Options

Chapter 2.5

Chapter 15

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Learning Objectives Understand key terms related to options

and options markets

Compute payoffs and profits to option holders and writers

Calculate potential profits from various options strategies

Describe the put-call parity relationship

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Derivative A derivative is a security who’s value is

dependent on another assetsBase Asset Ex: commodity prices, bond and stock

prices, or market index valuesDerivatives are contingent claims

Their payoffs depend on the value of another securities.

Options are a specific type of derivativeThe holder has the right, but not the obligation, to

buy or sell a given quantity of an asset on (or before) a given date, at a price agreed upon today.

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Options: Calls and Puts Call: The owner of a call has the right, but not

the obligation, to BUY an asset in the future at the strike or exercise priceValue decreases as the strike price increases

Put: The owner of a put has the right, but not the obligation, to SELL an asset in the future at the strike or exercise priceValue increases with the strike price

Value of both calls and puts increases with time to expiration, WHY?

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Rights and Obligations

Buyer:

Gets all the Rights

Seller: Gets all the Obligations

CallsRight to Buy the

assetIs obliged to Sell the

asset

PutsRight to Sell the

assetIs obliged to Buy the

asset

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Futures Contracts

An agreement made today regarding the delivery of an asset (or in some cases, its cash value) at a specified delivery or maturity date for an agreed-upon price (futures price) to be paid at contract maturityLong position: Take delivery at maturityShort position: Make delivery at maturity

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Comparison

Option Right, but not obligation,

to buy or sell; option is exercised only when it is profitable

Options must be purchased

Futures Contract Both sides have an

obligation Long position must buy at

the futures price Short position must sell at

futures price Futures contracts are

entered into without cost

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The Option Contract

The purchase price of the option is called the premium.Stock options cover 100 shares & premium is on a

per share basis Sellers (or Writers) of options receive

premium. If the Holder (or Buyer) exercises the option,

the Writer must deliver (call) or take delivery (put) of the underlying asset.

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Options Terminology Exercising the Option

Using the option Strike Price or Exercise Price

The price specified by the option Spot Price

The market price Expiration Date

The option’s maturity date European & American options

Europeans can only be exercised at expiration.Americans can be exercised at any time up to

expiration. BUT NEVER ARE

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Options Terminology

In-the-Money: Exercising the option results in a profitCall: exercise price < market pricePut: exercise price > market price

At-the-Money: Exercising the option results in 0 profitexercise price = market price

Out-of-the-Money: Exercising the option results in a lossCall: market price < exercise pricePut: market price > exercise price

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Call Payoffs Holder of the option

Pays the premium at time=0 Has the right to exercise the option at time=T

NotationStock Price at T = ST

T is maturity, t is any other time Exercise Price = X

Don’t Exercise ExerciseCall Payoff (Intrinsic Value) 0 ST - X

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Call Holder Payoff and Profit

Value of the Call at T: CT = Max [ST – X, 0]

What is the value of the Call at t?

ST >X ST < X

CT ST – X 0

Profit ST - X - Ct -Ct

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Payoff and Profit to Call Option at ExpirationWhat is the strike price?What is the premium?

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What is the value of a Call at t?

At maturity CT = Max [ST – X, 0]

Would you be willing to sell for St-X at t?Hint: What could happened to the stock price?

Time value is the premium a rational investor will pay above an options intrinsic valueThis base on the likelihood that the stock price will

move making the option more valuableTime Value = Option Price – Intrinsic Value

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Call Writer Payoff and Profit

ST >X ST < X

Exercise Cost -(ST – X) 0

Profit -(ST - X) + Ct Ct

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Payoff and Profit to Call Writers at Expiration

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Calls: A Zero Sum Game

Call Option If ST < X If ST > X

Decision No exercise Exercise

Option Payoff (holder) 0

ST – X

Option Profit (holder) -C

(ST – X) – C

Option Payoff (writer) 0 - (ST – X)

Option Profit (writer) +C C - (ST – X)

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Profit and Loss on a Call

A February 2013 call on IBM with an exercise price of $195 was selling on January 18, 2013, for $3.65.

The option expires on the third Friday of the month, or February 15, 2013.

If the price of IBM on Feb 15, 2013 is $194, what is the call worth?

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Profit and Loss on a Call (cont.)

Suppose IBM sells for $197 at expirationRemember: strike = $195, premium = $3.65

What is the value of the option?Call Intrinsic value = stock price-exercise price

Will the option be exercised? What is the Profit/Loss on this investment?

Profit = Final value – Original investment What must the price of IBM be for the option

to break-even?

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Call Option Problem

At time=0 you buy a call option on IBM for $3.00. The option gives you the right to buy 100 shares of IBM stock at time=T at $65

What is the payoff to you if ST = $70?

What is the payoff for the writer if ST = $70?

What is the payoff to you if ST = $60?

What is the payoff for the writer if ST = $60?

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Puts

Gives holder the right (but not the obligation) to sell an asset:At the exercise or strike priceOn or before the expiration date

Exercise the option to sell the underlying asset if market value < strike.

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Puts Payoffs Holder of the option

Pays the premium at time=0 Has the right to exercise the option at time=T

NotationStock Price at T = ST Exercise Price = X

Don’t Exercise ExercisePut Payoff (Intrinsic Value) 0 X - ST

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Put Holder Payoff and Profit

Value of the Call at T: PT = Max [X - ST , 0]

ST <X ST > X

Payoff X - ST 0

Profit X - ST - Pt -Pt

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Payoff and Profit to Put Option at Expiration

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Put Writer Payoff and Profit

ST <X ST > X

Exercise Cost -(X - ST) 0

Profit -(X - ST) + Pt Pt

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Puts: Another Zero Sum Game

Put Option ST < X If ST > X

Decision Exercise No Exercise

Option Payoff (holder)

X – ST 0

Option Profit (holder)

(X-ST) - P -P

Option Payoff (writer)

- (X-ST) 0

Option Profit (writer)

+P - (X-ST) +P

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Profit and Loss on a Put Consider a February 2013 put on IBM with an

exercise price of $195, selling on January 18, 2013 for $5.00.

Option holder can sell a share of IBM for $195 at any time until February 15.

If IBM sells for $196, what is the put worth?

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Profit and Loss on a Put

Suppose IBM’s price at expiration is $188. What is the value of the option?

Put value = Exercise price- Stock price Will the option be exercised? What is the Profit/Loss on this investment?

Profit = Final value – Original investment What is the HPR?

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Put Option Problem

At time=0 you buy a put option on ITT stock for $2.00. The option gives you the right to sell 100 shares of ITT stock at time=T at $50

What is the payoff to you if ST = $55?

What is the payoff to the put seller if ST = $55?

What is the payoff to you if ST = $45?

What is the payoff to the put seller if ST = $45?

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Option versus Stock Investments

Could a call option strategy be preferable to a direct stock purchase?

Suppose you think a stock, currently selling for $100, will appreciate.

A 6-month call costs $10 (contract size is 100 shares).

You have $10,000 to invest.

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Option versus Stock Investment

Investment Strategy Investment

Equity only Buy stock @ $100 (100 shares) $10,000

Options only Buy calls @ $10 (1,000 options) $10,000

Options + Buy calls @ $10 (100 options) $1,000T-Bills Buy T-bills @ 3% Yield $9,000

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Strategy Payoffs

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Option Versus Stock Investment

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Strategy Conclusions The all-option portfolio, B, responds more than

proportionately to changes in stock value; it is levered.

Portfolio C, T-bills plus calls, shows the insurance value of options.C ‘s T-bill position cannot be worth less than

$9270.Some return potential is sacrificed to limit

downside risk.

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Combining Options Puts and calls can serve as the building blocks

for more complex option contractsCan be used to manage risk

This is financial engineeringAllows you to tailor the risk-return profiles to meet

your client’s desires Ex: Protective Puts: Underlying asset and put

are combined to guarantee a minimum valuationPut is insurance against stock price declines.

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Protective Put at Expiration

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Covered Calls

Purchase stock and write calls against it. Call writer gives up any stock value above X

in return for the initial premium. If you planned to sell the stock when the price

rises above X anyway, the call imposes “sell discipline.”

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Covered Call Position at Expiration

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Straddle

The straddle is a bet on volatility. Long straddle: Buy call and put with same

exercise price and maturity.To make a profit, the change in stock price must

exceed the cost of both options.You make money on a large price shift in either

direction The writer of a straddle is betting the stock

price will not change much.

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Straddle Position at Expiration

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Spread

A spread is a combination of two or more calls (or two or more puts) on the same stock with differing exercise prices or times to maturity.

Some options are bought, whereas others are sold (written)

A bullish spread is a way to profit from moderate stock price increases EX. Buy a call with a strike of $20 and sell a call

with a strike price of $25

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Bullish Spread Position at Expiration

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Put Call Parity This is the relation between a put and call with

the same exercise price (E) and maturity It comes from replicating portfolios:

The payoffs from buying a call and selling a put is the same as the payoffs from buying the stock and borrowing the PV of the exercise price

P –C = PV (X) - St

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Payoff-Pattern of Long Call–Short Put Position

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Portfolios Portfolio 1: Buy a call and Write a put Portfolio 2: Buy the stock but Borrow PV (X)

Levered Equity positions If the payoffs are the same the price must be

the same

-C+P = -S0 + Xe-rT

C-P = S0 – Xe-rT

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Proof by Counter Example Assume that:

Stock Price = 110 Call Price = 14Put Price = 5 Risk Free = 5%Maturity = 6 months Strike Price = 105

Portfolio 1 costs-14 + 5 = -9

Portfolio 2 costs:-110 + 105e-5 = -7.59

Two different costs for the sample payoffs → ABRITRAGE

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Arbitrage Strategy Payoff

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Put Call Parity Example What is the value of a put with an exercise

price of $51, if the stock is currently trading at $49. The price of the corresponding call option is $4.65. According to put-call parity, if the effective annual risk-free rate of interest is 4% and there are three months until expiration, what should be the value of the put?FYI: It doesn’t matter if compounding is monthly

or quarterly