time to Two American puts differing only by time to

expiration, Texpiration, T11 and T and T22 where T where T11 < T < T22..

PPaa(S(S00,T,T22,X) ,X) P Paa(S(S00,T,T11,X),X) Proof/intuitionProof/intuition

See See Figure 3.9, p. 96Figure 3.9, p. 96 and and Table 3.7, p. 93Table 3.7, p. 93 Cannot be proven for European putsCannot be proven for European puts


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The Effect of Exercise PriceThe Effect of Exercise Price The Effect on Option ValueThe Effect on Option Value

Two European puts differing only by XTwo European puts differing only by X1 1 and Xand X22. .

Which is greater, PWhich is greater, Pee(S(S00,T,X,T,X11) or P) or Pee(S(S00,T,X,T,X22)?)? Construct portfolios A and B. See Construct portfolios A and B. See Table 3.8, p. 97Table 3.8, p. 97.. Portfolio A has non-negative payoff; therefore,Portfolio A has non-negative payoff; therefore,

• PPee(S(S00,T,X,T,X22) ) P Pee(S(S00,T,X,T,X11))

• Intuition: show what happens if not trueIntuition: show what happens if not true Prices of AOL options conformPrices of AOL options conform


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The Effect of Exercise Price (continued)The Effect of Exercise Price (continued) Limits on the Difference in PremiumsLimits on the Difference in Premiums

Again, note Again, note Table 3.8, p. 97Table 3.8, p. 97. We must have. We must have

• (X(X22 - X - X11)(1+r))(1+r)-T-T P Pee(S(S00,T,X,T,X22) - P) - Pee(S(S00,T,X,T,X11))

• XX22 - X - X1 1 PPee(S(S00,T,X,T,X22) - P) - Pee(S(S00,T,X,T,X11))

• XX22 - X - X1 1 PPaa(S(S00,T,X,T,X22) - P) - Paa(S(S00,T,X,T,X11))

• ImplicationsImplications See TSee Table 3.9, p. 99able 3.9, p. 99.. Prices of AOL options Prices of AOL options

conformconform


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The Lower Bound of a European PutThe Lower Bound of a European Put Construct portfolios A and B. See Construct portfolios A and B. See Table 3.10, p. 100Table 3.10, p. 100.. A dominates B. This implies that (after rearranging)A dominates B. This implies that (after rearranging)

PPee(S(S00,T,X) ,T,X) Max(0,X(1+r) Max(0,X(1+r)-T-T - S - S00)) This is the lower bound for a European putThis is the lower bound for a European put See See Figure 3.10, p. 101Figure 3.10, p. 101 for the price curve for for the price curve for

European putsEuropean puts Dividend adjustment: subtract present value of Dividend adjustment: subtract present value of

dividends from S to obtain S´dividends from S to obtain S´


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American Put Versus European PutAmerican Put Versus European Put PPaa(S(S00,T,X) ,T,X) P Pee(S(S00,T,X),T,X)

The Early Exercise of American PutsThe Early Exercise of American Puts There is always a sufficiently low stock price that will There is always a sufficiently low stock price that will

make it optimal to exercise an American put early.make it optimal to exercise an American put early. Dividends on the stock reduce the likelihood of early Dividends on the stock reduce the likelihood of early

exercise.exercise.


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Put-Call ParityPut-Call Parity Form portfolios A and B where the options are Form portfolios A and B where the options are

European. See European. See Table 3.11, p. 103Table 3.11, p. 103. . The portfolios have the same outcomes at the options’ The portfolios have the same outcomes at the options’

expiration. Thus, it must be true thatexpiration. Thus, it must be true that SS00 + P + Pee(S(S00,T,X) = C,T,X) = Cee(S(S00,T,X) + X(1+r),T,X) + X(1+r)-T-T

This is called put-call parity.This is called put-call parity. It is important to see the alternative ways the It is important to see the alternative ways the

equation can be arranged and their interpretations.equation can be arranged and their interpretations.


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Put-Call parity for American options can be stated only Put-Call parity for American options can be stated only as inequalities:as inequalities:

See See Table 3.12, p. 105Table 3.12, p. 105 for put-call parity for AOL for put-call parity for AOL optionsoptions

See See Figure 3.11, 106Figure 3.11, 106 for linkages between stock, risk- for linkages between stock, risk-free bond, call, and put through put-call parity.free bond, call, and put through put-call parity.

T'0a

'0a0

N

1j

tj

'0a

r)X(1X)T,,(SC

X)T,,(SPS

r)(1DXX)T,,(SC j


22


The Effect of Interest RatesThe Effect of Interest Rates The Effect of Stock VolatilityThe Effect of Stock Volatility

SummarySee See Table 3.13, p. 107Table 3.13, p. 107. .

Documents

Copyright © 2001 by Harcourt, Inc. All rights reserved.1 Chapter 3: Principles of Option Pricing Order and simplification are the first steps toward mastery