Chapter 3: Principles of Option Pricing
Well, it helps to look at derivatives like atoms. Split them
one way and you have heat and energy - useful stuff. Split
them another way and you have a bomb. You have to
understand the subtleties.
Kate Jennings
Moral Hazard, Fourth Estate, 2002, p. 8
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 3: 1
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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
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 3: 2
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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)
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 3: 3
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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
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 3: 4
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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
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 3: 5
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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
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 3: 6
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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
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 3: 7
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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 X1
and 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
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 3: 8
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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  Ce(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
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 3: 9
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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]
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 3: 10
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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 > S0 - 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
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 3: 11
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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
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 3: 12
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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
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 3: 13
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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
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 3: 14
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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
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 3: 15
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Principles of Put Option Pricing (continued)

Effect of Exercise Price
 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.
 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
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 3: 16
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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  Pe(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
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 3: 17
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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 - S0)
 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´
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 3: 18
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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.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 3: 19
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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.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 3: 20
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Principles of Put Option Pricing (continued)
 Put-Call
parity for American options can be stated only
as inequalities:
N
C a (S'0 , T, X)  X   D j (1  r)
t j
j1
 S0  Pa (S'0 , T, X)
 C a (S'0 , T, X)  X(1  r) T
 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.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 9th ed.
Ch. 3: 21
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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.
Appendix 3: The Dynamics of Option
Boundary Conditions: A Learning Exercise
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