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Econ 455/655 Options and Futures I

Lecture Notes

Professor Man-lui Lau

ECON 455/655 Options and Futures I Prof. Man-lui Lau 2

I Forward and Future Contracts

Definition: A DERIVATIVE (or derivative security) is a financial instrument whose value depends

on the values of other, more basic underlying variables.

Definition: A FORWARD CONTRACT is an agreement to buy or sell an asset at a certain future

time for a certain price.

The party who assumes a LONG POSITION agrees to buy the underlying asset on a certain specified

future date for a certain specified price.

The other party who assumes a SHORT POSITION agrees to sell the underlying asset on a certain

specified future date for a certain specified price.

Example:

On January 18, 2013, the forward prices for are as follows:

Spot 1.5864

1-month forward 1.5862 delivery price ( K ) 3-month forward 1.5858

6-month forward 1.5852

If a person takes a long position of the 1-month forward contract, then she agrees to buy the British

@ $1.5862 one month from January 18, 2013.

If a person takes a short position of the 6-month forward contract, then she agrees to sell the British

@ $1.5852 six months from January 18, 2013.

Example:

On January 18, 2013, the forward prices for are as follows:

Spot 0.01110

1-month forward 0.01110 delivery price ( K ) 3-month forward 0.01111

6-month forward 0.01112

If a person takes a long position of the 3-month forward contract, then she agrees to buy the @

$0.01110 three month from January 18, 2013.

If a person takes a short position of 6-month forward contract, then she agrees to sell the @

$0.01112 six month from January 18, 2013.


Note: A forward contract is settled at maturity.

The holder of the short position delivers the asset to the holder of the long position in return for

a cash amount equal to the delivery price.

Payoff Payoff K: delivery price ST: price of asset at maturity + +

ST ST K K

LONG POSITION SHORT POSITION

TPayoff S K TPayoff K S

Example:

Suppose that an investor entered into a long forward contract on January 18, 2013 to buy 1 million in

1 month @ USD 1.5862/.

If the spot exchange rate rises to 1.6500 at the end of one month, then

($1.6500 $1.5862) 1,000,000 $63,800

sell buy

If the spot exchange rate falls to 1.4000 at the end of one month, then

($1.4000 $1.5862) 1,000,000 $186,200

sell buy

Example:

Suppose that an investor entered into a short forward contract on January 18, 2013 to sell 1 million

in 3 month @ USD $1.5858/.

If the spot exchange rate rises to 1.6500 at the end of three months, then

($1.5858 $1.6500) 1,000,000 $64,200

sell buy

If the spot exchange rate falls to 1.4000 at the end of three months, then

($1.5858 $1.4000) 1,000,000 $185,800

sell buy


Definition: A Futures Contract is a forward contract trading in an exchange. The futures contract

has a fixed contract size with a fixed maturity date.

Example:

Delivery Month Commodity Exchange Contract Size

June Cattle CME 40000 pounds

September S&P Composite Index CME $250 Index

December IMM 12,500,000

Note:

1) In the case of commodities such as cotton, wheat, orange juice, the quality of the commodity is

specified.

2) The exchanges set the place and the time for delivery.

3) The last trading day of a contract is the second Friday of the delivery month.

Specification of the Futures Contract 1. The asset

2. Contract size

3. Delivery arrangement

4. Delivery months

5. Price quotes

6. Daily price movement limits (limit up and limit down)

7. Position limits

Reasons for trading futures:

1. Speculation

a) Suppose an investor believes that the price of gold will go up in the future, instead of buying

spot gold (which is very inconvenient), she can take a long position of gold futures.

1/18/2013

Month Settlement

Feb 13 1687.00

Mar 13 1688.00

Apr 13 1689.20

Jun 13 1691.30

Aug 13 1693.10

For example, on Jan 18, 2013, she buys a August Gold futures @$1693.10. [i.e. she promises

to buy 100 oz of gold @$1693.10 at closing on the third last business day of August] Once she

has a long position, she can make money in the following two ways:

i) She can wait until Aug 28, 2013 (third last business day of August). Suppose at the closing of that day, the spot price of gold is $1800, then she can buy the gold at the

contract price $1693.10 and immediately sell the gold in the spot market for $1800.

She will make ($1800 $1693.10) 100 $10,690 .


ii) Suppose on Jan 31, 2013, the August gold futures is trading @1750, then she can sell a

contract (and close her position) and make ($1750 $1693.10) 100 $5,690 .

Of course if she is wrong, the following may happen.

i) Suppose at the closing of Aug 28, 2013, the price of the spot gold is $1000, then her

profit is ($1000 $1693.10) 100 $69,310 (loss) .

ii) Suppose on Jan 31, 2013, the August gold futures is trading @1500, if she sells a contract (and closes her position), then her profit is

($1500 $1693.10) 100 $19,310 (loss) .

b) Suppose an investor believes that the is going to depreciate. Instead of shorting in the spot

market, she can take a short position on the futures.

1/18/2013

Month Settlement

Mar 13 0.011111

Jun 13 0.011121

Sep 13 0.011133

Dec 13 0.011147

Mar 14 0.011165

Jun 14 0.011185

For example, on Jan 18, 2013, she sells a June futures @0.011121. [i.e. she promise to sell

12.5 million @ 0.011121 per at closing on the second business day immediately preceding

the third Wednesday of June]

Once she has a short position, she can make money in the following two ways:

i) She can wait until June 17, 2013. Suppose at the closing of that day, the spot price of

contract is $0.01. She can buy the in the spot market @$0.01 and then sell the at

the futures contract price of $0.011121.

She will make ($0.011121 $0.01) 12,500,000 $14,012.50 .

ii) Suppose on January 31, 2013, the June futures is trading @0.009, then she can buy a

contract (and close her position) and make

($0.011121 $0.009) 12,500,000 $26,512.50

Of course if she is wrong, the following may happen.

i) Suppose at the closing of June 17, 2013, the spot price of is $0.012, then her profit is

($0.011121 $0.012) 12,500,000 $10,987.50 (loss) .


ii) Suppose on January 31, 2013, the June contract futures is trading @$0.0125, if she

buys a contract (and closes her position), then her profit is

($0.011121 $0.0125) 12,500,000 $17,237.50 (lo ) ss .

2. Hedging

a) Short Hedge: A company that knows it is due to sell an asset at a particular time in the future

can hedge by taking a short position in the futures market.

i) If a farmer knows that he will have wheat for sale in September, he can take a short

position in the futures market. Once he takes the short position, he no longer has to

worry about the price that he will get for the wheat in September.

ii) Suppose a company is going to receive 1 million in September. In order to avoid any

uncertainty about the value of by that time, he can short 16 September futures

(16 62500 1,000,000) @1.5827

Current September Futures price $1.5827

Not Hedged Hedged

September Spot Price Will receive Will receive

$1.5000 $1,500,000 $1,582,700

$1.5500 $1,550,000 $1,582,700

$1.6000 $1,600,000 $1,582,700

$1.6500 $1,650,000 $1,582,700

$1.7000 $1,700,000 $1,582,700

b) Long Hedge: A company that knows it is due to buy an asset at a particular time in the future

can hedge by taking a long position in the futures market.

i) In order to eliminate the uncertainty in the price of corn, a cereal company can buy corn

in the futures market.

ii) Suppose a company is going to pay 1 million Can$ in September. In order to avoid any

uncertainty about the value of Can$ by that time, he can long 10 September Can$

futures (101000001,000,000) @1.0019

Current September Futures price $1.0019

Not Hedged Hedged

September Spot Price Will pay Will pay

$0.9800 $980,000 $1,000,190

$0.9900 $990,000 $1,000,190

$1.0000 $1,000,000 $1,000,190

$1.0100 $1,010,000 $1,000,190

$1.0200 $1,020,000 $1,000,190

$1.0300 $1,030,000 $1,000,190


iii) If a mutual fund manager knows that there will be inflow of cash into her fund in June,

she can a long position in June Stock Index Futures to fix the purchasing price at the current level.

Operation of Margins in Trading Futures (marking to the market)

Example: buy 2 December Gold futures @$400/ounce. (100 ounces per contract)

Initial margin requirement: $2000/contract (non-current)

Maintenance margin requirement: $1500/contract (non-current)

Day Future price

($)

Daily gain

(loss) ($)

Cumulative gain

(loss) ($)

Margin account

balance ($)

Margin call ($)

400.00 4000 2 2000

6/3 397.00 (600) (600) 34003000 6/4 396.10 (180) (780)(600) (180) 32203000 6/5 398.20 420 (360) (780) 420 36403000 6/6 397.10 (220) (580) 34203000 6/7 396.70 (80) (660) 33403000 6/10 393.30 (680) (1340) 2660 3000 1340 4000 2660 6/11 396.30 600 (740) 4600

6/12 400.30 800 60 5400

6/13 405.30 1000 1060 6400

6/14 411.30 1200 2260 7600

Example: sell 1 December Yen futures @$0.009613 ( 12,500,000 per contract)

Initial margin requirement: $2025/contract

Maintenance margin requirement: $1500/contract

Day Future

price ($)

Daily gain

(loss) ($)

Cumulative gain (loss) ($) Margin account

balance ($)

Margin call ($)

0.009613 2025

6/1 0.009600 162.50 162.50 2187.5

6/2 0.009650 (625.00) (462.50)162.50 (625.00) 1562.501500 6/3 0.009700 (625.00) (1087.50)(462.50) (625.00) 937.50 1500 1087.50 2025 937.5 6/4 0.009700 0 (1087.50) 2025

6/5 0.009600 1250 162.50 3275

6/8 0.009400 2500 2662.50 5775


Example: sell 1 December S&P500 Index futures @1376.40 ($250 Index)

Initial margin requirement: $19687/contract

Maintenance margin requirement: $15750/contract

Day Future price

($)

Daily gain (loss)

($)

Cumulative gain

(loss) ($)

Margin account

balance ($)

Margin call ($)

1376.40 19687

6/1 1350.00 6600 6600 26287

6/2 1340.00 2500 9100 28787

6/3 1350.00 (2500) 6600 26287

6/4 1390.00 (10000) (3400) 1628715750 6/5 1360.00 7500 4100 23787

6/8 1390.00 (7500) (3400) 1628715750 6/9 1400.00 (2500) (5900) 13787 15750 5900 19687 13787 6/10 1405.00 (1250) (7150) 1843715750 6/11 1350.00 13750 6600 32187

6/12 1300.00 12500 19100 44687

Reasons why hedging using futures contracts works less than perfectly in practice

1. The asset whose price is to be hedged may not be exactly the same as the asset underlying the

futures contract.

2. The hedger may be uncertain as to the exact date when the asset will be bought or sold.

3. The hedge may require the futures contract to be closed out well before its expiration date.

These problems give rise to BASIS RISK.


Optimal Hedging Ratio

: change in spot price, , during a period of time equal to the life of the hedgeS S

: change in futures price, , during a period of time equal to the life of the hedgeF F

: standard deviation of S S

: standard deviation of F F

: coefficient of correlation between and

( , ) ( , ) [( ( ))( ( )]i i

S F

S F

Cov S FCov S F E S E S F E F

: hedge ratioh

When the hedger has a long position in the asset and a short position in the futures, the change in the

value of the hedgers position during the life of the hedge is S h F

When the hedger has a short position in the asset and a long position in the futures, the change in the

value of the hedgers position during the life of the hedge is h F S .

2 2 2( ) ( ) ( ) 2 ( , ) 2S F S FVar S h F Var S Var h F Cov S h F h h

Problem: 2 2 2min 2S F S Fh

v h h

FOC: 22 2 0 * SF S F

F

dvh h

dh

Optimal hedge ratio: * S

F

h

Note:

If 1 and * 1.0F S h

In this case, the futures price mirrors the spot price perfectly.

If 1 and 2 * 0.5F S h

The futures price always changes by twice as much as the spot price.


Example: (Optimal hedge ratio)

A company knows that it will buy 1 million gallons of jet fuels in 3 months. The standard deviation of

the change in the price per gallon of jet fuel over a 3-month period is calculated as 0.032. As there is

no futures contract on jet fuels, the company chooses to hedge by buying futures contracts on heating

oil. The standard deviation of the change in the futures price over a 3-month period is 0.040 and the

coefficient of correlation( ) between the 3-month change in the price of jet fuel and the 3-month

change in the futures price is 0.8.

optimal hedge ratio 0.032

* (0.8) 0.640.040

S

F

h

One heating oil futures is on 42000 gallons. The company should therefore buy

1000000(0.64) 15.2 15 contracts

42000

Example: (Optimal hedge ratio)

The standard deviation of monthly changes in the spot price of live cattle is (in cents per pound) 1.2.

The standard deviation of monthly changes in the futures price of live cattle for the closest contract is

1.4. The correlation between the futures price changes and the spot price changes is 0.7. It is now

October 15. A beef producer is committed to purchasing 200,000 pounds of live cattle on November

15. The producer wants to use the December live cattle futures contracts to hedge its risk. Each

contract is for the delivery of 40,000 pounds of cattle. What strategy should the beef producer follow?

1.2* (0.7) 0.6

1.4

S

F

h

The beef producer requires a long position in (200,000)(0.6) 120,000 pounds of cattle. The beef

producer should therefore take a long position in 120000

340000

contracts

Rolling the hedge forward

If the expiration date of the hedge is later than the delivery dates of all the futures contracts that can be

used, the hedger must then roll the hedge forward.

Example:

A company has to pay 1 million in December 2018.

In March 2014, buy 16 contracts of December 2014 . (16 62500 1000000 ) In December 2014, sell 16 contracts of December 2014 and buy 16 contracts of December 2015 .

In December 2015, sell 16 contracts of December 2015 and buy 16 contracts of December 2016 .




II Forward and Futures Prices

Continuous compounding

Principal: A Interest Rate: r

Compounding frequency Value at the end of 1 year Value at the end of thn year

1 (1 )A r (1 )nA r

2 2(1 )2

rA

2(1 )2

nrA

3 3(1 )3

rA

3(1 )3

nrA

m (1 )m

rA

m (1 )

mnrAm

lim (1 )m rm

rA Ae

m

r nAe

Example:

Suppose the semi-annual compounding interest rate is 10%. Find the corresponding continuous

compounding interest rate.

210%(1 ) : continuous compounding interest rate2

Cr

Ce r

2ln( ) ln(1.05) 2ln(1.05) 9.758%Cr

Ce r

Example:

Suppose the comtinuous compounding interest rate is 10%. Find the corresponding quarterly

compounding interest rate.

10% 444

1 1 1

10% 10% 10%4 44 4 44

(1 ) : quarterly compounding interest rate4

(1 ) ( ) ( ) 1 4 ( ) 1 10.126% 4 4

re r

r re e r e


Convergence of futures price to spot price

Claim: The futures price equals (or is very close) to the spot price @T

Proof:

(I) Suppose T TF S , this gives rise to an arbitrage opportunity for traders.

i) short a futures contract @TF

ii) buy the asset @TS

iii) make delivery

These trades will increase the spot price of the asset and decrease the price of the futures

contract, this process will go on until the equality is reached.

(II) Suppose T TF S , this gives rise to an arbitrage opportunity for traders.

i) buy a futures contract @TF

ii) accept delivery

iii) sell the asset @TS

These trades will increase the price of the futures contract and decrease the spot price of

the asset, this process will go on until the equality is reached.

spot price

$ $

futures price

futures price

spot price

time time

T T

T TF S

T TF S

Abritrage: buy low, sell high


Assumptions:

1. There are no transaction costs.

2. All trading profits (net of trading losses) are subject to the same tax rate.

3. The market participants can borrow money at the same risk-free rate of interest as they can lend

money.

4. The market participants take advantage of arbitrage opportunities as they occur.

Notation:

: time when forward contract matures (years)

: current time (years)

: price of asset underlying forward contract @

T

t

S t

: price of asset underlying forward contract @ (unknown at the current time)TS T

: delivery price (the specified price) in forward contract

: value of a long forward contract @

: forward price @

: risk-free interest rate (per annum) @ , with continuous compounding for an

K

f t

F t

r t investment maturing @T

[Note that the forward price @ t is the delivery price that would make the contract have a value 0 .

When a contract is initiated, the delivery price is normally set equal to the forward price so that

and 0F K f .]


Forward contracts on a security that provides no income (non-dividend-paying stocks and

discount bonds)

Claim: rTF Se

[The annual compounding interest rate verison of the formula is (1 )TF S r .]

Proof:

(I) Suppose rTF Se

An investor can earn a profit by carrying out the following trades:

i) borrows $ for (time period) at interest rate (per annum)S T r

ii) buys the asset (and pays $S )

iii) takes a short position in the forward contract (promises to sell the security for $ @F T )

cash flow @T :

i) $F (the asset is sold under the terms of the forward contract)

ii) $rTSe (repays the loan)

0rTF Se is realized @T

If a lot of traders carry out the above trades, and until the F S equality is restored.

(II) Suppose rTF Se


i) sells short the asset (and collects $S )

ii) deposits $ for at interest rate (per annum)S T r

iii) takes a long position in the forward contract (promises to buy the security for $ @F T )

cash flow @T :

i) $F (accepts delivery under the terms of the forward contract)

ii) $rTSe (the deposit)

0rTSe F is realized @T



Example: Arbitrage opportunity when forward price of a non-dividend-paying stock is too

high.

The forward price of a stock for a contract with delivery date in three months is $43. The 3-month

risk-free interest rate is 5 percent per annum, and the current stock price is $40. No dividends are

expected.

Opportunity:

Note that 3

(5%)( )1243 40 40.50rTF Se e

The forward price is too high relative to the stock price. An arbitrageur can make a profit by

i) borrowing $40 @ 5%r to buy 1 share, and

ii) taking a short position in the forward contract (and promised to sell for $43 three months later)

At the end of 3 months, the arbitrageur delivers the share and receives $43. He is required to pay back

the loan of the amount

3(5%)

12$40 $40.50e $43 $40.50 $2.50 .

[Note that if 3

(5%)12$43 43 $42.47rTF S Fe e

.]

Example: Arbitrage opportunity when forward price of a non-dividend-paying stock is too low

The forward price of a stock for a contract with a delivery date in three months is $39. The three-

month risk-free interest rate is 5 percent per annum and the current stock price is $40. No dividends

are expected.

Opportunity

Note that 3

(5%)( )1239 40 40.50rTF Se e

The forward price is too low relative to the stock price. An arbitrageur can make a profit by

i) shorting one share and loaning out the $40 received @ 5% for three monthsr , and

ii) taking a long position in the forward contract (and promises to buy @$39 three months later).

At the end of three months, the arbitrageur will have a total cash position of

3(5%)( )

1240 $40.50e . He will accept delivery according to the forward contract and buy 1 share @$39. This share will be used

to cover the short position established initially.

$40.50 $39 $1.50

[Note that if 3

(5%)12$39 39 $38.52rTF S Fe e

]


Forward contracts on a security that provides income (dividend-paying stocks and coupon-

bearing bonds)

I : the present value (using the risk-free interest rate to discount future incomes) of income to be received during the life of the forward contract

Claim: ( ) rTF S I e [Discrete time version: ( )(1 )TF S I r ]

Proof:

(I) Suppose ( ) rTF S I e


i) borrows $ for at interest rate (per annum)S T r

ii) buys the asset (and pay $S )

iii) takes a short position in the forward contract (promises to sell for $ @F T )

cash flow @T :

i) $ F the asset is sold under the terms of the forward contract

ii) $ rTSe repays the loan

iii) $ dividendrTIe

( ) 0rT rT rTF Se Ie F S I e is realized @T


(II) Suppose ( ) rTF S I e


i) sells short the asset (and collects $S )

ii) lends out $ for at interest rate (per annum)S T r

iii) takes a long position in the forward contract (promises to buy for $ @F T )

cash flow @T :

i) $ F accepts delivery under the terms of the forward contract

ii) $ rTSe gets back the loan

iii) $ pays the dividend (for the short stock position)rTIe

( ) 0rT rT rTSe Ie F S I e F is realized @T



Example Consider a 10-month forward contract on a stock with a price of $50. We assume that

the risk-free interest rate (continuously compounding) is 8% per annum for all maturities. We also

assume that dividends of $0.75 per share are expected after 3 months, 6 months, and 9 months.

Present value of the dividends: 3 6 9

(8%)( ) (8%)( ) (8%)( )12 12 120.75 0.75 0.75 2.162I e e e

r 3 months 6 months 9 months

10

(8%)( )12( ) (50 2.162) $51.14rTF S I e e

10 months

Example: Consider a 13-month contract on a coupon-paying bond. Let $900S . Coupon payments of $40 are expected in 6 months and 1 year respectively. Let the 6-month risk-free interest

rate 9% , the 12-month risk-free interest rate 10% , and the 13-month risk-free interest rate 11% .

Present value of the dividends: 6

(9%)( )(10%)(1)1240 40 74.43I e e

r 6 months 12 months

13

(11%)( )12( ) (900 74.43) $930.05rTF S I e e

13 months

Forward contracts on a security that provides a constant dividend yield

Claim: ( )r q TF Se

Example: Consider a 6-month forward contract on an investment asset that is expected to provide

a continuous dividend yield of 4% per annum. The risk-free interest rate (with continuous

compounding) is 10% per annum. The spot price of the asset is $25.

6

(10% 4%)( )( ) 1225 $25.76 r q TF Se e

Note: When the dividend yield is continuous, but varies throughout the life of the forward contract, q

should be set equal to the average dividend yield during the life of the contract.


Stock index futures

Stock indices

1. Capitalization-weighted index (S&P 500 Index)

Example:

i)

Stock Price Float Capitalization

A $30 175,000 $5,250,000

B $90 50,000 $4,500,000

C $50 100,000 $5,000,000

Total capitalization $14,750,000

Index 100

ii)


A $40 175,000 $7,000,000

B $80 50,000 $4,000,000

C $60 100,000 $6,000,000


Index 17000000100 ( ) 115.25

14750000

iii)


A $33 175,000 $5,775,000

B $90 50,000 $4,500,000

C $50 100,000 $5,000,000


Index 15275000100 ( ) 103.56

14750000

Note:

1. The index value will not be affected by the change in the number of shares floating.

2. As the prices of the stocks change, the weight will change as well.


2. Price-weighted index (Dow Jones Industrial Average)

Example:

i) Stock Price

A $30

B $90

C $50

Total $170

Index 100

ii) Stock Price

A $33

B $90

C $50

Total $173

Index 173100( ) 101.76

170

Note:

1. The stock with the highest weight is the one with the highest price.

2. A change in IBMs price will have more effect on a capitalization-weighted index than on a price-weighted index.

Major indices:

i) S&P500 index

A capitalization-weighted index ( 250 index ) Make up of a portfolio of the largest 500 companies. The index accounts for 80% of the market capitalization of all the stocks listed on the NYSE.

ii) Dow Jones Industrial Average

A price-weighted index.

Make up of a portfolio of 30 stocks.

iii) NYSE Composite index

A capitalization-weighted index.

Make up of a portfolio of all the stocks listed in NYSE.

iv) The Major Market Index

A price-weighted index.

Make up of a portfolio of 20 blue-chips stocks listed on NYSE.


Note:

1. The futures contracts on stock indices are settled in cash, not by delivery of the underlying

asset.

2. All contracts are marked to market on the last trading day and the positions are considered to be

closed.

3. For most contracts, the settlement price on the last trading day is set at the closing value of the

index on that day except the S&P500 index.

4. For the S&P500 index, the settlement price is set as the value of the index based on the opening

prices on the following trading day.

Future Price of Stock Indices

Most indices can be thought of as securities that pay dividends. The security is the portfolio of stocks

underlying the index and the dividends paid by the security are the dividends that would be received by

the holder of the portfolio.

Example

Consider a 3-month futures contract on the S&P500. Suppose that the stocks underlying the index

provide a dividend yield of 3% per annum.

Let 400 and 8%S r .

Then 3

(8% 3%)( )( ) 12400 405.03r q TF Se e

In practice, q is not constant over time. The average should be used. Or the actual dividend

date can be used.

Example

Consider the following data of S&P500 index:

June 675.80

September 681.50

Calculate the r q . (Assume that the term structure of interest rate is constant.)

Let T be the expiration date of the June contract. ( )( 0.25)r q T

SepF Se

( )r q T

JuneF Se

( )( 0.25)0.25( ) 0.25( )

( )

681.50 681.50ln( ) 0.25( )

675.80 675.80

r q TSep r q r q

r q T

June

F See e r q

F Se

1 681.50ln( ) 3.36%

0.25 675.80r q


Index Arbitrage

If ( )r q TF Se , the following profit opportunity arises:

i) buy the stocks underlying the index,

ii) sell the future contract.

[This is usually done by a corporation holding short-term money market instruments.]

If ( )r q TF Se , the following profit opportunity arises:

i) sell short the stocks underlying the index,

ii) buy the future contract.

[This is usually done by a pension fund that owns an index portfolio of stocks.]

Note:

1. In practice, index arbitrage is accomplished by trading a subset of the index stocks.

2. If it is done by a computer, it is called program trading.

3. In normal market conditions, ( ) is very close to r q TF Se .

However on Oct. 19, 1987, futures prices were at a significant discount. [At closing, the spot

S&P500 index was 225.06, but Dec. S&P500 index was 201.50.] The reason is as follows:

In order to take advantage of the arbitrage opportunity, traders need to buy futures and sell

short stocks. However, traders can only sell short stocks when the stocks are in an up-tick situation. In a down market such as the one on Oct. 19, 1987, it is impossible to find the

opportunity to sell short stocks. Hence even though there is a discrepancy in the futures price

and the spot price, the traders cannot take advantage of it.


Portfolio Analysis

Definition: end-of-period wealth beginning-of-period wealth

Returnbeginning-of-period wealth

Definition: Expected return on a portfolio of N securities: 1

N

p i i

i

r X r

where : expected return of the portfoliopr

: the proportion of the portfolio's initial value invested in security iX i

: expected return of security ir i

: the number of securities in the portfolioN

CALCULATING THE EXPECTED RETURN ON A PORTFOLIO

(a) Security and Portfolio Values

Security Name Number of

Shares in

Portfolio

Initial Market Price

Per Share

Total Investment Proportion of Initial

Market Value of Portfolio

A 100 $40 $4000 $4000/$17200=23.35%

B 200 $35 $7000 $7000/$17200=40.70%

C 100 $62 $6200 $6200/$17200=36.05% Total = $17200 100%

(b) Calculating The Expected Return for a Portfolio Using End-of-Period Values

Security Name Number of

Shares in

Portfolio

Security Expected

Returns

Expected End-

of-Period Value

Per Share

Aggregate Expected

End-of-Period Value

A 100 $46.48 $4016.2%

$40

$46.48 $46.48100 = $4648

B 200 $43.61 $3524.6%

$35

$43.61 $43.61200 = $8722

C 100 $76.14 $6222.8%

$62

$76.14 $76.14100 = $7614

Total = $20984

$20984 $17200Portfolio Expected Return 22.00%

$17200pr

(c) Calculating the Expected Return for a Portfolio Using Security Expected Returns Security Name Proportion of

Initial Market

Value at

Portfolio

Security Expected

Returns

Contribution to Portfolio Expected

Return

A 23.25% 16.2% 23.25%16.2% = 3.77% B 40.70% 24.6% 40.70%24.6% = 10.01% C 36.05% 22.8% 36.05%22.8% = 8.22%

Portfolio Expected Return 22.00%pr


The Capital Asset Pricing Model

Assumptions:

1. Investors evaluate portfolios by looking at the expected returns and standard deviations of the

portfolios over a one-period horizon.

2. Investors are never satiated, so when given a choice between two otherwise identical portfolios.

they will choose the one with the higher expected return.

3. Investors are risk-averse, so when given a choice between two otherwise identical portfolios,

they will choose the one with the lower standard deviation.

4. Individual assets are infinitely divisible, meaning that an investor can buy a fraction of a share

if he or she so desires.

5. There is a risk free rate at which an investor may wither lend (that is, invest) money or borrow

money.

6. Taxes and transaction costs are irrelevant.

7. All investors have the same one-period horizon.

8. The risk free rate is the same for all investors.

9. Information is freely and instantly available to all investors.

10. Investors have homogeneous expectations, meaning that they have the same perceptions in

regard to the expected returns, standard deviations, and covariance of securities.

Major implication of the model:

The expected return of an asset is related to a measure of risk for that asset known as Beta ( ).

Definition: The Market Portfolio is a portfolio consisting of all securities where the proportion

invested in each security corresponds to its relative market value.

aggregate market value of the securityrelative market value of a security

sum of the aggregate market value of all securities

expected return on portfolio risk free interest rate

return on index risk free interest rate

( , )

( )

M

M

Cov R R

Var R


Hedging using index futures

Define as a measure of the responsiveness of a security or portfolio to the market as a whole (in

term of excess return over risk-free interest rate).

When 1.0 , the return on the portfolio tends to mirror the return on the market.

When 2.0 , the return on the portfolio tends to be twice as great as the excess return on the market.

Suppose we wish to hedge against changes in the value of a portfolio during a period of time T .

Define : current value of the portfolioP

: current value of the stocks underlying one futures contractA

* : optimal number of contracts to short when hedging the portfolioN

*P

NA

Example

A company wishes to hedge a portfolio worth $2100000 over the next three months using an

S&P500 index futures contract with four months to maturity. The current level of the S&P500 index is

900 and the of the portfolio is 1.5.

The value of the assets underlying 1 futures contract is 900 $250 $225000 .

1.5(2100000)14

225000

PN

A

(should short 14 contracts)


Example:

200 10% 4% Value of portfolio $2,000,000 1.5S r q

Suppose an investor wants to use 4-month futures contract to hedge the value of the portfolio for

3 months.

The value of the stocks underlying 1 futures contract ($250)(200) $50000

$20000001.5 60

$50000

PN

A (should sell 60 contracts)

4

(10% 4%)( )( ) 0.0212200 200 204.04r q TF Se e e

Suppose the index turns out to be 180 3 months later (a 10% drop).

Futures: 1

(10% 4%)( )( ) 0.00512180 180 180.90r q TF Se e e

Gain from the short position 60(204.04 180.90)($250) $347100

Stocks:

index 10%

dividend 1% 4% per annum, or 1% in 3 months

Total return 9%

expected return on portfolio risk free interest rate

return on index risk free interest rate

expected return on portfolio (return on index )


r r

r r

2.5% 1.5[ 9% 2.5%] 2.5% 1.5[ 11.5%] 14.75%

Expected value of the portfolio (inclusive of dividends) at the end of 3 months

$2000000 (1 14.75%) $1705000

Aggregate position $347100 $1705000 $2,052,100


Suppose the index turns out to be 190 3 months later (a 5% drop).

Futures: 1

(10% 4%)( )( ) 0.00512190 190 190.95r q TF Se e e


Stocks:

index 5%


Total return 4%


2.5% 1.5[ 4% 2.5%] 2.5% 1.5[ 6.5%] 7.25%

r r


$2000000(1 7.25%) $1855000


Suppose the index turns out to be 200 3 months later (a 0% change).

Futures: 1

(10% 4%)( )( ) 0.00512200 200 201.00r q TF Se e e


Stocks: index 0%


Total return 1%


2.5% 1.5[ 1% 2.5%] 2.5% 1.5[ 1.5%] 0.25%

r r


$2000000(1 0.25%) $2005000



Suppose the index turns out to be 210 3 months later (a 5% increase)

Futures: 1

(10% 4%)( )( ) 0.00512210 210 211.05r q TF Se e e


Stocks:

index 5%


Total return 6%


2.5% 1.5[ 6% 2.5%] 2.5% 1.5[ 3.5%] 7.75%

r r


$2000000(1 7.75%) $2155000



Futures: 1

(10% 4%)( )( ) 0.00512220 220 221.10r q TF Se e e


Stocks: index 10%


Total return 11%


2.5% 1.5[ 11% 2.5%] 2.5% 1.5[ 8.5%] 15.25%

r r


$2000000(1 15.25%) $2305000



Performance of Stock Index Hedge (3 months)

Value of index in 3 months 180 190 200 210 220

Future prices of index in 3 months 180.90 190.95 201.00 211.05 221.10

Gain on futures position ($'000s) $347.1 $196.35 $45.6 -$105.15 -$255.9

Value of portfolio (including dividends) in 3

months ($'000s)

$1705.0 $1855.00 $2005.0 $2155.00 $2305.0

Aggregate position in 3 months ($'000s) $2052.1 $2051.35 $2050.6 $2049.85 $2049.1

Return 2.605% 2.5675% 2.53% 2.4925% 2.455%


Suppose the index turns out to be 180 4 months later (a 10% drop)

Futures: At maturity, F S


Stocks: index 10%

1 1dividend 1 % 4% per annum, or 1 % in 4 months

3 3

2Total return 8 %

3


1 2 1 1 23 % 1.5[ 8 % 3 %] 3 % 1.5[ 12%] 14 %

3 3 3 3 3

r r


2$2000000(1 14 %) $1706667

3


Suppose the index turns out to be 190 4 months later (a 5% drop)



Stocks: index 5%


3 3

2Total return 3 %

3


1 2 1 1 13 % 1.5[ 3 % 3 %] 3 % 1.5[ 7%] 7 %

3 3 3 3 6

r r


1$2000000(1 7 %) $1856667

6



Suppose the index turns out to be 200 4 months later (no change)


Gain from the short position 60(204.04 200)($250) $60600

Stocks: index 0%


3 3

1Total return 1 %

3


1 1 1 1 13 % 1.5[ 1 % 3 %] 3 % 1.5[ 2%] %

3 3 3 3 3

r r


1$2000000(1 %) $2006667

3





Stocks: index 5%


3 3

1Total return 6 %

3


1 1 1 1 53 % 1.5[ 6 % 3 %] 3 % 1.5[ 3%] 7 %

3 3 3 3 6

r r


5$2000000(1 7 %) $2156667

6






Stocks: index 10%


3 3

1Total return 11 %

3


1 1 1 1 13 % 1.5[ 11 % 3 %] 3 % 1.5[ 8%] 15 %

3 3 3 3 3

r r


1$2000000(1 15 %) $2306667

3


Performance of Stock Index Hedge (4 months)

Value of index in 4 months 180 190 200 210 220

Future prices of index in 4 months 180 190 200 210 220

Gain on futures position ($'000s) $360.6 $210.6 $60.6 -$89.4 -$239.4

Value of portfolio (including dividends) in 4

months ($'000s)

$1706.7 $1856.7 $2006.7 $2156.7 $2306.7

Aggregate position in 4 months ($'000s) $2067.3 $2067.3 $2067.3 $2067.3 $2067.3

Return 3.36% 3.36% 3.36% 3.36% 3.36%

Note:

1. A stock index hedge, if effective, should result in the hedger's position growing at

approximately the risk-free interest rate.

2. By hedging, the ( of the portfolios futures position ) becomes 0 .


Why hedging?

1. The hedger feels that the stocks in the portfolio have been chosen well and the portfolio will

out-perform the market.

2. The hedger is planning to hold a portfolio for a long period of time and requires short-term

protection in an uncertain market.

3. Hedging can eliminate "systematic" risk.

Remark: After hedging his portfolio, if the market actually goes up, the portfolio manager may

be fired!!

Changing

Stock index futures can be used to change the of a portfolio.

In general, to change the of the portfolio from to * .

If * , a short position in ( *)P

A contracts is required.

If * , a long position in ( * )P

A contracts is required.


Example: Stock Portfolio Hedge

Scenario: On January 2, a portfolio manager is concerned about the market over the next four

months. The portfolio has accumulated an impressive profit, which the manager wishes to protect over

the period ending July 27.

Stock Price (1/2) Number of

shares

Market Value Weight Beta

A $18.875 9000 $169875 4.4% 1.00

B $73.500 8000 $588000 15.2% 0.80

C $50.875 3500 $178063 4.6% 0.50

D $43.625 5400 $235575 6.1% 0.70

E $54.250 10500 $569625 14.7% 1.10

F $47.750 14400 $687600 17.8% 1.10

G $44.500 12500 $556250 14.4% 1.40

H $52.875 16600 $877725 22.7% 1.20

Portfolio $3862713 100% 1.06

S&P500 index:

Price on January 2: 1280

Multiple: $250

Value of 1 contract: ($250)(1280) $320000

S&P500 September futures:

Price on January 2: 1300

Optimal number of futures contract: $3862713

1.06 12.80 (#contracts shorted 13)$320000

PN

A

Results (July 27):

Stock Price (7/27) Market Value

A $21.625 $194625

B $81.500 $652000

C $43.875 $153563

D $47.125 $254475

E $45.875 $481688

F $48.125 $693000

G $40.000 $500000

H $50.000 $830000

Portfolio $3759351

S&P500 September futures contract:

Price on July 27: 1265

Gain on stocks $3759351 $3862713 $103362 Gain on futures contract (1300 1265)($250)(13) $113750

Aggregate position +$10388

portfolio 1 1 ... n nw w


Example: Anticipatory Hedge of a Takeover

Scenario: On November 17, a firm has decided to begin buying up shares of ABC Corporation

with the ultimate objective of obtaining controlling interest. The acquisition will be made by

purchasing lots of about 100000 shares until sufficient control is obtained. The first purchase of

100000 shares will take place on December 17. The stock is currently worth $54 and has a beta of

1.35.

Date Spot Market Futures Market

November 17 Current price of the stock is $54.

Current cost of shares:

(100000)($54) $5400000

1.35

March S&P500 is at 465.45.

Price per contract $116362.50 Approximate number of contracts:

$54000001.35 62.65

$116362.5

The number of contracts purchased 63 December 17 The stock is purchased at its

current price of $57.

Cost of shares:

(100000)($57) $5700000

March S&P500 is at 473.95.

When the 100000 shares are purchased on December 17, they cost $5700000, an additional $300000.

The profit on the futures contract (473.95 465.45)($250)(63) $133875

The hedge eliminated about 44% of the additional cost.

The shares end up effectively costing ($5700000 $133875) /100000 $55.66


Value of a forward contract of a Non-dividend paying stock

Claim: rTf S Ke

Proof:

Consider 2 portfolios.

Portfolio A: 1 long forward contract on the security $ cashrTKe

Portfolio B: 1 unit of the security

In Portfolio A, the cash will grow to [$ ] $ @rT rTKe e K T , which then can be used to buy 1 unit of

the security. Therefore, @T , value of Portfolio Avalue of Portfolio B.

@ , t value of Portfolio Avalue of Portfolio B (otherwise, there will be arbitrage opportunity)

rT rTf Ke S f S Ke

Note: When a forward contract is initiated, the forward price equals the delivery price specified in the

contract ( F K ) and is chosen so that 0f 0rT rTS Fe F Se

Example

Consider a long forward contract on a non-dividend-paying stock that matures in 12 months.

Let $40, 10%, $30K r S , then (10%)(1)30 40 $6.19rTf S Ke e

The owner of the long forward contract is willing to pay up to $6.19 to get rid of the contract.

Note: The short forward contract is worth $6.19 .

Example

Consider a long 6-month forward contract on a 1-year discount bond.

Let 6%, $950, $930r K S , then 6

(6%)( )12930 950 $8.08rTf S Ke e

The owner of the long forward contract will ask for $8.08 before he is willing to give up the contract.

Note: The value of the short forward contract is $8.08 .


Value of a forward contract of a security paying known dividend

Claim: rTf S I Ke

Proof:

Consider 2 portfolios.

Portfolio A: 1 long forward contract on the security $ cashrTKe

Portfolio B: 1 unit of the security borrowing of $ at the risk-free interest rateI

In Portfolio A, the cash will grow to [$ ] $ @rT rTKe e K T , which then can be used to buy 1 unit of

the security.

In Portfolio B, the investor will use the income from the security to pay back the loan, so that @T ,

the investor will hold 1 unit of the security.

Therefore, @T , the 2 portfolios are the samevalue of Portfolio Avalue of Portfolio B.

@ , t value of Portfolio Avalue of Portfolio B (otherwise, there will be arbitrage opportunity)

rT rTf Ke S I f S I Ke

Note: When a forward contract is initiated, the forward price equals the delivery price specified in the

contract ( F K ) and is chosen so that 0f 0 ( )rT rTS I Fe F S I e

Example

Consider a long forward contract on a 5-year bond with $900S has a maturity of 13 months. Let $910, 6-month 9%, 12-month 10%, 13-month 11%K r r r

Let the coupon payments of $60 are expected after 6 months and 12 months respectively.

Present value of the coupon payments: 6

(9%)( )(10%)(1)1260 60 $111.65I e e

r 6 months 12 months

Hence

13(11%)( )

12900 111.65 910 $19.42rTf S I Ke e




Value of a forward contract of a security that provides a known dividend yield

qT rTf Se Ke

Example

Consider a 6-month forward contract on a security that is expected to provide a continuous dividend

yield ( q ) of 4% per annum. The 6-month risk-free interest rate (with continuous compounding) is

10% per annum. Let $25 and $27S K .

6 6(4%)( ) (10%)( )

12 1225 27 $1.18qT rTf Se Ke e e



Also 6

(10% 4%)( )( ) 1225 $25.76r q TF Se e

.

Note:

i) ( )0 1r q Tr q r q e F S

This has nothing to do with whether the investors are optimistic about the future of the

stock or not!!!

ii) ( )0 1r q Tr q r q e F S


A General Result (Valuing Forward Contract)

The value of a forward contract at the time it is first entered into is $0. At a later stage, it may be either

positive or negative.

There is a general result, applicable to all forward contacts, that gives the value of a long forward

contract, f , in terms of the originally negotiated delivery price, K , and the current forward price, F .

Claim: ( ) rTf F K e

Proof:

Consider the following portfolio:

1) short forward contract A: delivery price @F T

2) long forward contract B: delivery price @K T

@ :T

security: accept the security (the long forward contract) and deliver the security (the short

forward contract)

Cash flow: F K

@ t :

Cash flow: ( ) rTF K e

Since the value contract A 0 value of contract B ( ) rTF K e

Note:

i) 0 0F K f

ii) 0 0F K f

Example: Consider a forward contract on a security, which will expire in 5 months with

$100K . Let 10% and $90r F . 5

(10%)( )12( ) (90 100) $9.59rTf F K e e




Note:

1. If ( ) rTf F K e , the following profit opportunity arises.

i) Take a long position in a forward contract with delivery price F .

ii) Take a short position in a forward contract with delivery price K .

@ t , the value of the first contract is $0 and the value of the second contract is quoted at f . With a

short position in the second contract, the investor collects $ f

@T , the investor's net cash flow the PV of the cash flow ( ) rTK F K F e

(@ today's dollar) ( ) ( ) 0rT rTf K F e f F K e .

2. If ( ) rTf F K e , the following profit opportunity arises.

i) Take a short position in a forward contract with delivery price F .

ii) Take a long position in a forward contract with delivery price K .

@ t , the value of the first contract is $0 and the value of the second contract is quoted at f . With a

long position in the second contract, the investor pays $ f .

@T , the investor's net cash flow the PV of the cash flow ( ) rTF K F K e

(today's dollar) ( ) 0rTf F K e .

Forward Prices versus Futures Prices

1. When the risk-free interest rate is constant and the same for all maturities, the forward price for

a contract with a certain delivery date is the same as the futures price for a contract with that

delivery date.

2. When interest rates vary unpredictably, forward and futures prices are in theory no longer the

same. As the life of a futures contract increases, the differences between forward and futures

contracts become more significant.


Forward and Futures Contracts on Currencies

: the current price in dollars of 1 unit of the foreign currency

: the delivery price agreed to in the forward contract

: the foreign risk-free interest rate with continuous compoundingf

S

K

r

Note: A foreign country has the property that the holder of the currency can earn interest at the risk-

free interest rate prevailing in the foreign country.

(For example, the holder can invest the currency in a foreign currency denominated bond.)

Value of forward contract: fr T rTf Se Ke

Note that @ 0t , ( )

0f f fr T r T r r TrT rTf Se Ke Se Fe F Se

Note:

1. The "formula" is very similar to the formula for a security with a constant yield.

2. ( )

0 1fr r T

f fr r r r e F S

3. ( )

0 1fr r T

f fr r r r e F S

Example

Suppose that the 6-month interest rates in the US and Japan are 5% and 1% per annum, respectively.

The current exchange rate is 100/USD. For a 6-month forward contract, we have 0.01, 5%S r ,

and 1%fr .

6(5% 1%)( ) 120.01 0.010202f

r r TF Se e


Example (arbitrage opportuity)

Let the 2 year interest rates in Australia and United States are 5% and 7% respectively.

Let $0.62/S AUD and $0.66/F AUD .

Since (7% 5%)(2)0.66 0.6453F Se , there is an arbitrage opportunity.

i) Borrow $1000 at 7% per annum for 2 years, convert to 1000

1612.900.62

AUD AUD and invest

the AUD at 5% per annum.

ii) Enter in a forward contract to sell (5%)(2) 1612.90 1782.53AUD e AUD for

1782.53 $0.66 $1176.47

2 years later

The investor will sell AUD 1782.53 and get $1176.47 (according to the forward contract). The $ will

be used to cover the initial loan of (7%)(2)$1000 $1150.27e

$1176.47 $1150.27 $26.20

Example (arbitrage opportunity)

Let the 2 year interest rates in Australia and United States are 5% and 7% respectively.

Let $0.62/S AUD and $0.63/F AUD .

Since (7% 5%)(2) 0.6453 0.63Se F , there is an arbitrage opportunity.

i) Borrow 1000AUD at 5% per annum for 2 years, convert to 1000 $0.62 $620 and invest the USD at 7% per annum.

ii) Enter in a forward contract to buy (5%)(2) 1000 1105.17AUD e for

1105.17 $0.63 $696.26

2 years later

The amount of USD on hand(7%)(2)620 $713.17e

AUD 1105.17 will be purchased ($696.26 will be paid, according to the forward contract) and the AUD

will be used to cover the initial short AUD position.

$713.17 $696.26 $16.91


Futures on Commodities

Investment Commodities (Gold and Silver)

3 cases:

i) storage cost 0 (analogous to securities paying no income)

rTF Se

ii) storage cost 0

storage cost can be treated as negative income

Let U be the PV of all the storage costs that will be incurred during the life of the futures contract.

( ) rTF S U e

iii) storage cost 0 and is proportional to the price of the commodity.

In this case, the storage cost can be treated as negative dividend yield.

Let u be the storage cost per annum as a percentage of the spot price.

( )r u TF Se

Example

Consider a one-year futures contract on gold. Suppose that it costs $2 per ounce per year to store gold,

with the payment being made at the end of the year. Assume that the spot price is $450 and the risk-

free interest rate is 7% per annum for all maturities.

We have 450, 7%, 1S r T

(7%)(1)2 1.865U e

(7%)(1)( ) (450 1.865) $484.63rTF S U e e

If 484.63F , an arbitrageur can make money by buying gold and shorting 1-year gold futures contracts.

If 484.63F , an arbitrageur can make money by selling gold and buying 1-year gold futures contracts.


Consumption Commodities

In the case of consumption commodities, because it is difficult to sell short the commodities even if

there is profit opportunity, hence the formulas become:

rTF Se

( ) rTF S U e

( )r u TF Se

Convenience yields

The benefits (including the ability to benefit from temporary local shortages or the ability to keep a

production process running) to the owners of the commodities are called convenience yield.

yT rTFe Se

( )yT rTFe S U e

( )yT r u TFe Se

The convenience yield reflects the markets expectations concerning the future availability of the commodity. The greater the possibility that shortages will occur during the life of the futures contract,

the higher the convenience yield. If users of the commodity have high inventories, there is very little

chance of shortages in the near future and the convenience yields tends to be low. On the other hand,

low inventories tend to lead to high convenience yields.


The Cost of Carry

The relationship between futures prices and spot prices can be summarized in terms of the cost of

carry.

cost of carry storage cost interst that is paid to finance the asset income earned on the asset

non-dividend stock: cost of carry r

stock index: cost of carry r q

currency: cost of carry fr r

commodity with storage cost: cost of carry r u

Define c cost of carry

For an investment asset, the futures price is cTF Se

For a consumption asset, the futures price is ( )c y TF Se where y is the convenience yield.


For agriultural products, the forward curve may not be upward sloping even there is cost of carry. It

may reflect the supply and demand condition of the crop in different part of the year.

Contango is a term used in the futures market to describe an upward sloping forward curve (Such a

forward curve is said to be "in contango" (or sometimes "contangoed").

Formally, it is the situation where, and the amount by which, the price of a commodity for future

delivery is higher than the spot price, or a far future delivery price higher than a nearer future

delivery. The opposite market condition to contango is known as backwardation.

A contango is normal for a non-perishable commodity which has a cost of carry. Such costs

include warehousing fees and interest forgone on money tied up, less income from leasing out the

commodity if possible.

The contango should not exceed the cost of carry, because producers and consumers can compare

the futures contract price against the spot price plus storage, and choose the better one. Arbitrageurs

can sell one and buy the other for a theoretically risk-free profit (see rational pricing futures).

If there is a near-term shortage, the price comparison breaks down and contango may be reduced or

perhaps even reverse altogether into a state called backwardation.

If there is lack of shorage space, then the warehouse cost will increase, hence the cost of carry will

go up and contango becomes more visible.

corn

1/18/2013

Month Settlement

Mar 13 7274

May 13 7292

July 13 7214

Sep 13 6134

Dec 13 5904

Mar 14 6004

May 14 6074

July 14 6090

July 14 5856

Contract size: 5000 bushels

Deliverable grade: #2 Yellow at contract Price,

#1 Yellow at a 1.5 cent/bushel premium

#3 Yellow at a 1.5 cent/bushel discount


III. Options

A CALL OPTION gives the holder the right to buy the underlying asset by a certain date (expiration

date) for a certain price (strike price).

A PUT OPTION gives the holder the right to sell the underlying asset by a certain date (expiration

date) for a certain price (strike price).

An AMERICAN OPTION can be exercised at any time up to the expiration date.

An EUROPEAN OPTION can only be exercised on the expiration date only.

Note:

1. Most of the options that are traded on exchanges are American options.

2. European options are generally easier to analyze than American options.

3. Unlike futures/forwards, the holder is not obligated to buy or sell the underlying asset.

4. Whereas it costs nothing to enter into a forward or futures contract (except commissions), an

investor must pay to purchase an option contract.

Different Types of Options

1. Exchange-traded options:

i) stock options: American

ii) currency options: American and European

iii) index options: S&P 500 is European (cash settlements)

S&P100 is American (cash settlements)

iv) futures options

2. Over-the-counter options: traded directly between financial institutions and corporations.

Because of the non-standard features, they are also called Exotic Options.


Trading Options

1. A trader purchased 5 IBM July11 170 Call @$1.75 on Jan 21, 2011. He closed his position 1

week later @$2.50.

($2.5 $1.75)(100)(5) $375

100 shares per option # of options

2. A trader sold 5 IBM July11 170 Call @$1.75 on Jan 21, 2011. He closed his position 1 week

later @$2.50.

($1.75 $2.5)(100)(10) $750

3. A trader purchased 20 IBM Jan12 145 Put @$8.70 on Jan 21, 2011. He closed his position 1

week later @$9.00.

($9.0 $8.7)(100)(20) $600

4. A trader sold 50 IBM Jan12 145 Put @$8.70 on Jan 21, 2011. He closed his position 1 week

@$8.00.

($8.7 $8.0)(100)(50) $3500

Note:

i) Options are traded like stocks.

ii) Whether a trader makes money or not only depends on the market price of the options, it has

nothing to do with the strike price etc.


: strike price

: final price of the underlying asset

: preium paid (buyer) or preium received (seller)

T

X

S

p

Long call position (European)

Example: $40, $5X p

TS $30 $35 $40 $45 $50 $55 $60

Payoff $0 $0 $0 $5 $10 $15 $20

-$5 -$5 -$5 $0 $5 $10 $15

payoff

X 0 TS X p

p

Short call position (European)

Example: $40, $5X p

TS $30 $35 $40 $45 $50 $55 $60

Payoff $0 $0 $0 -$5 -$10 -$15 -$20

$5 $5 $5 $0 -$5 -$10 -$15

p

X X p

0 TS

payoff

Payoff max( ,0)

max( ,0)

T

T

S X

S X p

Payoff max( ,0) min( ,0)

[max( ,0) ] min( ,0)

T T

T T

S X X S

S X p X S p

Expect increase in stock price

Unlimted gain, limited risk

Expect no increase in stock price

Limited gain, unlimited risk


Long put position

Example: $40, $5X p

TS $30 $35 $40 $45 $50 $55 $60

Payoff $10 $5 $0 $0 $0 $0 $0

$5 $0 -$5 -$5 -$5 -$5 -$5

X X p

X p

X payoff

0 TS p

Short put position

Example: $40, $5X p

TS $30 $35 $40 $45 $50 $55 $60

Payoff -$10 -$5 $0 $0 $0 $0 $0

-$5 $0 $5 $5 $5 $5 $5

p

X p X payoff

0 TS

X p

X

Note: : strike priceX

CALL OPTIONS PUT OPTIONS

in the money

at the money

out of money

S X S X

S X S X

S X S X

Payoff max( ,0)

max( ,0)

T

T

X S

X S p

Payoff max( ,0) min( ,0)

[max( ,0) ] min( ,0)

T T

T T

X S S X

X S p S X p

Expect decrease in stock price

Unlimited gain, limited loss

Expect no decrease in stock price

Limited gain, unlimited loss


Adjustment due to dividends

Over-the-counter options are usually dividend protected. If a company declares a dividend, the strike

price for options on the companys stock is reduced on the ex-dividend day by the amount of the dividend.

Exchange-traded options are not usually adjusted for cash-dividends. In other words, when a cash

dividend occurs, there are no adjustments to the terms of option contract.

[Exception is sometimes made for large cash dividends. On May 28, 2003, Gucci Group declared a

cash dividend of 13.50 euros ($15.88) per common share. In this case, the Options Clearing

Corporation (OCC) at the Chicago Board Options Exchange decided to adjust the terms of options. As

a result, exercise of an option contract required the delivery of 100 shares plus 100 15.88 $1588 of cash.

The holder of a call contract paid 100 times the strike price on exercise and received $1588 of cash in

addition to 100 shares.

The holder of a put contract received 100 times the strike price on exercise and delivered $1588 of

cash in addition to 100 shares.

These adjustments had the effect of reducing the strike price by $15.88.]

Adjustment due to stock split

Exchange-traded options are adjusted for stock splits. Because a stock split does not affect the wealth

of the company, a n-for-m stock split would cause the stock price to go down to m n of its previous

value. (For example, a 3-for-2 split would cause the stock price to go down to 2 3 of its previous

value.) The terms of option contracts are adjusted to reflect expected changes in a stock price arising

from a stock split. After the n-for-m stock split, the strike price is reduced to m n of its previous

value, and the number of shares covered by 1 contract is increased to n m of its previous value. If the

stock price declines in the way expected, the positions of both the writer and the purchaser of a

contract remain unchanged.

Example:

Consider a call option to buy 100 shares of ABC for $30 per share. Suppose that the company makes

for a 2-for-1 stock split. The terms of the option contract are then changed so that it gives the holder

the right to purchase 200 shares for $15 per share.

Adjustment due to stock dividends

A stock dividend involves a company issuing more shares to its existing shareholders. For example, a

20% stock dividend means that investors receive 1 new share for each 5 already owned. This is

effectively a 6-5 split and it should cause the stock price to decline to 5 6 of its previous value.


IV Trading Strategies Involving Options

Strategies involving a single option and a stock

Long position in stock short position in call (covered write) long stock

short

combined position put

TS

TS

S X

short call

Short position in stock long position in call

long call

TS

TS long put

S X

combined position

short stock

rT

rT

p S c Xe

S c p Xe

rT

rT

p S c Xe

p c S Xe


Long position in stock long position in put long stock

long call

combined position

TS

S X

TS

long put

Short position in stock short position in put

short put

TS

TS short call

S X

combined position

short stock

rTp S c Xe

rT

rT

p S c Xe

p S c Xe


Spreads

A spread strategy involves taking a position in 2 or more options of the same type.

Bull Spreads (calls)

i) buy a call with strike price 1X (higher premium)

ii) sell a call with a higher strike price 2 1X X (lower premium)

Payoff from a bull spread

Stock price

range

Payoff from long

call position

Payoff from short

call position

Total payoff Total

2TS X 1TS X 2 TX S 2 1X X 2 1 2 1( )rTX X c c e

2 1TX S X 1TS X 0 1TS X 1 2 1( )rT

TS X c c e

1 TX S 0 0 0 2 1( ) 0

(initial investment)

rTc c e

long call

combined position

1X 2X

short call

Example 1 1 2 2$3 ( $30), $1 ( $35)c X c X

Stock price

range

Payoff from long

call position

Payoff from short

call position

Total payoff Total

35TS 30TS 35 TS 35 30 5 35 30 3 1 $3

35 30TS 30TS 0 30TS 30 3 1 32T TS S

30 TS 0 0 0 3 1 2

time value is ignored


Bull Spreads (puts)

i) buy a put with strike price 1X (lower premium)

ii) sell a put with a higher strike price 2 1X X (higher premium)

Payoff from a bull spread

Stock price

range

Payoff from long

put position

Payoff from short

put position

Total payoff Total

2TS X 0 0 0 1 2( ) 0rTp p e

2 1TX S X 0 2 0TS X 2 0TS X 2 1 2( )rT

TS X p p e

1 TX S 1 0TX S 2 0TS X 1 2 0X X 1 2 1 2( )rTX X p p e

short put

combined position

1X 2X

long put

Note:

An investor who adopts a bull spread strategy is hoping that stock price will go up.


Bear Spread (calls)

i) sell a call with strike price 1X (higher premium)

ii) buy a call with a higher strike price 2 1X X (lower premium)

Stock price

range

Payoff from long

call position

Payoff from short

call position

Total payoff Total

2TS X 2TS X 1 TX S 1 2 0X X 1 2 1 2( )rTX X c c e

2 1TX S X 0 1 TX S 1 0TX S 1 1 2( )rT

TX S c c e

1 TX S 0 0 0 1 2( ) 0rtc c e

long call

1X 2X

combined position

short call

Bear Spread (puts)

i) sell a put with strike price 1X (lower premium)

ii) buy a put with a higher strike price 2 1X X (higher premium)

Stock price

range

Payoff from long

put position

Payoff from short

put position

Total payoff Total

2TS X 0 0 0 1 2( ) 0rTp p e

2 1TX S X 2 TX S 0 2 0TX S 2 1 2( )rT

TX S p p e

1 TX S 2 TX S 1TS X 2 1 0X X 2 1 1 2( )rTX X p p e

short put

1X 2X

combined position

long put


Butterfly Spreads (calls)

i) buy a call with strike price 1X

ii) buy a call with a higher strike price 3 1X X

iii) sell 2 calls with a strike price 1 32

2

X XX

(i)

(ii)

TS

1X 2X 3X combined position

(iii)


Butterfly Spreads (puts)

i) buy a put with strike price 1X

ii) buy a put with a higher strike price 3 1X X

iii) sell 2 puts with a strike price 1 32

2

X XX

(iii)

(i) TS

combined position

1X 2X 3X

(ii)


Combinations

Bottom Straddle

i) buy a call with strike price X ii) buy a put with strike price X

long call

combined position

TS

long put

Top Straddle

i) sell a call with strike price X ii) sell a put with strike price X

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