Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin

15

Option Valuation

16-2

Chapter 20 Con’t

• Black Scholes Option Pricing Model

16-3

The Black-Scholes-Merton Option Pricing Model

• The Black-Scholes option pricing model allows us to calculate the price of a call option before maturity (and, no put price is needed).– Dates from the early 1970s

– Created by Professors Fischer Black and Myron Scholes

– Made option pricing much easier—The CBOE was launched soon after the Black-Scholes model appeared.

• Today, many finance professionals refer to an extended version of the model as the Black-Scholes-Merton option pricing model.– To recognize the important theoretical contributions by professor

Robert Merton.

16-4

The Black-Scholes-Merton Option Pricing Model

• The Black-Scholes-Merton option pricing model says the value of a stock option is determined by six factors:

S, the current price of the underlying stock y, the dividend yield of the underlying stock K, the strike price specified in the option contract r, the risk-free interest rate over the life of the option contract T, the time remaining until the option contract expires , (sigma) which is the price volatility of the underlying stock

16-5

The Black-Scholes-Merton Option Pricing Formula

• The price of a call option on a single share of common stock is: C = Se–yTN(d1) – Ke–rTN(d2)

• The price of a put option on a single share of common stock is: P = Ke–rTN(–d2) – Se–yTN(–d1)

Tσdd

Tσ

T2σyrKSlnd

12

2

1

d1 and d2 are calculated using these two formulas:

16-6

Formula Details

• In the Black-Scholes-Merton formula, three common fuctions are used to price call and put option prices:

– e-rt, or exp(-rt), is the natural exponent of the value of –rt (in common terms, it is a discount factor)

– ln(S/K) is the natural log of the "moneyness" term, S/K.

– N(d1) and N(d2) denotes the standard normal probability for the values of d1 and d2.

• In addition, the formula makes use of the fact that:

N(-d1) = 1 - N(d1)

16-7

Example: Computing Pricesfor Call and Put Options

• Suppose you are given the following inputs:

S = $50

y = 2%

K = $45

T = 3 months (or 0.25 years)

= 25% (stock volatility)

r = 6%

• What is the price of a call option and a put option, using the Black-Scholes-Merton option pricing formula?

16-8

We Begin by Calculating d1 and d2

0.860380.250.250.98538Tσdd

0.98538

0.125

0.25 0.07125 0.10536

0.250.25

0.2520.250.020.064550ln

Tσ

T2σyrKSlnd

12

22

1

Now, we must compute N(d1) and N(d2). That is, the standard normal probabilities.

16-9

We Could Use a Table of Standard Normal Probabilities

16-10

Or, we could use the =NORMSDIST(x) Function in Excel

• If we use =NORMSDIST(0.98538), we obtain 0.83778.

• If we use =NORMSDIST(0.86038), we obtain 0.80521.

• Let’s make use of the fact N(-d1) = 1 - N(d1).

N(-0.98538) = 1 – N(0.98538) = 1 – 0.83778 = 0.16222.

N(-0.86038) = 1 – N(0.86038) = 1 – 0.80521 = 0.19479.

• We now have all the information needed to price the call and the put.

16-11

The Call Price and the Put Price:

• Call Price = Se–yTN(d1) – Ke–rTN(d2)

= $50 x e-(0.02)(0.25) x 0.83778 – 45 x e-(0.06)(0.25) x 0.80521

= 50 x 0.99501 x 0.83778 – 45 x 0.98511 x 0.80521

= $5.985.

• Put Price = Ke–rTN(–d2) – Se–yTN(–d1)

= $45 x e-(0.06)(0.25) x 0.19479 – 50 x e-(0.02)(0.25) x 0.16222

= 45 x 0.98511 x 0.19479 – 50 x 0.99501 x 0.16222

= $0.565.

16-12

We can Verify Our Results Using a Version of Put-Call Parity

$44.33$49.75$5.42

45e50e$0.565$5.985

KeSePC

0.25)(0.060.25)(0.02

rT-yT

Note: The options must have European-style exercise.

16-13

Varying the Option Price Input Values

• An important goal of this chapter is to show how an option price changes when only one of the six inputs changes.

• The table below summarizes these effects.

16-14

Varying the Underlying Stock Price

• Changes in the stock price has a big effect on option prices.

16-15

Varying the Time Remaining Until Option Expiration

16-16

Varying the Volatility of the Stock Price

16-17

Varying the Interest Rate

16-18

Calculating the Impact of Input Changes

• Option traders must know how changes in input prices affect the value of the options that are in their portfolio.

• Two inputs have the biggest effect over a time span of a few days:– Changes in the stock price (street name: Delta)– Changes in the volatility of the stock price (street name: Vega)

16-19

Calculating Delta

• Delta measures the dollar impact of a change in the underlying stock price on the value of a stock option.

Call option delta = e–yTN(d1) > 0

Put option delta = –e–yTN(–d1) < 0

• A $1 change in the stock price causes an option price to change by approximately delta dollars.

16-20

Implied Standard Deviations

• Of the six input factors for the Black-Scholes-Merton stock option pricing model, only the stock price volatility is not directly observable.

• A stock price volatility estimated from an option price is called an implied standard deviation (ISD) or implied volatility (IVOL).

• Calculating an implied volatility requires:– All other input factors, and

– Either a call or put option price

16-21

Implied Standard Deviations, Cont.

• Sigma can be found by trial and error, or by using the following formula.

• This simple formula yields accurate implied volatility values as long as the stock price is not too far from the strike price of the option contract.

rT-yT

22

Ke X SeY

π

XY

2

XYC

2

XYC

XY

T2πσ

16-22

CBOE Implied Volatilities for Stock Indexes

• The CBOE publishes data for two implied volatility indexes:– S&P 100 Index Option Volatility, ticker symbol VIX

– Nasdaq 100 Index Option Volatility, ticker symbol VXN

• Each of these volatility indexes are calculating using ISDs from eight options:– 4 calls with two maturity dates:

• 2 slightly out of the money • 2 slightly in the money

– 4 puts with two maturity dates:• 2 slightly out of the money• 2 slightly in the money

• The purpose of these indexes is to give investors information about market volatility in the coming months.

16-23

VIX vs. S&P 100 Index Realized Volatility

16-24

VXN vs. Nasdaq 100 Index Realized Volatility

16-25

Hedging a Portfolio with Index Options

• Many institutional money managers use stock index options to hedge the equity portfolios they manage.

• To form an effective hedge, the number of option contracts needed can be calculated with this formula:

• Note that regular rebalancing is needed to maintain an effective hedge over time. Why? Well, over time:– Underlying Value Changes

– Option Delta Changes

– Portfolio Value Changes

– Portfolio Beta Changes

100 ValueUnderlyingDelta Option

ValuePortfolioBeta Portfolio Contracts Option of Number

16-26

Example: Calculating the Number of Option Contracts Needed to Hedge an Equity Portfolio

• Your $45,000,000 portfolio has a beta of 1.10.

• You decide to hedge the value of this portfolio with the purchase of put options.– The put options have a delta of -0.31– The value of the index is 1050.

1,520.74 10010500.31

45,000,0001.10

100 ValueUnderlyingDelta Option

ValuePortfolioBeta Portfolio Contracts Option of Number

So, you buy 1,521 put options.

16-27

Implied Volatility Skews

• Volatility skews (or volatility smiles) describe the relationship between implied volatilities and strike prices for options.

– Recall that implied volatility is often used to estimate a stock’s price volatility over the period remaining until option expiration.

16-28

Implied Volatility Skews, Cont.

16-29

Graph of Volatility Skews

16-30

Implied Volatility Skews, Summary

• Logically, there can be only one stock price volatility, because price volatility is a property of the underlying stock.

• However, volatility skews do exist. There is widespread agreement that the major cause is stochastic volatility.

• Stochastic volatility is the phenomenon of stock price volatility changing randomly over time.

• Recall that the Black-Scholes-Merton option pricing model assumes that stock price volatility is constant over the life of the option.

• Nevertheless, the Black-Scholes-Merton option pricing model yields accurate option prices for options with strike prices close to the current stock price.

16-31

Useful Websites

• www.jeresearch.com (for more information on option formulas)

• www.cboe.com (for a free option price calculator)

• www.numa.com (for options trading strategies and a lot more on options)

• www.ivolatility.com (for applications of implied volatility)

• www.aantix.com (for stock option reports)

• www.pmpublishing.com (for free daily volatility summaries)

16-32

Chapter 21: Futures Contracts

• Our goal in this chapter is to discuss the basics of futures contracts and how their prices are quoted in the financial press.

• We will also look at how futures contracts are used, and the relationship between current cash prices and futures prices.

16-33

Forward Contract Basics, I.

• We begin by discussing a forward contract.

• A forward contract is an agreement made today between a buyer and a seller who are obligated to complete a transaction at a date in the future.

• The buyer and the seller know each other. – The negotiation process leads to customized agreements.– What to trade; Where to trade; When to trade; How much to

trade—all can be customized.

16-34

Forward Contract Basics, II.

• Important: The price at which the trade will occur is also determined when the agreement is made. – This price is known as the forward price.

• One party faces default risk, because the other party might have an incentive to default on the contract.

• To cancel the contract, both parties must agree. – One side might have to make a dollar payment to the other to

get the other side to agree to cancel the contract.

16-35

Futures Contract Basics, I.

• A Futures contract is an agreement made today between a buyer and a seller who are obligated to complete a transaction at a date in the future.

• The buyer and the seller do not know each other.– The "negotiation" occurs in the fast-paced frenzy of a futures pit.

• The terms of a futures contract are standardized.• What to trade; Where to trade; When to trade; How much to trade;

what quality of good to trade—all standardized under the terms of the futures contract.

16-36

Futures Contract Basics, II.

• The price at which the trade will occur is determined "in the pit." – This price is known as the futures price.

• No one faces default risk, even if the other party has an incentive to default on the contract.– The Futures Exchange where the contract is traded

guarantees each trade—no default is possible.

• To cancel the contract, an offsetting trade is made "in the pit." – The trader of a futures contract may experience a gain or a

loss.

16-37

Organized Futures Exchanges

• Established in 1848, the Chicago Board of Trade (CBOT) is the oldest organized futures exchange in the United States.

• Other Major Exchanges:– New York Mercantile Exchange (1872)– Chicago Mercantile Exchange (1874)– Kansas City Board of Trade (1882)

• Early in their history, these exchanges only traded contracts in storable agricultural commodities (i.e., oats, corn, wheat).

• Agricultural futures contracts still make up an important part of organized futures exchanges.

16-38

Financial Futures

• Financial futures are also an important part of organized futures exchanges.

• Some important milestones:– Currency futures trading, 1972.– Gold futures trading, 1974.

• Actually, on December 31, 1974.• The very day that ownership of gold by U.S. citizens was legalized.

– U.S. Treasury bill futures, 1976. – U.S. Treasury bond futures, 1977.– Eurodollar futures, 1981.– Stock Index futures, 1982.

• Today, financial futures are so successful that they constitute the bulk of all futures trading.

16-39

Futures Contracts Basics

• In general, futures contracts must stipulate at least the following five terms:

The identity of the underlying commodity or financial instrument. The futures contract size. The futures maturity date, also called the expiration date. The delivery or settlement procedure. The futures price.

16-40

Futures Prices

16-41

Futures Prices

16-42

Why Futures?

• A futures contract represents a zero-sum game between a buyer and a seller.– Gains realized by the buyer are offset by losses realized by the

seller (and vice-versa).– The futures exchanges keep track of the gains and losses every

day.

• Futures contracts are used for hedging and speculation. – Hedging and speculating are complementary activities.– Hedgers shift price risk to speculators.– Speculators absorb price risk.

16-43

Speculating with Futures, Long

• Buying a futures contract (today) is often referred to as “going long,” or establishing a long position.

• Recall: Each futures contract has an expiration date.

– Every day before expiration, a new futures price is established.

– If this new price is higher than the previous day’s price, the holder of a long futures contract position profits from this futures price increase.

– If this new price is lower than the previous day’s price, the holder of a long futures contract position loses from this futures price decrease.

16-44

Example I: Speculating in Gold Futures

• You believe the price of gold will go up. So,– You go long 100 futures contract that expires in 3 months.

– The futures price today is $400 per ounce.

– There are 100 ounces of gold in each futures contract.

• Your "position value" is: $400 X 100 X 100 = $4,000,000

• Suppose your belief is correct, and the price of gold is $420 when the futures contract expires.

• Your "position value" is now: $420 X 100 X 100 = $4,200,000

Your "long" speculation has resulted in a gain of $200,000

What would have happened if the gold price was $370?

16-45

Speculating with Futures, Short

• Selling a futures contract (today) is often called “going short,” or establishing a short position.

• Recall: Each futures contract has an expiration date.

– Every day before expiration, a new futures price is established.

– If this new price is higher than the previous day’s price, the holder of a short futures contract position loses from this futures price increase.

– If this new price is lower than the previous day’s price, the holder of a short futures contract position profits from this futures price decrease.

16-46

Example II: Speculating in Gold Futures

• You believe the price of gold will go down. So,– You go short 100 futures contract that expires in 3 months.– The futures price today is $400 per ounce.– There are 100 ounces of gold in each futures contract.

• Your "position value" is: $400 X 100 X 100 = $4,000,000

• Suppose your belief is correct, and the price of gold is $370 when the futures contract expires.

• Your “position value” is now: $370 X 100 X 100 = $3,700,000

Your "short" speculation has resulted in a gain of $300,000

What would have happened if the gold price was $420?

16-47

Hedging with Futures

• A hedger trades futures contracts to transfer price risk.

• Hedgers transfer price risk by adding a futures contract position that is opposite of an existing position in the commodity or financial instrument.

• When the hedge is in place:– The futures contract “throws off” cash when cash is needed.– The futures contract “absorbs” cash when cash is available.

16-48

Hedging with Futures, Short Hedge

• A company has a large inventory that will be sold at a future date.

• So, the company will suffer losses if the value of the inventory falls.

• Suppose the company wants to protect the value of their inventory.– Selling futures contracts today offsets potential declines in the value of

the inventory.– The act of selling futures contracts to protect from falling prices is

called short hedging.

16-49

Example: Short Hedging with Futures Contracts

• Suppose Starbucks has an inventory of about 950,000 pounds of coffee, valued at $0.57 per pound.

• Starbucks fears that the price of coffee will fall in the short run, and wants to protect the value of its inventory.

• How best to do this? You know the following: – There is a coffee futures contract at the New York Board of Trade.– Each contract is for 37,500 pounds of coffee.– Coffee futures price with three month expiration is $0.58 per pound.– Selling futures contracts provides current inventory price protection.

• 25 futures contracts covers 937,500 pounds.• 26 futures contracts covers 975,000 pounds.

Real world decision!

16-50

Example: Short Hedging with Futures Contracts, Cont.

• Starbucks decides to sell 25 near-term futures contracts.

• Over the next month, the price of coffee falls. Starbucks sells its inventory for $0.51 per pound.

• The futures price also falls, to $0.52. (There are two months left in the futures contract)

• How did this short hedge perform?

• That is, how much protection did selling futures contracts provide to Starbucks?

16-51

The Short Hedge Performance

• The hedge was not perfect. • But, the short hedge “threw-off” cash ($56,250) when Starbucks needed

some cash to offset the decline in the value of their inventory ($57,000).

What would have happened if prices had increased by $0.06 instead?

Date

Starbucks

Coffee Inventory

Price

Starbucks Inventory

Value

Near-Term Coffee

Futures Price

Value of 25 Coffee

Futures Contracts

Now $0.57 $541,500 $0.58 $543,750

1-Month

From now

$0.51 $484,500 $0.52 $487,500

Gain (Loss) $0.06 $57,000 $0.06 $56,250

16-52

Hedging with Futures, Long Hedge

• A company needs to buy a commodity at a future date.

• The company will suffer “losses” if the price of the commodity increases before then. (That is, they paid more than the could have)

• Suppose the company wants to "fix" the price that they will pay for the commodity.– Buying futures contracts today offsets potential increases in the price of

the commodity.– The act of buying futures contracts to protect from rising prices is called

long hedging.

16-53

Example: Long Hedging with Futures Contracts

• Suppose Nestles plans to purchase 750 metric tons of cocoa next month.

• Nestles fears that the price of cocoa (which is $1,400 per ton) will increase before they acquire the cocoa.

• Nestles wants to “fix” the price it will pay for cocoa.

• How best to do this? You know the following: – There is a cocoa futures contract at the New York Board of Trade.– Each contract is for 10 metric tons of cocoa.– Cocoa futures price with three months to expiration is $1,440 per ton.– Buying futures contracts provides inventory “acquisition” price

protection.

• 75 futures contracts covers 750 metric tons.

16-54

Example: Long Hedging with Futures Contracts, Cont.

• Nestles decides to buy 75 near-term futures contracts.

• Over the next month, the price of cocoa increases, and Nestles pays $1,490 per ton for its cocoa.

• The futures price also increases, to $1,525. (There are two months left on the futures contract)

• How did this long hedge perform?

• That is, how much protection did buying futures contracts provide to Nestles?

16-55

The Long Hedge Performance

• The hedge was not perfect. But, the long hedge “threw-off” cash ($63,750) when Nestles needed some extra cash to offset the increase in the cost of their cocoa inventory acquisition ($67,500).

What would have happened if cocoa prices fell by $85-90 instead?

Date

Nestles

Cocoa Price

Nestles Inventory

Acquisition

Near-Term Cocoa

Futures Price

Value of 75 Cocoa

Futures Contracts

Now $1,400 $1,050,000 $1,440 $1,080,000

1-Month

From now

$1,490 $1,117,500 $1,525 $1,143,750

Gain (Loss) $90 $67,500 $85 $63,750

This is a reference price to show the difference in what will be paid.

16-56

Futures Trading Accounts

• A futures exchange, like a stock exchange, allows only exchange members to trade on the exchange.

• Exchange members may be firms or individuals trading for their own accounts, or they may be brokerage firms handling trades for customers.

16-57

Important Aspects of Futures Trading Accounts

• The important aspects about futures trading accounts:

Margin is required - initial margin as well as maintenance margin.

The contract values are marked to market on a daily basis, and a margin call will be issued if necessary.

A futures position can be closed out at any time. This is done by entering a reverse trade.

– In the hedging examples, Starbucks and Nestles entered reverse trades at the time they adjusted inventories.

– In those examples, the futures contracts had two months left before expiration.

16-58

Futures Trading Accounts, Cont.

• Initial Margin (which is a “good faith” deposit) is required when a futures position is first established.

• Initial Margin levels:– Depend on the price volatility of the underlying asset

– Can differ by type of trader

• When the price of the underlying asset changes, the futures exchange adds or subtracts money from trading accounts (this is called marking to market).– When the balance in the trading account gets "too low," it violates

maintenance margin levels, and a margin call is issued.

– A margin call is simply a stern request by the broker that more money be deposited into the trading account.

16-59

Example: Margin and Marking to Market

• Molly opens a trading account with C and J Brokerage.– Molly believes gold prices will increase.

– She will take a long position in gold futures.

– Molly knows that there are 100 ounces of gold in a gold futures contract.

• Her broker requires an initial margin of $1,000 per contract, and requires a maintenance margin of $750 per contract.

• Suppose Molly deposits $1,000 into her trading account.

Note: A reputable brokerage firm would actually require more, perhaps as much as $10,000.

16-60

Molly’s Trading Account for a Long Position in One Gold Futures Contract

• The futures exchange keeps a daily record that marks all trading accounts to market.

Day Deposits

Closing

Futures Price

Equity Value of

Account

Maint. Margin Level

Diff. Action

0 $1,000 $1,000 $750 +$250

Initial Margin Deposit

1 $400 $1,000 $750 +$250 Molly buys at close

2 $398 $800 $750 +$50

3 $394 $400 $750 -$350

Margin Call for

$600

4 $600 $394 $1,000 $750 +$250

16-61

Cash Prices

• The Cash price (or spot price) of a commodity or financial instrument is the price for immediate delivery.

• The Cash market (or spot market) is the market where commodities or financial instruments are traded for immediate delivery.

• In reality, "immediate" delivery can be 2 or 3 days later.

16-62

Quoted Cash Prices

16-63

Quoted Cash Prices

16-64

Cash-Futures Arbitrage

• Earning risk-free profits from an unusual difference between cash and futures prices is called cash-futures arbitrage.– In a competitive market, cash-futures arbitrage has very slim

profit margins.

• Cash prices and futures prices are seldom equal.

• The difference between the cash price and the futures price for a commodity is known as basis.

basis = cash price – futures price

16-65

Cash-Futures Arbitrage, Cont.

• For commodities with storage costs, the cash price is usually less than the futures price, i.e. basis < 0. This is referred to as a carrying-charge market.

• Sometimes, the cash price is greater than the futures price, i.e. basis > 0. This is referred to as an inverted market.

• Basis is kept at an economically appropriate level by arbitrage.

16-66

Spot-Futures Parity

• The relationship between spot prices and futures prices that must hold to prevent arbitrage opportunities is known as the spot-futures parity condition.

• The equation for the spot-futures parity relationship is:

• In the equation, F is the futures price, S is the spot price, r is the risk-free rate per period, and T is the number of periods before the futures contract expires.

TT r1SF

16-67

Spot Futures Parity with Dividends

• Spot futures parity is particularly important for stock index futures—and stocks pay dividends.

• If D represents a dividend paid at or near the end of the futures contract’s life, the spot-futures parity formula is:

• If there is dividend yield (d = D/S), the spot-futures parity formula is:

Dr1SF TT

TT dr1SF

16-68

Stock Index Futures

• There are a number of futures contracts on stock market indexes. Important ones include:– The S&P 500 – The Dow Jones Industrial Average

• Because of the difficulty of actual delivery, stock index futures are usually cash settled.– That is, when the futures contract expires, there is no delivery of

shares of stock.– Instead, the positions are "marked-to-market" for the last time,

and the contract no longer exists.

16-69

Single Stock Futures

• OneChicago began trading single stock futures in November 2002.

• OneChicago is a joint venture of the Chicago Board Options Exchange (CBOE), Chicago Mercantile Exchange, Inc. (CME), and the Chicago Board of Trade (CBOT).

• Single Stock Futures contracts are listed on 80 stocks – The underlying asset for the single stock futures is 100 Shares of

common stock.– Unlike stock indexes, shares are delivered at futures expiration.

• In addition, futures contracts on about 14 industry sectors are also listed (i.e., Aerospace, Semiconductors, Software, etc.)– Each “basket” contains 5 stocks.– At expiration, the futures are cash settled.

16-70

Index Arbitrage

• Index arbitrage refers to trading stock index futures and underlying stocks to exploit deviations from spot futures parity.

• Index arbitrage is often implemented as a program trading strategy. – Program trading accounts for about 15% of total trading volume

on the NYSE.– About 20% of all program trading involves stock-index arbitrage.

16-71

Index Arbitrage, Cont.

• Another phenomenon often associated with index arbitrage (and more generally, futures and options trading) is the triple witching hour effect.

• S&P 500 futures contracts and options, and various stock options, all expire on the third Friday of four particular months per year.

• The closing out of all the positions held sometimes lead to unusual price behavior.

16-72

Cross Hedging

• Cross-hedging refers to hedging a particular spot position with futures contracts on a related, but not identical, commodity or financial instrument.

• For example, you may decide to protect your stock portfolio from a fall in value by establishing a short hedge using S&P 500 stock index futures.– This is a “cross-hedge” if changes in your portfolio value do not

move in tandem with changes in the value of the S&P 500 index.

16-73

Hedging Stock Portfolios with Stock Index Futures

• There is a formula for calculating the number of stock index futures contracts needed to hedge a portfolio.

• Suppose the beta of a portfolio, P, is calculated using the S&P 500 Index as the benchmark portfolio.

– We need to know the dollar value of the portfolio to be hedged, VP.

– We need the dollar value of one S&P 500 futures contract, VF (which is 250 X S&P 500 index futures price)

• The formula:F

PP

V

Vβ contracts of Number

16-74

Example: Hedging a Stock Portfolio with Stock Index Futures

• You want to protect the value of a $200,000,000 portfolio over the near term (so, you will "short-hedge").– The beta of this portfolio is 1.45.– The S&P futures contract with 3-months to expiration has a

price of 1,050.

• How many futures contracts do you need to sell?

1,050250

00$200,000,01.451,105

V

Vβ contracts of Number

F

PP

S&P 500 Multiplier:Suppose the beta was 1.15 instead.

16-75

Another Portfolio to Cross-Hedge

• To protect a bond portfolio against changing interest rates, we may cross-hedge using futures contracts on U.S. Treasury notes.– It is called a “cross-hedge” if the value of the bond portfolio held

does not move in tandem with the value of U.S. Treasury notes.

• A short hedge will protect your bond portfolio against the risk of a general rise in interest rates during the life of the futures contracts.– Bond prices fall when interest rates rise.– Selling bond futures throws off cash when bond prices fall.

16-76

Hedging Bond Portfolios with T-note Futures

• There is a formula for calculating the number of T-note futures contracts needed to hedge a bond portfolio.

• Information needed for the formula:– Duration of the bond portfolio, DP

– Value of the bond portfolio, VP

– Duration of the futures contract, DF

– Value of a single futures contract, VF

• The formula is: FF

PP

VD

VD needed contracts of Number

16-77

Handy Estimate for the Duration of an Interest Rate Futures Contract

• Rule of Thumb Estimate: The duration of an interest rate futures contract, DF, is equal to:

– The duration of the underlying instrument, DU, plus

– The time remaining until contract maturity, MF.

• That is:

FUF MDD

16-78

Example: Hedging a Bond Portfolio with T-note Futures

• You want to protect the value of a $100,000,000 bond portfolio over the near term (so, you will "short-hedge").– The duration of this bond portfolio is 8.– Suppose the duration of the underlying T-note is 6.5, and the

futures contract has 0.5 years to expiration.– Also suppose the T-note futures price is 98 (which is 98% of the

$100,000 par value.)

• How many futures contracts do you need to sell?

$100,000)(0.980.5)(6.5

00$100,000,081,166

VD

VD needed contracts of Number

FF

PP

What happens if futures contracts with 3 months to expiration are used?

16-79

Futures Contract Delivery Options

• The cheapest-to-deliver option refers to the seller’s option to deliver the cheapest instrument when a futures contract allows several instruments for delivery.

• For example, U.S. Treasury note futures allow delivery of any Treasury note with a maturity between 6 1/2 and 10 years. Note that the cheapest-to-deliver note may vary over time.

• Those deeply involved in using T-note futures to hedge risk are keenly aware of these features.

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Useful Websites

• Futures Exchanges:www.cbot.com www.nymex.com www.cme.com www.kcbt.com www.nybot.com

• For Futures Prices and Price Charts:www.futuresworld.com futures.pcquote.com www.thefinancials.com

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Useful Websites, Cont.

• To Learn More about Futures:www.futurewisetrading.com www.usafutures.com

• To Learn About Single-Stock Futures:www.nqlx.com www.onechicago.com

• Other Links:www.investorlinks.com (for a list of futures brokers)www.programtrading.com (for information on program trading)

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Chapter Review, I.

• Futures Contracts Basics– Modern History of Futures Trading– Futures Contract Features– Futures Prices

• Why Futures?– Speculating with Futures– Hedging with Futures

• Futures Trading Accounts

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Chapter Review, II.

• Cash Prices versus Futures Prices– Cash Prices– Cash-Futures Arbitrage– Spot-Futures Parity– More on Spot-Futures Parity

• Stock Index Futures– Basics of Stock Index Futures– Index Arbitrage– Hedging Stock Market Risk with Futures– Hedging Interest Rate Risk with Futures– Futures Contract Delivery Options