Introduction to options

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Basic and advanced concepts

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Today’s plan

Introduction 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 options

What 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 History

Options 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 Options

Financial 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 Options

European 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 date

The 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 Call

8070605040$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

opt

ion

valu

e

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

opt

ion

$ pa

yoff

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 Price

20 40 60 80 100

-50

-40

-30

-20

-10

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Shorting Puts

You 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 Price

20 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 examples

Please 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

Pos

itio

n V

alue

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

Pos

itio

n V

alue Protective Put

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Valuation of options

At 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 options

An 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 options

An 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 Parity

Let 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 concept

If 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 parity

We can first use position diagrams to show put-call parity

This 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

Pos

itio

n V

alue

K

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Position diagram

Payoff of long stock and long put

Share Price

Pos

itio

n V

alue

K

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The conclusion

Since 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 parity

Only 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 Calls

It 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 Puts

There 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 parity

One 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

Pos

itio

n V

alue

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 values

Since 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

ThenKTKP

STKC

),(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 option

Consider 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

tt

rtKSd

2

1)/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

tt

rtKSd

2

1)/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 example

Microsoft 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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Solution

The parameter values are

Then50,70

08.0,2,2.0

SK

rt

27.12$;63.2$

22.0)765.0()(

315.0685.01)(1)(

4825.02

1)/ln(

1

11

1

PC

NtdN

dNdN

tt

rtKSd