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Options and Futures:Risk Management

730g81Linköpings 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…

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Size of OTC and Exchange-Traded Markets

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Source: Bank for International Settlements. Chart shows total principal amounts for OTC market and value of underlying assets for exchange market

The Lehman BankruptcyLehman’s filed for bankruptcy on September 15, 2008. one of the biggest bankruptcy in US history• Lehman was an active participant in the OTC derivatives

markets and got into financial difficulties because it took high risks and found it was unable to roll over its short term funding

• It had hundreds of thousands of transactions outstanding with about 8,000 counterparties

• Unwinding these transactions has been challenging for both the Lehman liquidators and their counterparties

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

direction of the market)• To lock in an arbitrage profit (forwards)• 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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Foreign Exchange Quotes for GBP, ($/£) May 24, 2010

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Bid Offer

Spot 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 $1442200 for £1 million on November 24, 2010

• What are the possible outcomes?

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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

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• Whereas a forward contract is traded over the

counter, a futures contract is traded on an exchange.

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

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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

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Forward price

• If the spot price of a commodity is S and the forward price for a contract deliverable in T years is F, then

F = S (1+r )T

• Essentially a forward is a implicit loan.

Gold price: An Arbitrage Opportunity?

Suppose you observe: the spot price of gold is US$1400The 1-year forward price of gold is

US$1500The 1-year US$ interest rate is 5% per

annum, 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. S = 1400, T = 1, and r =0.05 so that

F = 1400(1+0.05) = 1470$

Yes, there is an arbitrage opportunity. Borrow at 5%, buy gold now and cover it with the forward

contract. You net 1500-1470=30 $ per contract!

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Options• A call option is the right but not the

obligation to buy a certain asset by a certain date for a certain price (the strike price)

• A put option is the right but not the obligation to sell a certain asset by a certain date for a certain price (the strike price)

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The writer of the option has obligations to buy and sell

Buyer WriterCall option Right to buy asset Obligation to sell assetPut option Right to sell asset Obligation to buy asset

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

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Google Call Option Prices(June 15, 2010; Stock Price is bid 497.07, offer 497.25)

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Strike Price

Jul 2010 Bid

Jul 2010 Offer

Sep 2010 Bid

Sep 2010 Offer

Dec 2010 Bid

Dec 2010Offer

460 43.30 44.00 51.90 53.90 63.40 64.80

480 28.60 29.00 39.70 40.40 50.80 52.30

500 17.00 17.40 28.30 29.30 40.60 41.30

520 9.00 9.30 19.10 19.90 31.40 32.00

540 4.20 4.40 12.70 13.00 23.10 24.00

560 1.75 2.10 7.40 8.40 16.80 17.70

Source: CBOE

Google Put Option Prices (June 15, 2010; Stock Price is bid 497.07, offer 497.25);

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Strike Price

Jul 2010 Bid

Jul 2010 Offer

Sep 2010 Bid

Sep 2010 Offer

Dec 2010 Bid

Dec 2010Offer

460 6.30 6.60 15.70 16.20 26.00 27.30

480 11.30 11.70 22.20 22.70 33.30 35.00

500 19.50 20.00 30.90 32.60 42.20 43.00

520 31.60 33.90 41.80 43.60 52.80 54.50

540 46.30 47.20 54.90 56.10 64.90 66.20

560 64.30 66.70 70.00 71.30 78.60 80.00

Source: CBOE

Options vs. Futures/Forwards

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

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

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Hedging Examples1. A German company will pay £10 million

for imports from Britain in 3 months and decides to hedge using a long position in a forward contract

2. 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

Stock Price ($)

Value of Holding ($)

The investor will do better when the price goes below 26,5$.

Example: Speculation vs. hedgingAn investor with $2,000 to invest feels that a stock price will increase over the next 2 months. The current stock price is $20 and the price of a 2-month call option with a strike of 22.50 is $1• What are the alternative strategies?

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Speculation 1. He buys 100 shares of the stock. 100*20$=20002. He buys 2000 call option 2000*1=2000$. If the

market price increases to more than 22,5 during the 2 months, the call option will have a higher payoff than the plain stocks.

3. Suppose the stock price is 25 at the expiration date: the call option has gained (25-22,5)*2000=5000$ or a 150% profit, while the first case gives 500/2000=25%

4. Suppose the market price is lower than 22,5$, the investor loses all, that is, -100%.

Hedge Funds • Hedge funds are not subject to the same rules as

mutual funds and cannot offer their securities publicly.

• Mutual funds must – disclose investment policies, – makes shares redeemable at any time,– limit use of leverageHedge funds are not subject to these constraints.

• Hedge funds use complex trading strategies are big users of derivatives for hedging, speculation and arbitrage

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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

• Margin call: when the margin is below the required minimum, it is subject to margin call. The client is obliged to increase the margin to the minimun.

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Example of a Futures Trade

• An investor takes a long position in 2 December gold futures contracts on

June 5• contract size is 100 oz.• futures price is US$1250• initial margin requirement is

US$6,000/contract (US$12,000 in total)• maintenance margin is US$4,500/contract

(US$9,000 in total)28

Options 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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Key Points About Futures• They are settled daily. (marked to the

market)• Closing out a futures position involves

entering into an offsetting trade• Most contracts are closed out before

maturity

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Delivery• If a futures contract is not closed out before maturity,

it is usually settled by delivering the assets underlying the contract. When there are alternatives about what is delivered, where it is delivered, and when it is delivered, the party with the short position chooses.

• A few contracts (for example, those on stock indices and Eurodollars) are settled in cash.

• When there is cash settlement, contracts are traded until a predetermined time. All are then declared to be closed out.

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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

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Option Value• The value of an option at expiration is a function of the

stock price and the exercise price (S-X for call and X-S for put).

Example Option values (exercise price = $720)

Stock Price $600 660 720 780 840Call Value 0 0 0 60 120Put Value 120 60 0 0 0

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Option ValueCall option value (graphic) on Google Stock on option expiration date, exercise price=$720.

Share Price

Call

o

ption

v

alue

720 840

$120

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Option ValuePut option value (graphic) on Google stock on option expiration date. ( exercise price=$720 )

Share Price

Put o

ption

val

ue

600 720

$120

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Option ValueCall option payoff (to the writer) on Google stock

($720 exercise price)

Share Price

Call

optio

n $

payo

ff

720

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Option ValuePut option payoff (to the writer ) on Google Stock

exercise price=$720 .

Share Price

Put

op

tion

$

pay

off

720

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Option Value

Protective Put - Long stock and long put

Share Price

Positi

on V

alue

Protective Put

Long Put

Long Stock

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Option Value

Protective Put - Long stock and long put

Share Price

Positi

on V

alue Protective Put

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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 putLong 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.

Combinations of Options (cont'd)

• Strangle– A portfolio that is long a call option and a put

option on the same stock with the same exercise date but the strike price on the call exceeds the strike price on the put

Textbook Example 20.5

Textbook Example 20.5

Combinations of Options

• Butterfly Spread

A portfolio that is long two call options with differing strike prices, and short two call options with a strike price equal to the average strike price of the first two calls

• While a straddle strategy makes money when the stock and strike prices are far apart, a butterfly spread makes money when the stock and strike prices are close.

Figure 20.6 Butterfly Spread

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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

Figure 20.7 Portfolio Insurance

The plots show two different ways to insure against the possibility of the price of Amazon stock falling below $45. The orange line in (a) indicates the value on the expiration date of a position that is long one share of Amazon stock and one European put option with a strike of $45 (the blue dashed line is the payoff of the stock itself). The orange line in (b) shows the value on the expiration date of a position that is long a zero-coupon riskfree bond with a face value of $45 and a European call option on Amazon with a strike price of $45 (the green dashed line is the bond payoff).

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Option ValueCall buyer profit diagram on Google stock–strike price = $720 and option price= $80.50

Share Price

Positi

on V

alue

Long call

720 800.50

-80.50

Break even

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Option ValuePut seller profit diagram withstrike price=$720 and option price of $71.20

Share Price

Positi

on V

alue Short put

648.80 720

+71.20

Break even

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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 Value

Components 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)

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

Market Makers

• Most exchanges use market makers to facilitate options trading

• A market maker quotes both bid and ask prices when requested.

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Margins• Margins are required when options are written.• When a naked option is written the margin is the greater

of:– A total of 100% of the proceeds of the sale plus 20% of

the underlying share price less the amount (if any) by which the option is out of the money

– A total of 100% of the proceeds of the sale plus 10% of the underlying share price (for call) or exercise price (for put)

• For other trading strategies there are special rules

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Put-Call Parity: No Dividends

• Consider the following 2 portfolios:Portfolio A: call option on a stock + a deposit that pays K at time TPortfolio B: Put option on the stock + the stock

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Values of Portfolios at the expiration

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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 at t=0. 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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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

Using Equity Prices: Merton’s Model

• Merton’s model regards the equity as an option on the assets of the firm

• In a simple situation the equity value ismax(VT −D, 0)

where VT is the value of the firm and D is the debt repayment required

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Risk Management

• Understand why companies hedge to reduce risk. Bigger risks can have potential disastrous effect on firms.

• Use options, futures, and forward contracts to devise simple hedging strategies: Managing currency exposure

• Explain how companies can use swaps to change the risk of securities that they have issued. Interest rate swaps, exchange rate swaps, etc

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Illustration of an Interest Rate SwapWe consider a floating and a fixed interest rate swap1. Suppose Company A can borrow at a floating rate of

ERIBOR plus 1% or at a fixed rate of 10%2. Company B can borrow at a floating rate of ERIBOR

plus 2% or at a fixed rate of 9,5%3. Company A desires a fixed rate, company B desires a

floating rate. 4. Through a swap dealer company A and B can get a

better rate than they could independently.ERIBOR: the Euro Interbank Offered Rate. The EURIBOR rates are based on the average interest rates at which a panel of more than 50 European banks borrow funds from one another. There are different maturities, ranging from one week to one year.

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An illustration of an Interest Rate Swap

Here the prime is ERIBOR: the Euro Interbank Offered Rate

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Reasons for Interest rate SWAP

• Note the interest rate SWAP has fulfilled two companies´ needs to hedge their interest rate exposure on their balance sheet. Other reasons for the swap can be:

• To reduce the cost of borrowing.• The company has a vision of where the

interest rate might go in the short run and tries to take advantage of it.

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Exercise You observe the spot exchange rate is 2,45$/£, the 3 Month

forward rate is 2,60$/£. 3 Month Call option premium for strike price 2,6 is 2 cents.

• A US company will pay £10 million for imports from Britain in 3 months and decides to hedge using a long position in a forward contract, he fixed his cost of £ 10 million with the forward exchange rate of 2,60$/£, total cost $26 M.

• If the 3-M call option premium is 0,02 dollars per call option,Then, the cost of buying call option on British Pounds is0,02*10M=0,2M. In this case, you have fixed your cost of £10M to

26+0,2=$26,2M, but with a freedom. If the market exchange rate is less than your strike price, you can take fully advantage of the lower market price. So you have effectively capped your payment.

E$/£2,6

Total Cost $M

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Quick Quiz1. Why is risk management more important now than it was in the 1960s?

Interest rates, energy prices, and exchange rates are more volatile today than in the 1960s due to structural changes in their respective markets.

2. What is a short-run exposure? A long-run exposure?A short-run exposure involves risk due to temporary price changes; a long-run exposure arises from permanent changes in economic fundamentals.

3. How does a forward contract differ from a futures contract?A forward contract is not standardized, not generally traded on an organized exchange, and doesn’t involve margin.A futures contract is highly standardized, traded on an organized exchange, and requires margin.

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The payoff to a long call and a short put is the same as owning the underlying assets by borrowed funds

• Ex: Suppose a financial manager buys call options on 50,000 barrels of oil with an exercise price of $30 per barrel. She simultaneously sells a put option on 50,000 barrels of oil with the same exercise price of $30 per barrel. Consider her gains and losses if oil prices are $25, $28, $30, $32, and $34. What do you notice about the payoff profile?

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Solution to ProblemSolution:

The call options give the manager the right to purchase oil futures contracts at a futures price of $30 per barrel. The manager will exercise the option if the price rises above $30.

• Writing put options obligates the manager to buy oil futures contracts at a futures price of $30 per barrel. The put holder will exercise the option if the price falls below $30. The payoffs are:

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Solution to Problem Oil price: $25 $28 $30 $32 $34Value of call option position: 0 0 0 2 4Value of put option position: -5 -2 0 0 0Total value -$5 -$2 $0 $2 $4

The payoff profile is identical to that of a forward contract with a $30 strike price. Or purchased the assets at 30$ by borrowed funds