18-1 copyright 2009 mcgraw-hill australia pty ltd ppts t/a business finance 10e by peirson slides...

18-1

Copyright 2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by PeirsonSlides prepared by Farida Akhtar and Barry Oliver, Australian National University

Chapter 18

Options and Contingent Claims

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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.

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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.

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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.

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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)

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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!

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Option Pricing

• At expiry, a call is worth:

• At expiry, a put is worth:

Component of option price

= intrinsic value + Time Value of Money

*0,Max S X *0,Max X S

*

where:

= the share price on the call's expiry date

= the exercise price of the option

S

X

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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.

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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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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.

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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.

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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:

1 '

Xp c S

r

where:

the price of the European put

the price of the corresponding European call

the share price

the exercise price

' the risk-free interest rate for borrowing or lending for

a period equal to th

p

c

S

X

r

e term of the put and the call

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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.

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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:

1 "

XC S P C S X

r

where:

the price of the American put

the price of the corresponding American call

the share price

the exercise price

" the risk-free interest rate for borrowing

P

C

S

X

r

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The Minimum Value of Callsand Puts

• The minimum value of a European call on a non-dividend paying share:

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:

– 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.

0,1 '

XMin c Max S

r

0,1 '

XMin p Max S

r

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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.

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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 risk-free 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.

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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.

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• 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:

$10.816 1 $10.00$10.40

1.015

p p

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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:

– Working back through the lattice to today (node A) gives the call’s price as $0.3658 or about 37 cents.

call price= 0.6814 $0.9986 0.3186 $0.15 1.015

$0.7175

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Lattice of share prices and call prices(possible future share prices in bold, call's payoffs in italic ) G $11.2486

10.82 $0.9986$0.7175

D

$10.4000$0.5133 B H $10.4000

10.00 $0.1500$0.1007

A E$10.0000$0.3658

I $9.6154$9.6154 C $0.00$0.0676

F$9.2456

$0.00 J $8.8900$0.00


Figure 18.3

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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.

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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.

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Black–Scholes Model of Call Option Pricing (cont.)• The Black–Scholes call option pricing formula is given by:

• N(d) indicates the cumulative standard normal density function with upper integral limit d. T is the term to expiry.

'1 2

2

1

2

2

1

ln ' 0.5

ln ' 0.5

r Tc SN d Xe N d

S X r Td

T

S X r Td

T

d T

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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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Black–Scholes Model of Call Option Pricing (cont.)• Example 18.4: Calculate d1 and d2:

1

2

ln 17.60 16.00 0.1 0.5 0.09 0.25

0.3 0.25

ln 1.1 0.03625

0.15 0.877

0.877 0.15 0.727

d

d

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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

– The Black–Scholes call price is, therefore, approximately $2.29.

' 0.025 0.9753r Te e

'1 2

$17.60 0.8098 $16.00 0.9753 0.7664

$2.293

r TC SN d Xe N d

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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:

• In this case, R is the observed effective risk-free interest rate for the term of the option.

1 2

2

1 2 1

1

ln 0.5 ,

c SN d PV X N d

XPV X

R

S X Td d d T

T

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Black–Scholes Model of Put Option Pricing (cont.)• The Black–Scholes put pricing model for European

puts on shares that do not pay dividends:

• 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.

'1 21 1r Tp S N d Xe N d

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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 dividend-adjusted model is used.

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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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Pricing Options on Foreign Currency

• A call to buy foreign currency may be priced as:

3 4

2

3

4 3

ln 0.5

iT rTc Se N d Xe N d

S X r i Td

T

d d T

2

where:

price of the call

spot price of one unit of foreign currency

exercise price (in domestic currency)

term of the call

volatility of the spot price

foreign currency risk-free interest rate

c

S

X

T

i

r

domestic currency risk-free interest rate

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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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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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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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Contingent Claims (cont.)

• Valuation of levered shares and risky coupon-paying 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 zero-coupon 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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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.

18-1 copyright 2009 mcgraw-hill australia pty ltd ppts t/a business finance 10e by peirson slides...

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