FIN 819: lecture 6

Introduction to options

Some basic concepts

FIN 819: lecture 6

Today’s plan Briefly review the case discussed last week. Review what we have learned so far Introduction of options

• Definition of options

• Position diagrams

• No arbitrage argument

• Put-call parity

• Application of put-call parity

• How does stock return volatility affect option values?

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What have we learned so far? So far we have discussed or reviewed some

fundamental concepts and ideas about valuation

• Present value concept

• Discounted cash flow approach and the NPV rule

• Free-cash flow calculation

• Cost of capital (WACC)

• CAPM

• Levered and unlevered betas

• Two cases related to these concepts

FIN 819: lecture 7

Introduction to options

What is an option in Finance? An option is a right or opportunity to do something at

a specified price or cost on or before some specified date.

An option, is a contract Options are everywhere.

• IBM offers its CEO a bonus that is related to the stock price (stock options)

• IBM postpones investment in a positive-NPV project, even if IBM has capital for taking this investment.

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Options

Financial options Real options It depends on what is the underlying

asset the option is written in

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Options in financial assets

What is an option written in financial assets? A financial option is a right to sell or buy

some financial asset at a specified price or cost on or before some specified date.

• (ex: option written on IBM or Dell)

A financial option can be regarded as a contract between sellers and buyers

The financial asset specified in the financial option contract is often called the underlying financial asset.

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Real Options

Real options

• Options that are written on real assets are called real options

• For example, the option to set up a factory is called a real option

In the next two lectures, we focus on financial options, since it is easier to price them.

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Now we focus on (financial) options…

Suppose the financial asset is common stock or stock

There are two basic types of options, • Call option

• the right to buy a share of stock at a specified price before or on some date.

• Put option• the right to sell a share of stock at a specified

price before or on some date.

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Strike Price, Expiration Date

The specified price is called the strike price or exercise price. (the price you would like to buy/sell the underlying stock)

The specified date is called the maturity date or expiration date. (the date by which you want to buy/sell the stock)

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Exercising an Option

An option is exercised when the buyer of the option decides to buy or sell the stock at the specified price, at which time the seller must sell or buy the stock at the specified price

Oh yes, according to expiration terms

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American vs. European Option Call options

• European call option• Buy the stock on the specified date

• American call option• Buy the stock on or before the specified date

Put options• European put option

• Sell the stock on the specified date

• American put option• Sell the stock on or before the specified date

Key Difference? • European (1 date), • American (many dates up until expiration 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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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 a

European call on stock ABC with the same strike price of $55. The payoffs of your options certainly depend on the price of ABC on expiration

00051525Put

25155000 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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Some examples

Please draw position diagrams for the following investment:• Buy a call and put with the same strike price

and maturity (straddle)

• Buy two calls with different strike prices (K1 and K2) and sell two calls with a strike price that equals the average strike price of the two calls you bought. (butterfly)

• Buy a stock and a put (protective put)

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Option payoff

Straddle - Long call and long put

Share Price

Pos

itio

n V

alue

Straddle

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Option Value

Butterfly

Share Price

Pos

itio

n V

alue

K1 K2

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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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Form your desired portfolios

Suppose you have access to risk-free securities, stocks, calls and puts. Can you form a portfolio now to have the following payoffs at time T?

Share Price

Pos

itio

n V

alue

K

K K1

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No arbitrage concept or one price rule

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

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Example

Two treasury bonds A and B both have the maturity of 10 years and coupon rate of 6%. Certainly, they should have the same price; otherwise suppose that A is more expensive than B. You can make money by buying B and short selling A. You can make money PA-PB.

Do you like this beautiful idea?

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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. Then we have

TfKRTKPSTKC ),(),( 0

),(),( 0 TKPSKRTKC Tf

or

ff rR 1Where

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Let’s show put-call parity

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

We can also simply use the payoffs in the future 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 TKPSKRTKC Tf

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Another way of showing put-call parity Consider the following portfolio:

• Buy the stock

• Buy a European put option

• Borrow the present value of the strike price • Where Rf= 1+ rf

The cost of this portfolio is The payoff of this portfolio

• If ST >= K, the payoff is ST-K

• If ST < K, the payoff is 0.

TfKR

TfKRPS 0

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Portfolio (continues)

Final payoff of the portfolio

ST<K ST>KPortfolio

Stock

Borrowing

Put

Total

S0

P

ST

K - ST

ST

-K -K

0

0 ST - KTfKRPS 0

TfKR

Position

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The conclusion

Since the portfolio and a call option have exactly the same payoff, their prices should be the same. That is,

TfKRTKPSTKC ),(),( 0

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

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

Who will bear the default cost: equity holders or debt holders?

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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The value of option

Since an option is a right to buy or sell securities, its price is non-negative.

If the option price is negative, what will happen?

If the option value must be non-negative, can you use what you have learned to value a call option or put option by considering the following two things• Expected cash flows

• The risk of options

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Volatility and option value

For options, the larger the volatility of the underlying asset, the larger the value of the option

For stocks, the larger the volatility of the underlying asset, the smaller the value of stocks

Why ?