Chance/Brooks
Kate Jennings
Moral Hazard, Fourth Estate, 2002, p. 8
CHAPTER 3: PRINCIPLES OF
OPTION PRICING
Ch.
3: 1
An Introduction to Derivatives and Risk Management,
9th ed.
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.
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.




An Introduction to Derivatives and Risk
Management, 9th ed.

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, riskfree rate and volatility on an option price
Difference between prices of European and
American options
Put-call parity
Chance/Brooks
IMPORTANT CONCEPTS IN CHAPTER 3
Ch.
3: 2
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
Symbols





An Introduction to Derivatives and Risk
Management, 9th ed.

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
BASIC NOTATION AND TERMINOLOGY
Ch.
3: 3
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
Computation of risk-free rate (r)
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
An Introduction to Derivatives and Risk
Management, 9th ed.
 Date:
Chance/Brooks
BASIC NOTATION AND TERMINOLOGY
(CONTINUED)
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
Ch.
3: 4
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
Minimum Value of a Call


Concept of intrinsic value: Max(0,S0 - X)


Proof of intrinsic value rule for DCRB calls
Concept of time value


Ca(S0,T,X)  Max(0,S0 - X)
See Table 3.2 for time values of DCRB calls
See Figure 3.1 for minimum values of calls
An Introduction to Derivatives and Risk
Management, 9th ed.
C(S0,T,X) 0 (for any call)
 For American calls:

Chance/Brooks
PRINCIPLES OF CALL OPTION PRICING
Ch.
3: 5
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
Maximum Value of a Call

Value of a Call at Expiration
C(ST,0,X) = Max(0,ST - X)
 Proof/intuition
 The same for American and European options
 See Figure 3.3

An Introduction to Derivatives and Risk
Management, 9th ed.
C(S0,T,X) S0
 Intuition
 See Figure 3.2, which adds this to Figure 3.1

Chance/Brooks
PRINCIPLES OF CALL OPTION PRICING
(CONTINUED)
Ch.
3: 6
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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
Cannot be proven (yet) for European calls
An Introduction to Derivatives and Risk
Management, 9th ed.

Chance/Brooks
PRINCIPLES OF CALL OPTION PRICING
(CONTINUED)
Ch.
3: 7
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
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

An Introduction to Derivatives and Risk
Management, 9th ed.

Chance/Brooks
PRINCIPLES OF CALL OPTION PRICING
(CONTINUED)
Ch.
3: 8
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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  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

An Introduction to Derivatives and Risk
Management, 9th ed.

Chance/Brooks
PRINCIPLES OF CALL OPTION PRICING
(CONTINUED)
Ch.
3: 9
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
Lower Bound of a European Call

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]
An Introduction to Derivatives and Risk
Management, 9th ed.

Construct portfolios A and B. See Table 3.5.
B dominates A. This implies that (after rearranging)
Chance/Brooks
PRINCIPLES OF CALL OPTION PRICING
(CONTINUED)
Ch.
3: 10
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An Introduction to Derivatives and Risk
Management, 9th ed.
23) Suppose the current stock price is $100, the
exercise price is $100, the annually compounded
interest rate is 5 percent, the stock pays a $1
dividend in the next instant, and the quoted call
price is $3.50 for a one-year option. Identify the
appropriate arbitrage opportunity
Chance/Brooks
PRINCIPLES OF CALL OPTION PRICING
(CONTINUED)
Ch.
3: 11
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
American Call Versus European Call

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
An Introduction to Derivatives and Risk
Management, 9th ed.

Ca(S0,T,X)  Ce(S0,T,X)
But S0 - X(1+r)-T > S0 - X prior to expiration so
Chance/Brooks
PRINCIPLES OF CALL OPTION PRICING
(CONTINUED)
Ch.
3: 12
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
Early Exercise of American Calls on DividendPaying Stocks
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
An Introduction to Derivatives and Risk
Management, 9th ed.
 If
Chance/Brooks
PRINCIPLES OF CALL OPTION PRICING
(CONTINUED)
Ch.
3: 13
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
Minimum Value of a Put


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
An Introduction to Derivatives and Risk
Management, 9th ed.
P(S0,T,X) 0 (for any put)
 For American puts:

Chance/Brooks
PRINCIPLES OF PUT OPTION PRICING
Ch.
3: 14
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
Maximum Value of a Put

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

An Introduction to Derivatives and Risk
Management, 9th ed.
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

Chance/Brooks
PRINCIPLES OF PUT OPTION PRICING
(CONTINUED)
Ch.
3: 15
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An Introduction to Derivatives and Risk
Management, 9th ed.
18) Suppose a European put has an exercise price of
$110 on February 5. The put expires in 45 days.
Suppose the appropriate discount rate on Treasury
bills maturing in 44 days is 7.615. What is the
maximum value of the European put? If the put
were instead an American put, what would be its
maximum value?
Chance/Brooks
PRINCIPLES OF PUT OPTION PRICING
(CONTINUED)
Ch.
3: 16
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
Effect of Time to Expiration



Proof/intuition
See Figure 3.9 and Table 3.7
Cannot be proven for European puts
An Introduction to Derivatives and Risk
Management, 9th ed.
Two American puts differing only by time to
expiration, T1 and T2 where T1 < T2.
 Pa(S0,T2,X)  Pa(S0,T1,X)

Chance/Brooks
PRINCIPLES OF PUT OPTION PRICING
(CONTINUED)
Ch.
3: 17
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
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

An Introduction to Derivatives and Risk
Management, 9th ed.

Chance/Brooks
PRINCIPLES OF PUT OPTION PRICING
(CONTINUED)
Ch.
3: 18
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
Effect of Exercise Price (continued)

Limits on the Difference in Premiums
An Introduction to Derivatives and Risk
Management, 9th ed.
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
PRINCIPLES OF PUT OPTION PRICING
(CONTINUED)
Ch.
3: 19
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
Lower Bound of a European Put
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´
An Introduction to Derivatives and Risk
Management, 9th ed.
Construct portfolios A and B. See Table 3.10.
 A dominates B. This implies that (after rearranging)

Chance/Brooks
PRINCIPLES OF PUT OPTION PRICING
(CONTINUED)
Ch.
3: 20
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
American Put Versus European Put

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.

An Introduction to Derivatives and Risk
Management, 9th ed.

Pa(S0,T,X)  Pe(S0,T,X)
Chance/Brooks
PRINCIPLES OF PUT OPTION PRICING
(CONTINUED)
Ch.
3: 21
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
Put-Call Parity
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.

An Introduction to Derivatives and Risk
Management, 9th ed.
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

Chance/Brooks
PRINCIPLES OF PUT OPTION PRICING
(CONTINUED)
Ch.
3: 22
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 Put-Call
parity for American options can be stated only
as inequalities:
C a (S , T, X) + X + å D j (1 + r)
-t j
j=1
³ S0 + Pa (S'0 , T, X)
³ 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.
An Introduction to Derivatives and Risk
Management, 9th ed.
N
'
0
Chance/Brooks
PRINCIPLES OF PUT OPTION PRICING
(CONTINUED)
Ch.
3: 23
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
Summary
See Table 3.13.
An Introduction to Derivatives and Risk
Management, 9th ed.

The Effect of Interest Rates
The Effect of Stock Volatility
Chance/Brooks
PRINCIPLES OF PUT OPTION PRICING
(CONTINUED)
Ch.
3: 24
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An Introduction to Derivatives and Risk
Management, 9th ed.
2) Suppose someone offers you the following gamble:
You pay $7 and toss a coin. If the coin comes
up heads, he pays you $10, and if tails comes up,
h e p a y s y o u $ 5 . Y o u i n tu r n g e t t h e id e a o f
offering another person a coin toss in which he
pays you $7 and tosses another coin. You tell him
that if heads comes up, you will pay him $9 and
if tails comes up, you will pay him $5. You think
you see an opportunity to earn an arbitrage profit
by engaging in both transactions at the same
time. Why is this not an arbitrage opportunity?
How could you make it one assuming that you
could get two people to engage in these gambles?
Chance/Brooks
PRINCIPLES OF OPTION PRICING
Ch.
3: 25
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An Introduction to Derivatives and Risk
Management, 9th ed.
10) Suppose a European put price exceeds the
value predicted by put-call parity. How could an
investor profit? Demonstrate that your strategy is
correct by constructing a payoff table showing the
outcomes at expiration.
Chance/Brooks
PRINCIPLES OF OPTION PRICING
Ch.
3: 26
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An Introduction to Derivatives and Risk
Management, 9th ed.
25) On December 9 of a particular year, a January
Swiss franc call option with an exercise price of 46 had
a price of 1.63. The January 46 put was at 0.14. The
spot rate was 47.28. All prices are in cents per Swiss
franc. The option expired on January 13. The U.S. riskfree rate was 7.1 percent, and the Swiss risk-free rate
was 3.6 percent. Do the following:
a. Determine the intrinsic value of the call.
b. Determine the lower bound of the call.
c. Determine the time value of the call.
d. Determine the intrinsic value of the put.
e. Determine the lower bound of the put.
f. Determine the time value of the put.
g. Determine whether put-call parity holds.
Chance/Brooks
PRINCIPLES OF OPTION PRICING
Ch.
3: 27
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An Introduction to Derivatives and Risk
Management, 9th ed.
26) Suppose Congress decides that investors should
not profit when stock prices go down, so it outlaws
short selling. Congress has not figured out options,
however, so there are no restrictions on option
trading. Explain how to accomplish the equivalent
of a short sale by using options.
Chance/Brooks
PRINCIPLES OF OPTION PRICING
Ch.
3: 28
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Chance/Brooks
An Introduction to Derivatives and Risk
Management, 9th ed.
(Return to text slide)
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Chance/Brooks
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Management, 9th ed.
(Return to text slide 5)
(Return to text slide 7)
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(Return to text slide)
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(Return to text slide 8)
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Ch.
3: 52
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
10
An Introduction to Derivatives and Risk
Management, 9th ed.
(Put-Call Parity)
18 (Maximum Value of a Put)
23 (Lower Bound of a European Call)
25 (Principles of Call/Put Option Pricing)
26 (Put-Call Parity)
Chance/Brooks
MY PROBLEMS
Ch.
3: 53
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
5
An Introduction to Derivatives and Risk
Management, 9th ed.
(Lower bound of European Call)
 12 (Intrinsic Value, Time Value,
Arbitrage)
 15 (Put-Call Parity)
 21 (Lower Bound of a European Put)
 22 (Put-Call Parity)
Chance/Brooks
TA PROBLEMS
Ch.
3: 54
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.