Ch. 3: 1

1
Chapter 3 Principles of Option Pricing

• Asking a fund manager about arbitrage
  opportunities is akin to asking a fisherman where
  his favorite hole is. He will be glad to tell
  you a fish story from long ago, but he will not
  tell you where he caught the trout that in our
  analogy can be translated into millions of
  dollars, lest there will be hundreds of fishermen
  in his spot pulling in their own trout and
  reducing the inefficiency that made that
  arbitrage opportunity profitable in the first
  place.
• Daniel P. Collins
• Futures, December, 2001, p. 66

2
Important Concepts in Chapter 3

• 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

3
Basic Notation and Terminology

• Symbols
• S0 (stock price)
• X (exercise price)
• T (time to expiration (days until
  expiration)/365)
• r (see below)
• ST (stock price at expiration)
• C(S0,T,X), P(S0,T,X)

4
Basic Notation and Terminology (continued)

• Computation of risk-free rate
• Date May 14. Option expiration May 21
• T-bill bid discount 4.45, ask discount 4.37
• Average T-bill discount (4.454.37)/2 4.41
• T-bill price 100 - 4.41(7/360) 99.91425
• T-bill yield (100/99.91425)(365/7) - 1 .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, p. 58 for prices of AOL options

5
Principles of Call Option Pricing

• The Minimum Value of a Call
• C(S0,T,X) ³ 0 (for any call)
• For American calls
• Ca(S0,T,X) ³ Max(0,S0 - X)
• Concept of intrinsic value Max(0,S0 - X)
• Proof of intrinsic value rule for AOL calls
• Concept of time value
• See Table 3.2, p. 59 for time values of AOL calls
• See Figure 3.1, p. 60 for minimum values of calls

6
Principles of Call Option Pricing (continued)

• The Maximum Value of a Call
• C(S0,T,X) S0
• Intuition
• See Figure 3.2, p. 61, which adds this to Figure
  3.1
• The 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, p. 63

7
Principles of Call Option Pricing (continued)

• The Effect of Time to Expiration
• Two American calls differing only by time to
  expiration, T1 and T2 where T1 lt T2.
• Ca(S0,T2,X) ³ 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, p. 64 and Table 3.2, p. 59
• Cannot be proven (yet) for European calls

8
Principles of Call Option Pricing (continued)

• The Effect of Exercise Price
• The Effect on Option Value
• Two European calls differing only by strikes of
  X1 and X2. Which is greater, Ce(S0,T,X1) or
  Ce(S0,T,X2)?
• Construct portfolios A and B. See Table 3.3, p.
  65.
• Portfolio A has non-negative payoff therefore,
• Ce(S0,T,X1) ³ Ce(S0,T,X2)
• Intuition show what happens if not true
• Prices of AOL options conform

9
Principles of Call Option Pricing (continued)

• The Effect of Exercise Price (continued)
• Limits on the Difference in Premiums
• Again, note Table 3.3, p. 65. We must have
• (X2 - X1)(1r)-T ³ Ce(S0,T,X1) - Ce(S0,T,X2)
• X2 - X1 ³ Ce(S0,T,X1) - Ce(S0,T,X2)
• X2 - X1 ³ Ca(S0,T,X1) - Ca(S0,T,X2)
• Implications
• See Table 3.4, p. 67. Prices of AOL options
  conform

10
Principles of Call Option Pricing (continued)

• The Lower Bound of a European Call
• Construct portfolios A and B. See Table 3.5, p.
  68.
• B dominates A. This implies that (after
  rearranging)
• Ce(S0,T,X) ³ Max0,S0 - X(1r)-T
• This is the lower bound for a European call
• See Figure 3.5, p. 69 for the price curve for
  European calls
• Dividend adjustment subtract present value of
  dividends from S adjusted stock price is S
• For foreign currency calls,
• Ce(S0,T,X) ³ Max0,S0(1?)-T - X(1r)-T

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Principles of Call Option Pricing (continued)

• American Call Versus European Call
• Ca(S0,T,X) ³ Ce(S0,T,X)
• But S0 - X(1r)-T gt S0 - X prior to expiration so
• Ca(S0,T,X) ³ Max(0,S0 - X(1r)-T)
• Look at Table 3.6, p. 70 for lower bounds of AOL
  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

12
Principles of Call Option Pricing (continued)

• The 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
• The Effect of Interest Rates
• The Effect of Stock Volatility

13
Principles of Put Option Pricing

• The Minimum Value of a Put
• P(S0,T,X) ³ 0 (for any put)
• For American puts
• Pa(S0,T,X) ³ Max(0,X - S0)
• Concept of intrinsic value Max(0,X - S0)
• Proof of intrinsic value rule for AOL puts
• See Figure 3.6, p. 74 for minimum values of puts
• Concept of time value
• See Table 3.7, p. 75 for time values of AOL puts

14
Principles of Put Option Pricing (continued)

• The Maximum Value of a Put
• Pe(S0,T,X) X(1r)-T
• Pa(S0,T,X) X
• Intuition
• See Figure 3.7, p. 76, which adds this to Figure
  3.6
• The 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, p. 77

15
Principles of Put Option Pricing (continued)

• The Effect of Time to Expiration
• Two American puts differing only by time to
  expiration, T1 and T2 where T1 lt T2.
• Pa(S0,T2,X) ³ Pa(S0,T1,X)
• Proof/intuition
• See Figure 3.9, p. 78 and Table 3.7, p. 75
• Cannot be proven for European puts

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Principles of Put Option Pricing (continued)

• The Effect of Exercise Price
• The Effect on Option Value
• Two European puts differing only by X1 and X2.
  Which is greater, Pe(S0,T,X1) or Pe(S0,T,X2)?
• Construct portfolios A and B. See Table 3.8, p.
  79.
• Portfolio A has non-negative payoff therefore,
• Pe(S0,T,X2) ³ Pe(S0,T,X1)
• Intuition show what happens if not true
• Prices of AOL options conform

17
Principles of Put Option Pricing (continued)

• The Effect of Exercise Price (continued)
• Limits on the Difference in Premiums
• Again, note Table 3.8, p. 79. We must have
• (X2 - X1)(1r)-T ³ Pe(S0,T,X2) - Pe(S0,T,X1)
• X2 - X1 ³ Pe(S0,T,X2) - Pe(S0,T,X1)
• X2 - X1 ³ Pa(S0,T,X2) - Pa(S0,T,X1)
• Implications
• See Table 3.9, p. 81. Prices of AOL options
  conform

18
Principles of Put Option Pricing (continued)

• The Lower Bound of a European Put
• Construct portfolios A and B. See Table 3.10, p.
  81.
• A dominates B. This implies that (after
  rearranging)
• Pe(S0,T,X) ³ Max(0,X(1r)-T - S0)
• This is the lower bound for a European put
• See Figure 3.10, p. 82 for the price curve for
  European puts
• Dividend adjustment subtract present value of
  dividends from S to obtain S

19
Principles of Put Option Pricing (continued)

• American Put Versus European Put
• Pa(S0,T,X) ³ Pe(S0,T,X)
• The 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.

20
Principles of Put Option Pricing (continued)

• Put-Call Parity
• Form portfolios A and B where the options are
  European. See Table 3.11, p. 84.
• 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(1r)-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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Principles of Put Option Pricing (continued)

• Put-Call parity for American options can be
  stated only as inequalities
• See Table 3.12, p. 86 for put-call parity for AOL
  options
• See Figure 3.11, p. 87 for linkages between
  underlying asset, risk-free bond, call, and put
  through put-call parity.

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Principles of Put Option Pricing (continued)

• The Effect of Interest Rates
• The Effect of Stock Volatility

Summary See Table 3.13, p. 90.
Appendix 3 The Dynamics of Option Boundary
Conditions A Learning Exercise
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