Chapter 18
Options and
Contingent Claims
Copyright  2009 McGraw-Hill Australia Pty Ltd
PPTs t/a Business Finance 10e by Peirson
Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
18-1
Learning Objectives
• Understand the major types and characteristics of
options and distinguish between options and futures.
• Identify and explain the factors that affect option
prices.
• Understand and apply basic option pricing theories,
including put–call parity.
• Understand the binomial model and Black–Scholes
model of option pricing and calculate option prices
using these models.
• Explain the characteristics and uses of foreign
currency options and options on futures.
• Define a contingent claim and identify the option-like
features of several contingent claims.
Copyright  2009 McGraw-Hill Australia Pty Ltd
PPTs t/a Business Finance 10e by Peirson
Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
18-2
Options and Option Markets
• An ‘option’ is the right to force a transaction to occur at
some future time on terms and conditions decided now.
• It is a contract that gives the purchaser the right, but not
the obligation, to assume a long (buy) or short (sell)
position in the relevant underlying financial instrument or
future at a predetermined exercise (strike) price, at a time
in the future.
• In return for this right the purchaser pays the option price
to the seller (‘writer’) of the option.
• Unlike forward-rate agreements (FRAs) and futures
contracts, options allow the benefits of favourable price
movements and provide protection against unfavourable
price movements.
Copyright  2009 McGraw-Hill Australia Pty Ltd
PPTs t/a Business Finance 10e by Peirson
Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
18-3
Option Terminology
• Call option
– Purchaser has the right but not the obligation to buy an
asset at the specified exercise price.
– Purchaser’s risk is limited to the price paid.
– Writer’s profit is limited to the price received and has
unlimited upside risk should prices rise.
• Put option
– Purchaser has the right, but not the obligation, to sell a
specified asset at a specified exercise price.
– The purchaser’s risk is limited to the price paid.
– The writer of the put option has limited risk (up to the
‘exercise’ price of the put option) should prices fall, and
profit is limited to the price received.
Copyright  2009 McGraw-Hill Australia Pty Ltd
PPTs t/a Business Finance 10e by Peirson
Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
18-4
Option Terminology (cont.)
• Exercise or strike price
– The price at which a particular option can be exercised.
• American-type options
– Buyer exercises option at any time up to the expiry date.
• European-type options
– Buyer may only exercise option on the expiry date.
• In the money
– Call option: exercise price (Xp) < Spot Price (Sp)
– Put option: exercise price (Xp) > Spot Price (Sp)
• Out-of-the-money
– Call option: exercise price (Xp) > Spot Price (Sp)
– Put option: exercise price (Xp) < Spot Price (Sp)
Copyright  2009 McGraw-Hill Australia Pty Ltd
PPTs t/a Business Finance 10e by Peirson
Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
18-5
Option Contracts and Futures
Contracts
• Both involve delivery of some underlying asset at a
future date and at a predetermined price.
• However, a futures contract requires delivery, whereas
an option buyer chooses whether or not delivery will
occur.
• Payment of the futures price is not required until the
expiry date, but when an option contract is created, the
buyer must immediately pay the option price to the
writer.
• If the option is exercised, there is a further transaction
where the exercise price is paid.
• Draw payoff diagram for put and call options!
Copyright  2009 McGraw-Hill Australia Pty Ltd
PPTs t/a Business Finance 10e by Peirson
Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
18-6
Option Pricing
• At expiry, a call is worth:
Max 0, S *  X 
• At expiry, a put is worth:
Max 0, X  S * 
where:
S * = the share price on the call's expiry date
X = the exercise price of the option
Component of option price
= intrinsic value + Time Value of Money
Copyright  2009 McGraw-Hill Australia Pty Ltd
PPTs t/a Business Finance 10e by Peirson
Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
18-7
Factors Affecting Call Options
• The current share price
– The higher the current share price, the greater is the
probability that the share price will increase above the
exercise price; hence, the higher the call price.
• The exercise price
– The higher the exercise price, the lower is the probability
that the share price will increase above the exercise
price; hence, the lower the call price.
• The term to expiry
– A longer-term option dominates a short-term option
because there is more time for the share price to increase
above the exercise price.
Copyright  2009 McGraw-Hill Australia Pty Ltd
PPTs t/a Business Finance 10e by Peirson
Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
18-8
Factors Affecting Call Options
(cont.)
• The volatility of the share
– Higher share price volatility increases the chance of both
large increases and large decreases in the share price.
• The risk-free interest rate
– The buyer of a call option can defer paying for the shares.
The right to defer payment is valuable because of TVM.
Therefore, the higher the risk-free interest rate, the higher
the price of a call, other things being equal.
• Expected dividends
– If a company pays a dividend to its ordinary shareholders,
the share price will fall on the ex-dividend date.
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Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
18-9
Factors Affecting Call Options
(cont.)
• Other things being equal, call prices should be
higher (lower):
– The higher (lower) the current share price.
– The lower (higher) the exercise price.
– The longer (shorter) the term to expiry.
– The more (less) volatile the underlying share.
– The higher (lower) the risk-free interest rate.
– The lower (higher) the expected dividend to be paid
following an ex-dividend date that occurs during the term
of the call.
Copyright  2009 McGraw-Hill Australia Pty Ltd
PPTs t/a Business Finance 10e by Peirson
Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
18-10
Basic Features of Put Option
Pricing
• Other things being equal, put prices should be
higher (lower):
– The lower (higher) the current share price.
– The higher (lower) the exercise price.
– The longer (shorter) the term to expiry (for American
puts).
– The more (less) volatile the underlying share.
– The lower (higher) the risk-free interest rate.
– The higher (lower) the expected dividend to be paid
following an ex-dividend date that occurs during the term
of the put.
Copyright  2009 McGraw-Hill Australia Pty Ltd
PPTs t/a Business Finance 10e by Peirson
Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
18-11
Put–Call Parity
• For European options on shares that do not pay
dividends, there is an equilibrium relationship between the
prices of puts and calls that have the same underlying
share, exercise price, term to expiry, and are traded
simultaneously.
• The put–call parity relationship can be written as:
where:
X
p cS 
1 r '
p  the price of the European put
c  the price of the corresponding European call
S  the share price
X  the exercise price
r '  the risk-free interest rate for borrowing or lending for
a period equal to the term of the put and the call
Copyright  2009 McGraw-Hill Australia Pty Ltd
PPTs t/a Business Finance 10e by Peirson
Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
18-12
Put–Call Parity (cont.)
• The put–call parity relationship is based on the idea of
arbitrage between a portfolio that includes a put option and
another portfolio that includes a call option.
• These two portfolios are constructed such that the payoff
structure is identical for all possible share price movements.
• The first, portfolio A, comprises a call option and a zero
coupon bond that will pay out the exercise price, X, at
maturity of bond and option.
• The second, portfolio B, comprises a put option plus the
share over which the options are held.
• The payoffs to these portfolios are the same for all possible
share price movements; thus, the values of the portfolios
must be equal, giving rise to put–call parity.
Copyright  2009 McGraw-Hill Australia Pty Ltd
PPTs t/a Business Finance 10e by Peirson
Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
18-13
Put–Call Parity (cont.)
• While there is no simple equation linking the
values of American puts and calls, the following
upper and lower bounds have been established:
X
CS 
1 r "
 PCS X
where:
P  the price of the American put
C  the price of the corresponding American call
S  the share price
X  the exercise price
r "  the risk-free interest rate for borrowing
Copyright  2009 McGraw-Hill Australia Pty Ltd
PPTs t/a Business Finance 10e by Peirson
Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
18-14
The Minimum Value of Calls
and Puts
• The minimum value of a European call on a nondividend paying share:
X 

Min c  Max 0, S 
1  r ' 

In the absence of dividends, American call options should not be
exercised before expiry. Unlike calls, it can be rational to exercise an
American put before expiry.
• The minimum value of a European put is:
 X

Min p  Max 0,
 S
 1 r ' 
–
In some circumstances, the benefit of receiving an early
cash flow from early exercise will outweigh the cost of
forfeiting some of the option’s time value.
Copyright  2009 McGraw-Hill Australia Pty Ltd
PPTs t/a Business Finance 10e by Peirson
Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
18-15
Binomial Option Pricing
• Binomial option pricing was developed by Cox,
Ross and Rubinstein (1979).
• Assumption that after each time period, the price
of the underlying asset can only take one of two
values.
• Risk-neutrality: enables us to price options
regardless of the risk preferences of traders in the
market:
– Situation in which investors are indifferent to risk; assets
are therefore priced such that they are expected to yield
the risk-free interest rate.
Copyright  2009 McGraw-Hill Australia Pty Ltd
PPTs t/a Business Finance 10e by Peirson
Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
18-16
Binomial Option Pricing (cont.)
• Multi-period binomial option pricing
– Stage 1: Building up a lattice of share prices.
– Stage 2: Calculating the option payoffs at expiry from
the expiry share prices.
– Stage 3: Calculating option prices by calculating
expected values and then discounting at the risk-free
interest rate.
• Example 18.3
We wish to value a 3-month call option with an exercise
price of $10.25. The current share price is $10 and the riskfree interest rate is 1.5% p.m. Three time periods of
1 month each. It is assumed that at the end of each month,
the share price can move to only one of two values.
Copyright  2009 McGraw-Hill Australia Pty Ltd
PPTs t/a Business Finance 10e by Peirson
Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
18-17
Binomial Option Pricing (cont.)
• Stage 1: The lattice of share prices
– The objective is to lay out all the future share prices that
can arise, given our assumptions.
– In this example, it is assumed that each month the share
price can rise by 4% or fall by 3.846% (a rise in 1 month
will be exactly offset if there is a fall in the following month,
and vice versa).
– If the share price increases in the first month:
$10.00 × 1.04, then the price will be $10.40.
Copyright  2009 McGraw-Hill Australia Pty Ltd
PPTs t/a Business Finance 10e by Peirson
Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
18-18
Binomial Option Pricing (cont.)
• Stage 2: Option payoffs at expiry
– As we know the expiry share prices from Stage 1, it is a
simple matter to calculate the matching call option payoffs.
– If after three raises in price, the share is $11.2486, and the
payoff is $0.9986 because the exercise price is $10.25.
•
Stage 3: Discounting
–
We first need to find the probabilities of a rising and falling
share price. Using node B as an example:
$10.40 
p  $10.816   1  p  $10.00
1.015
Copyright  2009 McGraw-Hill Australia Pty Ltd
PPTs t/a Business Finance 10e by Peirson
Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
18-19
Binomial Option Pricing (cont.)
– This equation solves to give
p = 0.6814 and
(1 – p) = 0.3186
We can now work back through the lattice from expiry
to the present, at each node calculating the present
value of the expected payoff.
– For example, at node D, the call’s price is:
call price=  0.6814  $0.9986    0.3186  $0.151.015
 $0.7175
– Working back through the lattice to today (node A)
gives the call’s price as $0.3658 or about 37 cents.
Copyright  2009 McGraw-Hill Australia Pty Ltd
PPTs t/a Business Finance 10e by Peirson
Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
18-20
Binomial Option Pricing (cont.)
Lattice of share prices and call prices
(possible future share prices in bold,
call's payoffs in italic )
G
10.82
$0.7175
D
Figure 18.3
$10.4000
$0.5133 B
$0.9986
H
$10.4000
$0.1500
I
$9.6154
$0.00
J
$8.8900
$0.00
10.00
$0.1007
E
A
$10.0000
$0.3658
$9.6154 C
$0.0676
F
$9.2456
$0.00
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PPTs t/a Business Finance 10e by Peirson
Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
$11.2486
18-21
Black–Scholes Model of Call
Option Pricing
• Black and Scholes presented a model that
determines the price of a call as a function of five
variables:
– The current price of the underlying share.
– The exercise price of the call.
– The call’s term to expiry.
– The volatility of the share (as measured by the
variance of the distribution of returns on the share).
– The risk-free interest rate.
Copyright  2009 McGraw-Hill Australia Pty Ltd
PPTs t/a Business Finance 10e by Peirson
Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
18-22
Black–Scholes Model of Call
Option Pricing (cont.)
• Assumptions:
– Constant risk-free interest rate at which investors can
borrow and lend unlimited amounts.
– Share returns follow random walk in continuous time
with a variance proportional to the square of the share
price. The variance rate is a known constant.
– No transaction costs or taxes.
– Short selling is allowed, with no restrictions or penalties.
– There are no dividends or rights issues.
– The call is of the European type.
Copyright  2009 McGraw-Hill Australia Pty Ltd
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Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
18-23
Black–Scholes Model of Call
Option Pricing (cont.)
• The Black–Scholes call option pricing formula is given by:
c  SN  d1   Xe r 'T N  d 2 
d1 
d2 
ln  S X  
ln  S X 


r ' 0.5 2 T
 T

  r ' 0.5 2  T
 T
 d1   T 
• N(d) indicates the cumulative standard normal density
function with upper integral limit d. T is the term to expiry.
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Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
18-24
Black–Scholes Model of Call
Option Pricing (cont.)
• Example 18.4:
–
Current share price:
S = $17.60
–
Exercise price:
X = $16.00
–
Term to expiry:
T = 3 months = 0.25 yrs
–
Volatility (variance):
 = 0.09 p.a.
–
Risk-free interest rate:
r  = 0.1 p.a.
continuously
compounding
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18-25
Black–Scholes Model of Call
Option Pricing (cont.)
• Example 18.4: Calculate d1 and d2:
d1 

 0.1
ln 17.60 16.00  
ln 1.1

 0.5  0.09   0.25
0.3 0.25
 0.03625
0.15

 0.877
d 2  0.877  0.15  0.727
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Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
18-26
Black–Scholes Model of Call
Option Pricing (cont.)
•
Example 18.4:
–
N (0.877) from normal density function is 0.8098
–
N (0.727) from normal density function is 0.7664
–
The discounting factor
e r 'T  e0.025  0.9753
C  SN  d1   Xe  r 'T N  d 2 
  $17.60  0.8098    $16.00  0.9753 0.7664 
 $2.293
– The Black–Scholes call price is, therefore, approximately
$2.29.
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Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
18-27
Black–Scholes Model of Call
Option Pricing (cont.)
• The Black–Scholes call option pricing formula is
specified for continuous compounding.
• It can be modified to assume discrete compounding:
c  SN  d1   PV  X  N  d 2 
X

PV  X  
1 R
ln  S X   0.5 2T
d1 
, d 2  d1   T
 T 
• In this case, R is the observed effective risk-free
interest rate for the term of the option.
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18-28
Black–Scholes Model of Put
Option Pricing (cont.)
• The Black–Scholes put pricing model for European
puts on shares that do not pay dividends:
r 'T


p  S  N  d1   1  Xe 1  N  d 2  
• This exploits the put–call parity formula.
• The Black–Scholes formula has been combined
with put–call parity to arrive at the above formula
for pricing a put.
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Slides prepared by Farida Akhtar and Barry Oliver, Australian National University
18-29
Black–Scholes Model: Brief
Assessment
• No assumption concerning the attitude of investors
towards risk is required or implied.
• The variables that are required are for the most
part observable, and have reliable data available.
• Measuring the share’s volatility is, however,
subject to error (the big unknown).
• Empirical evidence suggests that the Black–
Scholes model can price exchange-traded call
options accurately, provided that a dividendadjusted model is used.
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18-30
Options on Foreign Currency
• What is an option on foreign currency?
– A contract that confers the right to buy (sell) an agreed
quantity of foreign currency at a given exchange rate.
– Options on foreign currency are traded in organised
markets, as well as in ‘over-the-counter’ markets and by
privately arranged contracts.
– A company selling USD in exchange for AUD can be said
to be purchasing AUD in exchange for USD.
– A put option to sell USD in exchange for AUD can be
described as a call to buy AUD in exchange for USD.
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18-31
Pricing Options on Foreign
Currency
• A call to buy foreign currency may be priced as:
c  Se  iT N  d3   Xe rT N  d 4 
d3 
ln  S X  
 r  i  0.5  T
2
 T
d 4  d3   T
where:

c  price of the call
S  spot price of one unit 
of foreign currency
X  exercise price (in domestic currency)
T  term of the call
 2  volatility of the spot price
i  foreign currency risk-free interest rate
r  domestic currency risk-free interest rate
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18-32
Options, Forwards and Futures
• Simple relationship between European-type options and
forward prices.
• Combining a put and call with identical exercise price
and expiry date can replicate a forward/futures contract.
• If prices of such puts and calls are equal, then
futures/forward price should equal exercise price of
options.
• Low Exercise Price Options (LEPO) are options with an
exercise price of 1 cent.
• Options will certainly be exercised if they are written on
shares worth more than 1 cent.
• On purchase, option price is not paid; instead traders
pay margin calls. This appears like a futures contract on
shares.
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18-33
Options on Futures
• A call option on futures confers on the buyer of the call
the right to enter into a futures contract as a buyer.
• A put option on futures confers on the buyer of the put
the right to enter into a futures contract as a seller.
• Uses of options on futures: The key is the right feature
of an option rather than the obligation feature of a future.
– Open futures positions entail very high risks for a
speculator, particularly if those positions are held for a long
time.
– Hedgers may not be certain enough of their own
circumstances to justify accepting the obligations of a
futures contract.
– The deposit/margin system is simpler for option buyers
than for futures traders.
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18-34
Contingent Claims
• A ‘contingent claim’ is an asset whose value depends on the
given value of some other asset. A call option is perhaps the
simplest type of contingent claim.
• Rights issue
– A shareholder is given the right to purchase new shares
in the company at an issue price set by the company.
– The rights must be sold or taken up by a specified date.
– Simply a call option issued by the company.
• Convertible bonds
– A type of debt security that, in addition to paying interest,
gives the investor the right to convert the security into
shares of the company.
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18-35
Contingent Claims (cont.)
• Valuation of levered shares and risky couponpaying debt:
– The approach can also be applied to the problems of
valuing risky coupon-paying debt and valuing the shares
of a company that has issued this type of debt.
• Valuation of levered shares and risky zerocoupon debt:
– Shareholders must make a choice that resembles the
choice facing the holder of a call option.
• Project evaluation and ‘real’ options:
– The NPV approach is based on an analogy between a
proposed investment project and a bond.
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18-36
Summary
• Option — right to force a transaction at future date on
terms decided now. For example, call, put, American and
European.
• Option/contingent claim value depends on the value
of another asset.
• Put–call parity — European put can be replicated by
share, risk-free deposit and counterpart call.
• Options can be priced using:
– Binomial option and Black–Scholes option pricing model.
• Instruments such as rights, convertible bonds, shares in
levered companies, and default risk of interest rates can
all be analysed as options.
• Real options, such as to delay, consider further, or
expand projects, have value just as the options on
financial assets do.
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18-37