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Role of arbitrage in pricing options Minimum value, maximum value, value at expiration and lower bound of an option price Effect of exercise price, time to expiration, risk free

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Important Concepts

Role of arbitrage in pricing options

Minimum value, maximum value, value at expiration

and lower bound of an option price

Effect of exercise price, time to expiration, risk-free

rate and volatility on an option price

Difference between prices of European and American

options

Put-call parity

2

Role of arbitrage in pricing options

Minimum value, maximum value, value at expiration

and lower bound of an option price

Effect of exercise price, time to expiration, risk-free

rate and volatility on an option price

Difference between prices of European and American

options

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Basic Notation and

Terminology

Symbols

 S0 – stock price

 X – exercise price

 T – time to expiration = (days until expiration)/365

 r – risk-free rat, see below

 ST – stock price at expiration

 C(S0,T,X) – value of a call

 P(S0,T,X) – value of a put

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Symbols

 S0 – stock price

 X – exercise price

 T – time to expiration = (days until expiration)/365

 r – risk-free rat, see below

 ST – stock price at expiration

 C(S0,T,X) – value of a call

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Basic Notation and

Terminology (continued)

Computation of risk-free rate (r)

 Date: May 14. Option expiration: May 21

 T-bill bid discount = 4.45, ask discount = 4.37

 Average T-bill discount = (4.45+4.37)/2 = 4.41

 T-bill price = 100 - 4.41(7/360) = 99.91425

 T-bill yield = (100/99.91425)(365/7) - 1 = 0.0457

 So 4.57 % is risk-free rate for options expiring May 21

 Other risk-free rates: 4.56 (June 18), 4.63 (July 16)

See

Table 3.1

for prices of DCRB options

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Computation of risk-free rate (r)

 Date: May 14. Option expiration: May 21

 T-bill bid discount = 4.45, ask discount = 4.37

 Average T-bill discount = (4.45+4.37)/2 = 4.41

 T-bill price = 100 - 4.41(7/360) = 99.91425

 T-bill yield = (100/99.91425)(365/7) - 1 = 0.0457

 So 4.57 % is risk-free rate for options expiring May 21

 Other risk-free rates: 4.56 (June 18), 4.63 (July 16)

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Principles of Call Option

Pricing

Minimum Value of a Call

 C(S0,T,X)  0 (for any call)

 For American calls:

 Ca(S0,T,X)  Max(0,S0 - X)

 Concept of intrinsic value: Max(0,S0 - X)

 Proof of intrinsic value rule for DCRB calls

 Concept of time value

 See Table 3.2 for time values of DCRB calls

 See Figure 3.1 for minimum values of calls

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Minimum Value of a Call

 C(S0,T,X)  0 (for any call)

 For American calls:

 Ca(S0,T,X)  Max(0,S0 - X)

 Concept of intrinsic value: Max(0,S0 - X)

 Proof of intrinsic value rule for DCRB calls

 Concept of time value

 See Table 3.2 for time values of DCRB calls

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Principles of Call Option

Pricing (continued)

Maximum Value of a Call

 C(S0,T,X) S0

 Intuition

 See Figure 3.2, which adds this to Figure 3.1

Value of a Call at Expiration

 C(ST,0,X) = Max(0,ST - X)

 Proof/intuition

 For American and European options

 See Figure 3.3

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Maximum Value of a Call

 C(S0,T,X) S0

 Intuition

 See Figure 3.2, which adds this to Figure 3.1

Value of a Call at Expiration

 C(ST,0,X) = Max(0,ST - X)

 Proof/intuition

 For American and European options

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Principles of Call Option

Pricing (continued)

Effect of Time to Expiration

 Two American calls differing only by time to expiration, T1 and T2 where T1 < T2.

 Ca(S0,T2,X)  Ca(S0,T1,X)

 Proof/intuition

 Deep in- and out-of-the-money

 Time value maximized when at-the-money

 Concept of time value decay

 See Figure 3.4 and Table 3.2

 Cannot be proven (yet) for European calls

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Effect of Time to Expiration

 Two American calls differing only by time to expiration, T1 and T2 where T1 < T2.

 Ca(S0,T2,X)  Ca(S0,T1,X)

 Proof/intuition

 Deep in- and out-of-the-money

 Time value maximized when at-the-money

 Concept of time value decay

 See Figure 3.4 and Table 3.2

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Principles of Call Option

Pricing (continued)

Effect of Exercise Price

 Effect on Option Value

 Two European calls differing only by strikes of X1and X2. Which is greater, Ce(S0,T,X1) or Ce(S0,T,X2)?

 Construct portfolios A and B. See Table 3.3.

 Portfolio A has non-negative payoff; therefore,

 Ce(S0,T,X1)  Ce(S0,T,X2)

 Intuition: show what happens if not true

 Prices of DCRB options conform

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Effect of Exercise Price

 Effect on Option Value

 Two European calls differing only by strikes of X1and X2. Which is greater, Ce(S0,T,X1) or Ce(S0,T,X2)?

 Construct portfolios A and B. See Table 3.3.

 Portfolio A has non-negative payoff; therefore,

 Ce(S0,T,X1)  Ce(S0,T,X2)

 Intuition: show what happens if not true

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Principles of Call Option

Pricing (continued)

Effect of Exercise Price (continued)

 Limits on the Difference in Premiums

 Again, note Table 3.3. We must have

 (X2 - X1)(1+r)-T  C

e(S0,T,X1) - Ce(S0,T,X2)

 X2 - X1 Ce(S0,T,X1) - Ce(S0,T,X2)

 X2 - X1 Ca(S0,T,X1) - Ca(S0,T,X2)

 Implications

 See Table 3.4. Prices of DCRB options conform

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Effect of Exercise Price (continued)

 Limits on the Difference in Premiums

 Again, note Table 3.3. We must have

 (X2 - X1)(1+r)-T  C

e(S0,T,X1) - Ce(S0,T,X2)

 X2 - X1 Ce(S0,T,X1) - Ce(S0,T,X2)

 X2 - X1 Ca(S0,T,X1) - Ca(S0,T,X2)

 Implications

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Principles of Call Option

Pricing (continued)

Lower Bound of a European Call

 Construct portfolios A and B. See Table 3.5.

 B dominates A. This implies that (after rearranging)

 Ce(S0,T,X)  Max[0,S0 - X(1+r)-T]

 This is the lower bound for a European call

 See Figure 3.5 for the price curve for European calls

 Dividend adjustment: subtract present value of dividends from S0; adjusted stock price is S0´

 For foreign currency calls,

 Ce(S0,T,X)  Max[0,S0(1+)-T - X(1+r)-T]

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Lower Bound of a European Call

 Construct portfolios A and B. See Table 3.5.

 B dominates A. This implies that (after rearranging)

 Ce(S0,T,X)  Max[0,S0 - X(1+r)-T]

 This is the lower bound for a European call

 See Figure 3.5 for the price curve for European calls

 Dividend adjustment: subtract present value of dividends from S0; adjusted stock price is S0´

 For foreign currency calls,

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Principles of Call Option

Pricing (continued)

American Call Versus European Call

 Ca(S0,T,X)  Ce(S0,T,X)

 But S0 - X(1+r)-T > S

0 - X prior to expiration so  Ca(S0,T,X)  Max(0,S0 - X(1+r)-T)

 Look at Table 3.6 for lower bounds of DCRB calls

 If there are no dividends on the stock, an American call will never be exercised early. It will always be better to sell the call in the market.

 Intuition

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American Call Versus European Call

 Ca(S0,T,X)  Ce(S0,T,X)

 But S0 - X(1+r)-T > S

0 - X prior to expiration so  Ca(S0,T,X)  Max(0,S0 - X(1+r)-T)

 Look at Table 3.6 for lower bounds of DCRB calls

 If there are no dividends on the stock, an American call will never be exercised early. It will always be better to sell the call in the market.

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Principles of Call Option

Pricing (continued)

Early Exercise of American Calls on Dividend-Paying

Stocks

 If a stock pays a dividend, it is possible that an American call will be exercised as close as possible to the

ex-dividend date. (For a currency, the foreign interest can induce early exercise.)

 Intuition

Effect of Interest Rates

Effect of Stock Volatility

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Early Exercise of American Calls on Dividend-Paying

Stocks

 If a stock pays a dividend, it is possible that an American call will be exercised as close as possible to the

ex-dividend date. (For a currency, the foreign interest can induce early exercise.)

 Intuition

Effect of Interest Rates

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Principles of Put Option Pricing

Minimum Value of a Put

 P(S0,T,X)  0 (for any put)

 For American puts:

 Pa(S0,T,X)  Max(0,X - S0)

 Concept of intrinsic value: Max(0,X - S0)

 Proof of intrinsic value rule for DCRB puts

 See Figure 3.6 for minimum values of puts

 Concept of time value

 See Table 3.7 for time values of DCRB puts

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Minimum Value of a Put

 P(S0,T,X)  0 (for any put)

 For American puts:

 Pa(S0,T,X)  Max(0,X - S0)

 Concept of intrinsic value: Max(0,X - S0)

 Proof of intrinsic value rule for DCRB puts

 See Figure 3.6 for minimum values of puts

 Concept of time value

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Principles of Put Option Pricing

(continued)

Maximum Value of a Put

 Pe(S0,T,X)  X(1+r)-T

 Pa(S0,T,X)  X

 Intuition

 See Figure 3.7, which adds this to Figure 3.6

Value of a Put at Expiration

 P(ST,0,X) = Max(0,X - ST)

 Proof/intuition

 For American and European options

 See Figure 3.8

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Maximum Value of a Put

 Pe(S0,T,X)  X(1+r)-T

 Pa(S0,T,X)  X

 Intuition

 See Figure 3.7, which adds this to Figure 3.6

Value of a Put at Expiration

 P(ST,0,X) = Max(0,X - ST)

 Proof/intuition

 For American and European options

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Principles of Put Option Pricing

(continued)

Effect of Time to Expiration

 Two American puts differing only by time to expiration, T1 and T2 where T1 < T2.

 Pa(S0,T2,X)  Pa(S0,T1,X)

 Proof/intuition

 See Figure 3.9 and Table 3.7

 Cannot be proven for European puts

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Effect of Time to Expiration

 Two American puts differing only by time to expiration, T1 and T2 where T1 < T2.

 Pa(S0,T2,X)  Pa(S0,T1,X)

 Proof/intuition

 See Figure 3.9 and Table 3.7

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Principles of Put Option Pricing

(continued)

Effect of Exercise Price

 Effect on Option Value

 Two European puts differing only by X1and X2. Which is greater, Pe(S0,T,X1) or Pe(S0,T,X2)?

 Construct portfolios A and B. See Table 3.8.

 Portfolio A has non-negative payoff; therefore,

 Pe(S0,T,X2)  Pe(S0,T,X1)

 Intuition: show what happens if not true

 Prices of DCRB options conform

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Effect of Exercise Price

 Effect on Option Value

 Two European puts differing only by X1and X2. Which is greater, Pe(S0,T,X1) or Pe(S0,T,X2)?

 Construct portfolios A and B. See Table 3.8.

 Portfolio A has non-negative payoff; therefore,

 Pe(S0,T,X2)  Pe(S0,T,X1)

 Intuition: show what happens if not true

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Principles of Put Option Pricing

(continued)

Effect of Exercise Price (continued)

 Limits on the Difference in Premiums

 Again, note Table 3.8. We must have

 (X2 - X1)(1+r)-T  P

e(S0,T,X2) - Pe(S0,T,X1)

 X2 - X1 Pe(S0,T,X2) - Pe(S0,T,X1)

 X2 - X1 Pa(S0,T,X2) - Pa(S0,T,X1)

 Implications

 See Table 3.9. Prices of DCRB options conform

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Effect of Exercise Price (continued)

 Limits on the Difference in Premiums

 Again, note Table 3.8. We must have

 (X2 - X1)(1+r)-T  P

e(S0,T,X2) - Pe(S0,T,X1)

 X2 - X1 Pe(S0,T,X2) - Pe(S0,T,X1)

 X2 - X1 Pa(S0,T,X2) - Pa(S0,T,X1)

 Implications

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Principles of Put Option Pricing

(continued)

Lower Bound of a European Put

 Construct portfolios A and B. See Table 3.10.

 A dominates B. This implies that (after rearranging)

 Pe(S0,T,X)  Max(0,X(1+r)-T - S 0)

 This is the lower bound for a European put

 See Figure 3.10 for the price curve for European puts

 Dividend adjustment: subtract present value of dividends from S to obtain S´

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Lower Bound of a European Put

 Construct portfolios A and B. See Table 3.10.

 A dominates B. This implies that (after rearranging)

 Pe(S0,T,X)  Max(0,X(1+r)-T - S 0)

 This is the lower bound for a European put

 See Figure 3.10 for the price curve for European puts

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Principles of Put Option Pricing

(continued)

American Put Versus European Put

 Pa(S0,T,X)  Pe(S0,T,X)

Early Exercise of American Puts

 There is always a sufficiently low stock price that will make it optimal to exercise an American put early.

 Dividends on the stock reduce the likelihood of early exercise.

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American Put Versus European Put

 Pa(S0,T,X)  Pe(S0,T,X)

Early Exercise of American Puts

 There is always a sufficiently low stock price that will make it optimal to exercise an American put early.

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Principles of Put Option Pricing

(continued)

Put-Call Parity

 Form portfolios A and B where the options are European. See Table 3.11.

 The portfolios have the same outcomes at the options’ expiration. Thus, it must be true that

 S0 + Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T  This is called put-call parity.

 It is important to see the alternative ways the equation can be arranged and their interpretations.

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Put-Call Parity

 Form portfolios A and B where the options are European. See Table 3.11.

 The portfolios have the same outcomes at the options’ expiration. Thus, it must be true that

 S0 + Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T  This is called put-call parity.

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Principles of Put Option Pricing

(continued)

Put-Call parity for American options can be stated

only as inequalities:

See

Table 3.12

for put-call parity for DCRB options

See

Figure 3.11

for linkages between underlying

asset, risk-free bond, call, and put through put-call

parity.

T ' 0 a ' 0 a 0 N 1 j t j ' 0 a

r)

X(1

X)

T,

,

(S

C

X)

T,

,

(S

P

S

r)

(1

D

X

X)

T,

,

(S

C

j   

42

Put-Call parity for American options can be stated

only as inequalities:

See

Table 3.12

for put-call parity for DCRB options

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Principles of Put Option Pricing

(continued)

The Effect of Interest Rates

The Effect of Stock Volatility

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Summary

See Table 3.13

Appendix 3: The Dynamics of Option Boundary

Conditions: A Learning Exercise

See Table 3.13

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Homework

Read Chapter 6 and answer questions and problems:

2,4,5, 8-14

Read Chapter 7 and answer questions and problems:

4-17,20,21, CC2-3

Read Chapter 9 and answer questions and problems:

1,8-14,17,18,21, CC5

Read Chapter 3 and answer questions and problems:

1,5,10-18,22, CC4

Read Chapter 6 and answer questions and problems:

2,4,5, 8-14

Read Chapter 7 and answer questions and problems:

4-17,20,21, CC2-3

Read Chapter 9 and answer questions and problems:

1,8-14,17,18,21, CC5

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