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Options and Futures

Finansiell ekonomi 723g28Linköpings University

What is a Derivative?

• A derivative is an instrument whose value depends on, or is derived from, the value of another asset.

• Examples: futures, forwards, swaps, options, exotics…

Obs: you may jump to slide #21 to start direct with options.

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How Derivatives are Used• To hedge risks• To speculate (take a view on the future

direction of the market)• To lock in an arbitrage profit• To change the nature of a liability• To change the nature of an investment

without incurring the costs of selling one portfolio and buying another

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Options vs. Futures/Forwards

• A futures/forward contract gives the holder the obligation to buy or sell at a certain price at a certain date in the future

• An option gives the holder the right, but not the obligation to buy or sell at a certain price at a certain date in the future

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Foreign Exchange Quotes for GBP, (£) May 24, 2010

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Bid OfferSpot 1.4407 1.4411

1-month forward 1.4408 1.4413

3-month forward 1.4410 1.4415

6-month forward 1.4416 1.4422

The forward price may be different for contracts of different maturities (as shown by the table)

Long position and short position

• The party that has agreed to buy has a long position

• The party that has agreed to sell has a short position

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Example

• On May 24, 2010 the treasurer of a corporation enters into a long forward contract to buy £1 million in six months at an exchange rate of 1.4422

• This obligates the corporation to pay $1,442,200 for £1 million on November 24, 2010

• What are the possible outcomes?

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Svenska termer

Forwards och Terminer• Spotkontrakt: en överenskommelse mellan tvåparter att utbyta något idag för ett specificeratpris, spotpriset. à vista marknad.• Terminskontrakt: en överenskommelse(skyldighet) mellan två parter att utbyta någotför ett specificerat pris, terminspriset, vid enspecifik framtida tidpunkt, lösendagen.

Profit from a Long Forward Position (K= delivery price=forward price at the time contract is entered into)

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Profit

Price of Underlying at Maturity, ST

K

Payoff diagram

Profit from a Short Forward Position (K= delivery price=forward price at the time contract is entered into)

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Profit

Price of Underlying at Maturity, ST

K

Futures Contracts • Agreement to buy or sell an asset for a certain price

at a certain time• Similar to forward contract• a forward contract is traded over the counter (OTC) (Skräddarsydd)• a futures contract is standardized and traded on an

exchange. CME Group NYSE Euronext, BM&F (Sao Paulo, Brazil) TIFFE (Tokyo)

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Key Points About Futures

• They are settled daily• Closing out a futures position

involves entering into an offsetting trade

• Most contracts are closed out before maturity

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Margins

• A margin is cash or marketable securities deposited by an investor with his or her broker

• The balance in the margin account is adjusted to reflect daily settlement

• Margins minimize the possibility of a loss through a default on a contract

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Pricing of forward

• Guld (commodities): F = (1 + rf + s) · S0

• Finansiella tillgångar: F = (1 + rf) · S0

S0 is the spot price.S is the storage costrf is risk free interest rateF is the forward price

Examples of Futures Contracts

Agreement to:– Buy 100 oz. of gold @ US$1400/oz. in

December – Sell £62,500 @ 1.4500 US$/£ in March– Sell 1,000 bbl. of oil @ US$90/bbl. in

April

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Oz: ounceBbl: barrel

Example : An Arbitrage Opportunity?

Suppose that:The spot price of gold is US$1,400The 1-year forward price of gold is US$1,500The 1-year US$ interest rate is 5% per annum

Q: What should be the 1-year forward price? Is there an arbitrage opportunity?

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The Forward Price of Gold If the spot price of gold is S and the

forward price for a contract deliverable in T years is F, then

F = S (1+r )T

where r is the 1-year (domestic currency) risk-free rate of interest.In our examples, S = 1400, T = 1, and r =0.05 so that

F = 1400(1+0.05) = 1470

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Hedging Examples1. An investor owns 1,000 Microsoft

shares currently worth $28 per share. A two-month put with a strike price of $27.50 costs $1. The investor decides to hedge by buying 10 contracts

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Value of Microsoft Shares with and without Hedging

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20 22 24 26 28 30 32 34 36 3820,000

25,000

30,000

35,000

40,000

No Hedging

Hedging

Stock Price ($)

Value of Holding ($)

Some Terminology

• Open interest: the total number of contracts outstanding – equal to number of long positions or number of

short positions• Settlement price: the price just before the

final bell each day – used for the daily settlement process

• Trading Volume : the number of trades in one day

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Forward Contracts vs Futures Contracts

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Contract usually closed out

Private contract between 2 parties Exchange traded

Non-standard contract Standard contract

Usually 1 specified delivery date Range of delivery dates

Settled at end of contract Settled daily

Delivery or final cashsettlement usually occurs prior to maturity

FORWARDS FUTURES

Some credit risk Virtually no credit risk

Options

The right but not the obligation…

Options• A call option is an option to buy a certain

asset by a certain date for a certain price (the strike price)

• A put option is an option to sell a certain asset by a certain date for a certain price (the strike price)

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Option Obligations: the writer of the option

assetbuy toObligationasset sell Right tooptionPut asset sell toObligationassetbuy Right tooption Call

WriterBuyer

American vs. European Options

• An American option can be exercised at any time during its life

• A European option can be exercised only at maturity

• The time value will be lost when you exercise prematurely.

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Option Value: ExampleOption values given an exercise price of $720

00060$120ValuePut 1206000$0Value Call840780720660$600PriceStock

What are the payoff limits for call option buyers? Sellers?What are the payoff limits for put option buyers? Sellers?

Call Option Value

Call option value (buyer) given a $720 exercise price.

Share Price

Call

optio

n va

lue

720 840

$120

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Call Option Profit

$20 call option (buyer) given a $720 exercise price

Share Price

Call

optio

n va

lue

720 840

$100

Profit (buyer): Current Price - Exercise Price - Cost of Call

Profit = ($840 $720) $20 $100

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Call Option ValueCall option payoff (seller) given a $720 exercise price.

Share Price

Call

optio

n $

payo

ff 720 840

$-120

Call Option Profit

$20 call option (seller) given a $720 exercise price:

Share Price

Call

optio

n $

payo

ff 720 840

$-120

Profit (Seller): Exercise Price - Current Price + Cost of Call

$-100

Profit = $720 $840 $20 $100

Call Option: Example

How much must the stock be worth at expiration in order for a call holder to break even if the exercise price is $50 and the call premium

was $4?

Put Option ValuePut option value (buyer) given a $720 exercise price:

Share Price

Put o

ption

val

ue

600 720

$120

Put Option Profit

$30 put option (buyer) given a $720 exercise price:

Share Price

Put o

ption

val

ue

600 720

$90

Profit (buyer): Exercise Price - Current Price - Cost of Put

Profit = $720 $600 $30 $90

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Put Option Value

Put option payoff (seller) given a $720 exercise price.

Share Price

Put o

ption

$ p

ayoff

600 720

-$120

Put Option Profit

$30 put option (seller) given a $720 exercise price.

Share Price

Put o

ption

$ p

ayoff

600 720

-$90

Profit (Seller): Current Price - Exercise Price + Cost of Put

Profit = $600 $720 $30 $90

Put Options: Example

What is your return on exercising a put option which was purchased for $10 with an exercise price of $85? The stock price at expiration is

$81.

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Options ValueStock Price

Upper Limit

Lower Limit

(Stock price - exercise price) or 0which ever is higher

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Option Value

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Option Value

• Point A -When the stock is worthless, the option is worthless.

• Point B -When the stock price becomes very high, the option price approaches the stock price less the present value of the exercise price.

• Point C -The option price always exceeds its minimum value (except at maturity or when stock price is zero).

• The value of an option increases with both the variability of the share price and the time to expiration.

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Option ValueComponents of the Option Price1 - Underlying stock price2 - Strike or Exercise price3 - Volatility of the stock returns (standard deviation of annual

returns)4 - Time to option expiration5 - Time value of money (discount rate)

Call Option Value

Put-Call Parity: No Dividends

• Consider the following 2 portfolios:– Portfolio A: call option on a stock + zero-coupon

bond (or a deposit) that pays K at time T– Portfolio B: Put option on the stock + the stock

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Values of Portfolios are the same at expiration (förfalldag)

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ST > K ST < K

Portfolio A Call option ST − K 0

Zero-coupon bond K K

Total ST K

Portfolio B Put Option 0 K− ST

Share ST ST

Total ST K

The Put-Call Parity Result

• Both are worth max(ST , K ) at the maturity of the options

• They must therefore be worth the same today. This means that

c + Ke -rT = p + S0

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Suppose that

• What are the put option price? c + Ke -rT = p + S0

p = c-S0 +Ke -rT

=3-31+30*EXP(-0,1*0,25) = 1,259

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Ex: put-call parity

c= 3 S0= 31 T = 0.25 r = 10% K =30

Bounds for European and American Put Options (No Dividends)

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Synthetic options

Two or more options combines together creates exotic options

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Option Value: profit diagram for a straddle

Straddle - Long call and long put - Strategy for profiting from high volatility

Share Price

Positi

on V

alue

Straddle

Long put

Long call

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Option ValueStraddle - Long call and long put - Strategy for profiting from high volatility

Share Price

Positi

on V

alue

Straddle

An investor may take a long straddle position if he thinks the market is highly volatile, but does not know in which direction it is going to move.

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Exotic options: a butterfly option

• A butterfly

A long butterfly position will make profit if the future volatility is lower than the implied volatility. The spread is created by buying a call with a relatively low strike (x1), buying a call with a relatively high strike (x3), and shorting two calls with a strike in between (x2).

x2x3

x1

Long Call Profit from buying one European call option: option

price = $5, strike price = $100, option life = 2 months

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30

20

10

0-5

70 80 90 100

110 120 130

Profit ($)

Terminalstock price ($)

Short Call Profit from writing one European call option: option

price = $5, strike price = $100

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-30

-20

-10

05

70 80 90 100

110 120 130

Profit ($)

Terminalstock price ($)

Long Put Profit from buying a European put option: option

price = $7, strike price = $70

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30

20

10

0

-770605040 80 90 100

Profit ($)

Terminalstock price ($)

Short Put Profit from writing a European put option: option

price = $7, strike price = $70

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-30

-20

-10

70

70

605040

80 90 100

Profit ($)Terminal

stock price ($)

Payoffs from OptionsWhat is the Option Position in Each Case?

K = Strike price, ST = Price of asset at maturity

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Payoff Payoff

ST STKK

PayoffPayoff

ST STK

K

The Black-Scholes-Merton Formulas

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TdT

TrKSd

TTrKSd

dNSdNeKp

dNeKdNScrT

rT

10

2

01

102

210

)2/2()/ln(

)2/2()/ln(

)()(

)()(

where

Real options • With the limited liability of the modern corporations,

the shareholders´ equity can be regarded as a real option on the assets of the firm.

• The shareholder value of equity value is

max(VT −D, 0)where VT is the value of the firm and D is the debt repayment required.

Thus the company can be considered as a call option

on the firm value V at the strike price of D.57

Options on Real Assets

Real Options - Options embedded in real assets

Option to ExpandOption to Abandon

Options on Financial Assets

Executive Stock Options

Warrants

Convertible Bonds

Callable Bonds