Introduction to options

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Basic and advanced concepts

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Today’s planIntroduction of options

Definition of options Position diagrams No arbitrage argument Put-call parity Application of put-call parity How parameter values affect option

values?

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Introduction to optionsWhat is an option? An option is a right to do something at a specified

price or cost on or before some specified date. An option, is a contract, and is therefore “written”

– just means it exists Options are everywhere.

AOL offers its CEO a bonus (stock options) if its stock price exceeds $65 per share

You have the option to come to my office hours at the cost of walking several extra steps.

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Brief HistoryOptions are a form of insurance, so

in that sense they have been around for quite some time.

The first organized exchange on which options were traded was opened in Chicago in 1973. Before that, options were traded over-the-counter.

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Brief History (cont’d)In the same year, the Black-Scholes

formulae for option prices was published. The prices predicted by the formulae turned out to be extremely close to actual option prices.

The popularity of options skyrocketed. They are arguably the most successful derivative security ever!

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Financial Options vs. Real OptionsFinancial Options

Options written on financial asset are called financial options, or simply “options” (ex: option written on IBM or Dell)

Real options Options written on real assets are called real

options For example, the option to set up a factory or

discontinue a division is called real option

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Now we focus on two types of (financial) options…Call

An option to buy an underlying security (for example, a stock) for a fixed price (that is, the strike or exercise price) on or before a certain date (expiration date or maturity date).

Put An option to sell the underlying security (for

example, a stock) for a fixed price (that is the strike or exercise price) on or before a certain date (expiration date or maturity date).

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Option TermsExercising the Option

Enforcing the contract, i.e., buy or selling the underlying asset using the option

Striking, Strike, or Exercise Price The fixed price specified in the option contract for

which the holder can buy or sell the underlying asset.

Expiration Date The last date on which the contract is still valid.

After this date the contract no longer exists.

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Option terminology In-the-money call – a call option whose exercise

price is less than the current price of the underlying stock.

Out-of-the-money call – a call option whose exercise price exceeds the current stock price.

Another way to remember whether an option is in the money: if you can make money by immediately exercising your option, the option is in the money. (You may not be able to exercise it, though.)

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European vs. American OptionsEuropean

A European option can only be exercised on the exercise date.

American An American option can be exercised on

any date up to the exercise date.

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Option Obligations Options are rights (to the buyer), and are obligations

(to the seller) This means that:

the buyer of an option may or may not exercise the option. However, the seller of the option must sell or buy the

underlying assets if the buyer decides to exercise the option.

assetbuy toObligationasset sell Right tooptionPut asset sell toObligationassetbuy Right tooption Call

SellerBuyer

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What is a short position in an option?

In this case the other party has the option.

Is a long position in a call the same as a short position in a put?

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Payoff or cash flows from options at expiration dateThe payoff of a call option with a strike

price K at the expiration date T is

Where S(T) is the stock price at time T The payoff of a put option with a strike price K

at the expiration date T is

Where S(T) is the stock price at time T

)0,)(max( KTS

)0),(max( TSK

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Example on payoffs Suppose that you have bought one European

put and an European call on AOL with the same strike price of $55. The payoffs of your options certainly depend on the price of AOL on expiration

00051525ValuePut 25155000Value Call8070605040$30PriceStock

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Option payoff at expiration

Call option value (graphic) given a $55 exercise price.

Share Price

Cal

l opt

ion

$ pa

yoff

55 75

$20

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Option payoff

Put option value (graphic) given a $55 exercise price.

Share Price

Put o

ptio

n va

lue

50 55

$5

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Option payoff

Call option payoff (to seller) given a $55 exercise price.

Share Price

Cal

l opt

ion

$ pa

yoff

55

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Option payoff

Put option payoff (to seller) given a $55 exercise price.

Share Price

Put o

ptio

n $

payo

ff

55

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Let's do some examples.Going short, selling an option you do not

own, or writing an option are all the same thing.

You have written a call with a strike of $50 on GM stock. What is your position if, on the expiration date, GM closes at

$55$45

Who has the option in this case?

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Value of the position at expiration

Stock Price20 40 60 80 100

-50

-40

-30

-20

-10

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Shorting PutsYou have written a put with a strike

of $50 on GM stock. What is your position if, on the expiration date, GM closes at

$55$45

Who has the option in this case?

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Value of the position at expiration

Stock Price20 40 60 80 100

-50

-40

-30

-20

-10

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What is the payoff if you go long a call and short a put, both with a strike of $50?

Say I add $50, what is another name for this position?

20 40 60 80 100

-40

-20

20

40

Stock Price

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Some examplesPlease draw position diagrams for

the following investment: Buy a call and put with the same strike

price and maturity (straddle)

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Option payoff

Straddle - Long call and long put

Share Price

Posi

tion

Val

ue

Straddle

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More examples Buy a stock and a put (protective put)

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Option payoff

Protective Put - Long stock and long put

Share Price

Posi

tion

Val

ue Protective Put

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Valuation of optionsAt expiration an option must be

worth its exercise value or zero.An American option's value is as least

as large as its immediate exercise value (why?) and since it gives an extra right (which can always be ignored) is always at least as valuable as its European counterpart.

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Valuation of optionsAn American call's (put's) value can

never exceed the value of the stock (strike price)

Why?Does this principle hold for European

options? Yes.

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Valuation of options

Everything else equal, the longer maturity for An American option, the more valuable.

Why? Does this principle hold for European

options?

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Valuation of optionsAn American call (put) with a higher

exercise price will be worth less (more).

Why?Does this principle apply to European

options?Yes.

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Put-Call ParityLet P(K,T) and C(K,T) be the prices of

a European put and a call with strike prices of K and maturity of T. S0 is current stock price. Then we have

TfKRTKPSKTC ),(),( 0

),(),( 0 TKPSKRKTC Tf

or

ff rR 1Where

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No arbitrage conceptIf two securities have the exactly the

same payoff or cash flows in every state of each future period, these two securities should have the same price; otherwise there is an arbitrage opportunity or money making opportunity.

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Let’s show put-call parityWe can first use position diagrams to

show put-call parityThis exercise is a good way of

getting used to the ideas of the single price rule or no arbitrage argument.

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Position diagram

Payoff of investing PV(K) in risk-free security and buying a call

Share Price

Posi

tion

Val

ue

K

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Position diagram

Payoff of long stock and long put

Share Price

Posi

tion

Val

ue

K

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The conclusionSince both portfolios in the previous

two slides give you exactly the same payoff, they must have the same price. That is,

),(),( 0 TKPSKRKTC Tf

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In the above we have assumed that the stock will not pay any dividend.

Consider dividend payment D before expiration date. For European options:

0( ) ( ) ( ) ( , ) ( , )TfKR PV D PV K PV D S P K T C T K

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Things to note about Put-Call parityOnly works for European options. Based on arbitrage so it works

exactly. This is how brokers created puts out of calls when options were traded over the counter.

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European vrs American CallsIt turns out that you would never

want to exercise an American call on a non-dividend paying stock early.

Why might you wish to exercise an American call early when the stocks pays dividend?

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European vs American PutsThere are times when you will want

to exercise an American put on a non-dividend paying stock early.

Why?

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Applications of option concepts and put-call parityOne important application of option

concepts and put-call parity is the valuation of corporate bonds.

For example, suppose that a firm has issued $K million zero-coupon bonds maturing at time T. Let the market value of the firm asset at time t be V(t).

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Applications of option concepts and put-call parity (continue)Payoff of equity

Market value of asset

Posi

tion

Val

ue

K

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Applications of option concepts and put-call parity (continue)So based on the position payoff diagram

in the previous slide, we can see that the value of equity is just the value of a call option with strike price K.

Then bond value =Asset value –equity value (value of call: C(K,T)

Using the put-call parity, we have Bond value=V(A)-(V(A)+P(K,T)-

PV(K))=PV(K)- P(K,T) (value of put )

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Applications of option concepts and put-call parity (continue)What does this result mean?The value of risky corporate bonds is

equal to the value of the safe corporate bonds minus the cost of default.

When will the firm default? At time T, if the value of asset is less than

K, the firm will default. P(K,T) is the cost of this default to bond holders.

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Some bounds about option valuesSince an option is a right to buy or sell

securities, its price is always non-negative.Since at expiration, we have payoff Max(S(T)-K,0) for a call with a price C(K,T) at

time 0Max(K-S(T),0) for a put with a price P(K,T) at time 0

Then KTKPSTKC

),(0)0(),(0

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Some bounds about option values (continue)From put-call parity, we have

Thus

KSKRSKRTKPSKTC Tf

Tf

000 ),(),(

0),( KTC

)0,max(),( 0 KSKTC

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The impact of volatility of the stock price on the call optionConsider the following two call options

written on stocks A and B with the same strike price of $50 and same maturity, respectively: current price SA=SB=$40 and stock A is much more volatile than stock B. Then At maturity, stock A has a much larger chance that the stock price is larger than $50 than stock B. Thus, the payoff from the option on stock A is expected to be larger than from the option A. Thus the option on stock A is more valuable than the option on stock B.

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Volatility and option values.For call options, the larger the volatility

of the underlying asset, the larger the value of the option.

Suppose a firm has both debt and equity. If the managers are to take riskier projects

than bond holders expect, should the bond holders or equity holders benefit from this?

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How option values are affected by variables?

If this variable increases

The value of an American or European call

The value of an European put

The value of an American put

Stock price (S) Increase Decrease Decrease

Exercise price (K)

Decrease Increase Increase

Volatility (σ) Increase ? Increase

Time to expiration (T)

Increase ? Increase

Interest rate (rf)

Increase Decrease Decrease

Dividend payout

Decrease Increase Increase

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The Black-Scholes formula for a call option The Black-Scholes formula for a European call is

Where)()(),,,,( 11 tdNKedSNrtKSC rt

ttrtKSd

21)/ln(

1

optiontheofpricestrikeK

pricestockstodayS 'periodpervolatilityreturnstock

ratefreeriskcompoundedlycontinuousr irationtotimet exp

functionondistributinormalcumulativedN )(

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The Black-Scholes formula for a put option The Black-Scholes formula for a European put is

Where)()(),,,,( 11 dSNdtNKertKSP rt

ttrtKSd

21)/ln(

1

optiontheofpricestrikeK

pricestockstodayS 'periodpervolatilityreturnstock

ratefreeriskcompoundedlycontinuousr

irationtotimet expfunctionondistributinormalcumulativedN )(

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Of course, if you already know a call with same maturity and expiration…

You can get the put price by put-call parity.

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Intuition for the Black-Scholes formula

One way to understand the Black-Scholes formula is to find the present value of the payoff of the call option if you are sure that you can exercise the option at maturity, i.e., S - exp(-rt)K.

Comparing this present value of this payoff to the Black-Scholes formula, we know that N(d1) can be regarded as the probability that the option will be exercised at maturity

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An exampleMicrosoft sells for $50 per share. Its

return volatility is 20% annually. What is the value of a call option on Microsoft with a strike price of $70 and maturing two years from now suppose that the risk-free rate is 8%?

What is the value of a put option on Microsoft with a strike price of $70 and maturing in two years?

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SolutionThe parameter values are

Then50,70

08.0,2,2.0

SK

rt

27.12$;63.2$22.0)765.0()(

315.0685.01)(1)(

4825.021)/ln(

1

11

1

PCNtdN

dNdN

ttrtKSd