financial derivatives how do futures and options markets

2006-4-14

北航 1

2006年春北航金融系李平 1

Financial Derivatives

Reference：John Hull著，张陶伟译，《期权期货入门》，第三版，中国人民大学出版社。


Aim of this course

How do futures and options markets run?How to use futures and options?What determine the prices of futures and options?


Chapter 1 Introduction

1. Derivatives

2. Futures Contracts

3. Forward Contracts

4. Options

5. Types of Traders

6. Other Derivatives


1. The Nature of Derivatives

A derivative is an instrument whose value depends on the values of other more basic underlying variables


Examples of Derivatives

• Futures Contracts• Forward Contracts• Swaps• Options


Ways Derivatives are Used

To hedge risksTo speculate (take a view on the future direction of the market)To lock in an arbitrage profitTo change the nature of a liabilityTo change the nature of an investment without incurring the costs of selling one portfolio and buying another

2006-4-14

北航 2


2. Futures Contracts

A futures contract is an agreement to buy or sell an asset at a certain time in the future for a certain priceBy contrast in a spot contract there is an agreement to buy or sell the asset immediately (or within a very short period of time)


Exchanges Trading Futures

Chicago Board of TradeChicago Mercantile ExchangeLIFFE (London)Eurex (Europe)BM&F (Sao Paulo, Brazil)TIFFE (Tokyo)and many more (see list at end of book)


Futures Price

The futures prices for a particular contract is the price at which you agree to buy or sellIt is determined by supply and demand in the same way as a spot price


Electronic Trading

Traditionally futures contracts have been traded using the open outcry system where traders physically meet on the floor of the exchangeIncreasingly this is being replaced by electronic trading where a computer matches buyers and sellers


Examples of Futures Contracts

Agreement to:buy 100 oz. of gold @ US$400/oz. in December (NYMEX) sell £62,500 @ 1.5000 US$/£ in March (CME)sell 1,000 bbl. of oil @ US$20/bbl. in April (NYMEX)


Terminology

The party that has agreed to buyhas a long positionThe party that has agreed to sellhas a short position

2006-4-14

北航 3


Example

January: an investor enters into a longfutures contract on COMEX to buy 100 oz of gold @ $300 in April

April: the price of gold $315 per oz

What is the investor’s profit?


Over-the-Counter Markets

The over-the-counter market is an important alternative to exchangesIt is a telephone and computer-linked network of dealers who do not physically meetTrades are usually between financial institutions, corporate treasurers, and fund managers


Size of OTC and Exchange Markets

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

0

50

100

150

200

Jun-98 Jun-99 Jun-00 Jun-01 Jun-02 Jun-03

Size of Market ($ trillion)

OTCExchange


3. Forward Contracts

Forward contracts are similar to futures except that they trade in the over-the-countermarketForward contracts are popular on currencies and interest rates


Foreign Exchange Quotes for GBP

1.61001.60946-month forward



1.62851.6281SpotOfferBid


4. Options

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

2006-4-14

北航 4


American vs European Options

An American option can be exercised at any time during its lifeA European option can be exercised only at maturity


Intel Option Prices (May 29, 2003; Stock Price=20.83)

2.852.201.851.150.450.2022.50

1.500.850.452.401.601.2520.00

Oct Put

July Put

June Put

Oct Call

July Call

June Call

Strike Price


Exchanges Trading Options

Chicago Board Options ExchangeAmerican Stock ExchangePhiladelphia Stock ExchangePacific ExchangeLIFFE (London)Eurex (Europe)and many more (see list at end of book)


Options vs Futures/Forwards

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


5. Types of Derivative Traders

• Hedgers: use derivatives to reduce the riskthat they face from potential future movements in a market variable• Speculators: use derivatives to bet on the future direction of a market variable• Arbitrageurs: lock in a riskless profit by simultaneously entering into two or more transactions


Hedging Examples (1)

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 contractThe price is locked, but the outcome may be worse

2006-4-14

北航 5


Hedging Examples (2)

An investor owns 1,000 Microsoft shares currently worth $28 per share. A two-month put with a strike price of $27.5 costs $1. The investor decides to hedge by buying 10 contractsThe difference between the use of forward and options for hedging:

Forward: fix the priceOption: provide insurance


Value of Microsoft Shares with and without Hedging

20,000

25,000

30,000

35,000

40,000

20 25 30 35 40

Stock Price ($)

Value of Holding ($)

No HedgingHedging


Speculation Example

An investor with $4,000 to invest feels that Cisco’s 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 25 is $1


Speculation Example (cont.)

Two possible alternative strategies: buy calls and shares. The use of futures and options for speculation:

Both obtain leverageThe potential loss and gain are different


Arbitrage Example

A stock price is quoted as £100 in London and $172 in New YorkThe current exchange rate is 1.7500What is the arbitrage opportunity?Arbitrage opportunities can’t last for long.


1) An Arbitrage Opportunity?

Suppose that:The spot price of gold is US$390The quoted 1-year futures price of gold is US$425The 1-year US$ interest rate is 5% per annumNo income or storage costs for gold

Is there an arbitrage opportunity?

2006-4-14

北航 6


2) Another Arbitrage opportunity?

Suppose that:The spot price of gold is US$390The quoted 1-year futures price of gold is US$390The 1-year US$ interest rate is 5% per annumNo income or storage costs for gold



The Futures Price of Gold

If the spot price of gold is S, the price for a futures contract deliverable in T years is F, then

F = S (1+r )T

where r is the 1-year (domestic currency) risk-free rate of interest.In our examples, S=390, T=1, and r=0.05 so that

F = 390(1+0.05) = 409.50


6. Other Derivatives

Plain vanilla/ standard derivativesExoticsCredit derivatives: creditworthiness of a

companyWeather derivatives: average temperatureInsurance derivatives: dollar value of

insurance claimElectricity derivatives: spot price of electricity


Mechanics of Futures Markets

Chapter 2


Futures Contracts

Available on a wide range of underlyingsExchange tradedSpecifications need to be defined:

What can be delivered,Where it can be delivered, & When it can be delivered

Settled daily


Price and Position Limits

Many futures exchanges set limits on daily price changes and holdings. Limits are set to prevent excessive volatility and market manipulation. Limits are often removed in the last month of the contract.

2006-4-14

北航 7


Margins

A margin is cash or marketable securities deposited by an investor with his or her brokerThe balance in the margin account is adjusted to reflect daily settlementMargins minimize the possibility of a loss through a default on a contract


Margin Requirement (cont.)

清算所(clearing house): track all the transactions to calculate the positions经纪人也需在清算所存入保证金(clearing margin) 。但数额小于等于客户交给经纪人的保证金

变动保证金必须以现金支付，初识保证金中的一

部分可以以生息债券存入。


Margin Requirement

Initial Margin - funds deposited to provide capital to absorb losses, generally 5%-15%.Maintenance Margin - an established value below which a trader’s margin may not fall.Marking to marketWhen the maintenance margin is reached, the trader will receive a margin call from her broker to add variation margin to reach the level of initial margin.


An 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$400margin requirement is US$2,000/contract (US$4,000 in total)maintenance margin is US$1,500/contract (US$3,000 in total)


A Possible Outcome

Daily Cumulative MarginFutures Gain Gain Account MarginPrice (Loss) (Loss) Balance Call

Day (US$) (US$) (US$) (US$) (US$)

400.00 4,000

5-Jun 397.00 (600) (600) 3,400 0. . . . . .. . . . . .. . . . . .

13-Jun 393.30 (420) (1,340) 2,660 1,340. . . . . .. . . . .. . . . . .

19-Jun 387.00 (1,140) (2,600) 2,740 1,260. . . . . .. . . . . .. . . . . .

26-Jun 392.30 260 (1,540) 5,060 0

+

= 4,000

3,000+

= 4,000

<


Example

An investor enters into two long futures contracts on frozen orange juice. Each contract is for the delivery of 15,000 pounds. The current futures price is 160 cents per pound, the initial margin is $6,000 per contract, and the maintenance margin is $4,500 per contract. What price change would lead to a margin call? Under what circumstances could $2,000 be withdrawn from the margin account?

Falls by 10 cents and rises by 6.67 cents

2006-4-14

北航 8


Other Key Points About Futures

They are settled dailyClosing out a futures position involves entering into an offsetting tradeMost contracts are closed out before maturity


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


Newspaper quotes

Open interest: the total number of contracts outstanding

equal to number of long positions or number of short positionsOne trading older

Settlement price: the price just before the final bell each day

used for the daily settlement processVolume of trading: the number of trades in 1 day


Convergence of Futures to Spot

Time Time

(a) (b)

FuturesPrice

FuturesPrice

Spot Price

Spot Price


Regulation of Futures

Regulation is designed to protect the public interestRegulators try to prevent questionable trading practices by either individuals on the floor of the exchange or outside groups


Forward Contracts

A forward contract is an OTC agreement to buy or sell an asset at a certain time in the future for a certain priceThere is no daily settlement. At the end of the life of the contract one party buys the asset for the agreed price from the other party

2006-4-14

北航 9


Profit from a Long Forward or Futures Position

Profit

Price of Underlyingat Maturity


Profit from a Short Forward or Futures Position

Profit

Price of Underlyingat Maturity


Forward Contracts vs Futures Contracts

Private contract between 2 parties Exchange traded

Not standardized Standard contract

Usually 1 specified delivery date Range of delivery dates

Settled at maturity Settled daily

Delivery or final cash settlement usually occurs

Contract usually closed out prior to maturity

FORWARDS FUTURES

Some credit risk Virtually no credit risk


Foreign Exchange Quotes

Futures exchange rates are quoted as the number of USD per unit of the foreign currencyForward exchange rates are quoted in the same way as spot exchange rates. This means that GBP, EUR, AUD, and NZD are USD per unit of foreign currency. Other currencies (e.g., CAD and JPY) are quoted as units of the foreign currency per USD.


Chapter 3

Determination of Forward and Futures

Prices


Measuring Interest Rates

The compounding frequency used for an interest rate is the unit of measurementIn the limit as we compound more and more frequently we obtain continuously compounded interest rates$100 grows to $100eRT when invested at a continuously compounded rate R for time T$100 received at time T discounts to $100e-RT at time zero when the continuously compounded discount rate is R

2006-4-14

北航 10


Conversion FormulasDefineRc : continuously compounded rateRm: same rate with compounding m times per

year

( )R m

Rm

R m e

cm

mR mc

= +

= −

ln

/

1

1


Example

1. Consider an interest rate that is quoted as 10% per annum with semiannual compounding. What is the equivalent rate with continuous compounding?

09758.0)2/1.01ln(2 =+


Example (cont.)

2. A deposit account pays 12% per annum with continuous compounding, but interest is actually paid quarterly. How much interest will be paid each quarter on a $10,000 deposit?

10000×0.1218/4=304.55


Forward vs Futures Prices

Forward and futures prices are usually assumed to be the same. When interest rates are uncertain they are, in theory, slightly different.


Consumption Assets vs Investment

Investment assets are assets held by significant numbers of people purely for investment purposes (Examples: gold, silver)Consumption assets are assets held primarily for consumption (Examples: copper, oil)


Short Selling

Short selling involves selling securities you do not ownYour broker borrows the securities from another client and sells them in the market in the usual way

2006-4-14

北航 11


Short Selling (continued)

At some stage you must buy the securities back so they can be replaced in the account of the clientYou must pay dividends and other benefits the owner of the securities receives


Notation

S0: Spot price todayF0: Futures or forward price todayT: Time until delivery dater: Risk-free interest rate for maturity T


Gold: An Arbitrage Opportunity?

Suppose that:The spot price of gold is US$390The quoted 1-year forward price of gold is US$425The 1-year US$ interest rate is 5% per annumNo income or storage costs for gold



Gold: Another Arbitrage Opportunity?

Suppose that:The spot price of gold is US$390The quoted 1-year forward price of gold is US$390The 1-year US$ interest rate is 5% per annumNo income or storage costs for gold



The Forward/Futures Price of Gold

If the spot price of gold is S & the price for a futures contract deliverable in T years is F, then

F = S (1+r )T

where r is the 1-year (domestic currency) risk-free rate of interest.In our examples, S=390, T=1, and r=0.05 so that

F = 390(1+0.05) = 409.50


For investment asset

For any investment asset that provides no income and has no storage costs

F0 = S0(1 + r )T

If r is compounded continuously

F0 = S0erT

2006-4-14

北航 12


For Investment Asset Providing Known Cash Income

stocks paying known dividendscoupon bond

When an Investment Asset Provides a Known Dollar Income

F0 = (S0 – I )erT

where I is the present value of the income


For investment asset (cont.)

When an Investment Asset Provides a Known Yield

F0 = S0 e(r–q )T

where q is the average yield during the life of the contract (expressed with continuous compounding)


Example

1. Consider a 10-month forward contract on a stock with a price of $50. The risk-free interest rate (continuous compounded) is 8% per annum for all maturities. Assume that dividends of $0.75 per share are expected after three months, six months and nine months. What is the forward price?

51.14


Example (cont.)

2. Consider a six-month futures contract on an asset that is expected to provide income equal to 2% of the asset price once during the six-month period. The risk-free rate of interest (continuous compounded) is 10% per annum. The asset price is $25. What is the futures price?

25e(0.1-0.0396)/2=25.77


Stock Index

S&P500, $250Nikkei 225, $5Nasdaq100, $100Dow Jones(30), $10DAX-30(Germany)FT-SE100(London)


Stock Index (cont.)

Can be viewed as an investment asset paying a dividend yieldThe investment asset is the portfolio of stocks underlying the indexThe dividends paid are the dividends that would be received by the holder of the portfolioIt is usually assumed that the dividends provide a known yield rather than a known cash income

2006-4-14

北航 13


Pricing for Stock Index Futures

The futures price and spot price relationship is therefore

F0 = S0 e(r–q )T

where q is the dividend yield on the portfolio represented by the index


For Stock Index Futures (cont.)

In practice, the dividend yield on the portfolio underlying the index variesweek by week throughout the year. The chosen value of q should represent the average annualized dividend yieldduring the life of the contract.


ExampleThe risk-free interest rate is 9% per annum with continuously compounding The dividend on the stock index varies throughout the year. In February, May, August and November, dividends are paid at a rate of 5% per annum. In other months, dividends are paid at a rate of 2% per annum. The value of the index on July 31, 2002 is 300. What is the futures price for a contract deliverable on December 31, 2002?

307.34


Index arbitrage

If F0 > S0 e(r–q )T , profits can be made by buying the stocks underlying the index and shorting futures contract;If F0 < S0 e(r–q )T , profits can be made by shorting or selling the stocks underlying the index and taking a long position in futures contract.


Forward and Futures on Currencies

A foreign currency is analogous to a security providing a dividend yieldThe continuous dividend yield is the foreign risk-free interest rateIt follows that if rf is the foreign risk-free interest rate

F S e r r Tf0 0= −( )


Why the Relation Must Be True1000 units of

foreign currency at time zero

units of foreign currency at time T

Trfe1000

dollars at time T

TrfeF01000

1000S0 dollars at time zero

dollars at time T

rTeS01000

1000 units of foreign currency

at time zero

units of foreign currency at time T

Trfe1000

dollars at time T

TrfeF01000

1000S0 dollars at time zero

dollars at time T

rTeS01000

2006-4-14

北航 14


Example

Suppose that the two-year interest rate in Australia and the United States are 5% and 7%, respectively, The spot exchange rate between the Australian dollar and the US dollar is US$0.62/AUD. What is two-year forward exchange rate?

0.6453


Futures on Commodities

Storage cost for investment asset is regarded as negative income, so

F0 = (S0+U )erT

where U is the present value of the storage costs.Alternatively,

F0 = S0 e(r+u )T

where u is the storage cost per annum as a percent of the spot price.


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 rate is 7% per annum with continuous compounding. Then the futures price is

484.63


Consumption Assets

One keep the commodity for consumption, so he won’t sell the commodity and buy futures, which influence the arbitrage argument.

F0 ≤ S0 e(r+u )T

or

F0 ≤ (S0+U )erT


The convenience yieldThe convenience yield on the consumption asset

ability to profit from temporary local shortagesability to immediately keep a production process running

The convenience yield, y, is defined so that

F0 eyT= S0 e(r+u )T

Or F0 = S0 e(r+u-y )T


Cost of Carry

Cost of carry refers to the cost and benefit of holding the asset, including:

interest rate paid to finance the assetstorage costsdividends or other income

2006-4-14

北航 15


Cost-of-Carry (cont.)

Non-divident-paying stock (no storage cost and no income): c =r

Stock index: c =r-q

Currency: c =r-rf

Commodities: c =r+u


Cost-of-Carry and futures price

For an investment asset

F0 = S0ecT

For a consumption asset

F0 = S0e(c-y)T


Futures Prices & Expected Future Spot Prices

Suppose k is the expected return required by investors on an assetWe can invest F0e–r T now to get ST back at maturity of the futures contractThis shows that

F0 = E (ST )e(r–k )T


Valuing a Forward ContractSuppose that K is delivery price in a forward contract F0 is forward price that would apply to the

contract todayThe value of a long forward contract, ƒ, is

ƒ = (F0 – K )e–rT

Similarly, the value of a short forward contract is

(K – F0 )e–rT


Example

A long forward contract on a non-dividend paying stock was entered into some time ago. It currently has six months to maturity. The risk-free interest rate (with continuous compounding) is 10% per annum, the stock price is $25 and delivery price is $24. What is the value of the forward contract?

$2.172006年春北航金融系李平 90

Chapter 4

Hedging Strategies Using Futures

2006-4-14

北航 16


long futures hedge: involves a long position in futures, appropriate when you know you will purchase an asset in the future and want to lock in the priceshort futures hedge: involves a shortposition in futures, appropriate when you know you will sell an asset in the future & want to lock in the price

Long & Short Hedges


Example of short hedge

On May 15, X has contracted to sell 1 millionbarrels of oil on August 15 at the spot price of that dayMay 15 quotes:S1= $19.00 /barrel, F1= $18.75 /barrelHedging actions:Contract size: 1000 barrelsOn May 15, short 1000 August oil futuresOn August 15, close out futures position


Example (cont.)

August 15: S2= F2=$17.50 /barrel, X receives $17.50 per barrel per contractGains from futures=F1-F2=$(18.75 - 17.50) = $1.25 per barrel Price realized=$17.50+ $1.25=$18.75= F1+( S2- F2)Alternatively if S2=$19.50 /barrel


Basis Risk

Basis is the difference between spot & futures pricesBasis risk arises because of the uncertainty about the basis when the hedge is closed out

The asset to be hedged may not be the same as the asset underlying the futuresThe hedger is uncertain about the precise date of buying or selling the asset


Choice of Contract

Choose a delivery month that is as close as possible to, but later than, the end of the life of the hedgeWhen there is no futures contract on the asset being hedged, choose the contract whose futures price is most highly correlated with the asset price.


Long Hedge

Suppose thatF1 : Initial Futures PriceF2 : Final Futures PriceS2 : Final Asset Priceb2 : Basis at time t2

You hedge the future purchase of an asset by entering into a long futures contractCost of Asset=S2 –(F2 – F1) = F1 + Basis

2006-4-14

北航 17


Example 2

It is June 8 and a company knows that it will need to purchase 20,000 barrels of crude oil at some time in October or November. Oil futures contracts are currently traded for delivery every month on NYMEX and the contract size is 1,000 barrels. The company therefore decides to take a longposition in 20 December contracts for hedging (Assuming that the hedge ratio is 1).


Example 2 (cont.)

The futures price on June 8 is F1=$18 /barrel. The company finds that it is ready to purchase the crude oil on November 10. It therefore closes out its futures contract on that date. The pot price and futures price on November 10 are S2=$20 and F2=$ 19.10 per barrel.The gain on the futures contract is 19.10-18=$1.10 per barrel. The effective price paid is 20-1.10=$18.90


Short Hedge

Suppose thatF1 : Initial Futures PriceF2 : Final Futures PriceS2 : Final Asset Price

You hedge the future sale of an asset by entering into a short futures contractPayoff Realized =S2+ (F1 –F2) = F1 + Basis


Basis riskSince F1 is known at t1, hedging risk is the basis risk b2

when asset to be hedged is different from asset underlying futures Effective price at t2 is(S2 + F1 - F2) = F1 +(S2

* - F2) + (S2 - S2*)

where S2* is the spot price of the asset

underlying the futures contractThe term (S2 - S2

*) arises due to the difference between the two assets


Optimal Hedge RatioHedge ratio: the ratio of the position taken in futures contract to the size of the exposureOptimal hedge ratio: proportion of the exposure that should optimally be hedged is (hedgeratio)

where σS and σF are the standard deviations of ∆S and ∆F, the change in the spot price and futures price during the hedging period, ρ is the coefficient of correlation between ∆S and ∆F.

h S

F

* = ρσσ


Example 2 (cont.)

If the company decides to use a hedge ratio of 0.8, how does the decision affect the way in which the hedge is implemented and the result?If the hedge ratio is 0.8, the company takes a long position in 16 NYM December oil futures contracts on June 8 and closes out its position on November 10.

2006-4-14

北航 18


Example 2 (cont.)

The gain on the futures position is(19.10-18)×16,000=17,600

The effective cost of the oil is therefore 20,000×20-17,600=382,400

or $19.12 per barrel. (This compares with $18.90 per barrel when

the hedge ratio is 1.)


Hedging Using Index Futures

To hedge the risk in a portfolio the numberof index futures contracts that should be used is

where P is the value of the portfolio, β is its beta, and A is the value of the index underlying one futures contract

βPA


Example 3

Value of S&P 500 is 1,000Value of Portfolio is $5 millionBeta of portfolio is 1.5

What position in futures contracts on the S&P 500 is necessary to hedge the portfolio?


Example 3 (cont.)

Dollar value of S&P 500 underlying the futures contract:

No. of futures contract

$250 1000 $250,000× =

5,000,0001.5 30250,000

=


Chapter 6

Swaps


Outline A swap is an agreement to exchange cash flows at specified future times according to certain specified rulesContents:

How swaps are defined How they are be used How they can be valued

Two plain vanilla swap: Interest-rate swap, fixed-for-fixed currency swap

2006-4-14

北航 19


1. An Example of a “Plain Vanilla”Interest Rate Swap

An agreement by Microsoft to receive6-month LIBOR & pay a fixed rate of 5% per annum every 6 months for 3 years on a notional principal of $100 millionNext slide illustrates cash flows


---------Millions of Dollars---------LIBOR FLOATING FIXED Net

Date Rate Cash Flow Cash Flow Cash FlowMar.5, 2001 4.2%

Sept. 5, 2001 4.8% +2.10 –2.50 –0.40Mar.5, 2002 5.3% +2.40 –2.50 –0.10

Sept. 5, 2002 5.5% +2.65 –2.50 +0.15Mar.5, 2003 5.6% +2.75 –2.50 +0.25

Sept. 5, 2003 5.9% +2.80 –2.50 +0.30Mar.5, 2004 6.4% +2.95 –2.50 +0.45

Cash Flows to Microsoft


Typical Uses of an Interest Rate Swap

Converting a liability from

fixed rate to floating rate floating rate to fixed rate

Converting an investment from

fixed rate to floating ratefloating rate to fixed rate


Intel and Microsoft (MS) Transform a Liability

Intel MS

LIBOR

5%

LIBOR+0.1%

5.2%


Financial Institution is Involved

F.I.

LIBOR LIBORLIBOR+0.1%

4.985% 5.015%

5.2%Intel MS


Intel and Microsoft (MS) Transform an Asset

Intel MS

LIBOR

5%

LIBOR-0.25%

4.7%

2006-4-14

北航 20


Financial Institution is Involved

Intel F.I. MS

LIBOR LIBOR

4.7%

5.015%4.985%

LIBOR-0.25%


Quotes By a Swap Market Maker

6.8506.876.8310 years

6.6656.686.657 years

6.4906.516.475 years

6.3706.396.354 years

6.2256.246.213 years

6.0456.066.032 years

Swap Rate (%)Offer (%)Bid (%)Maturity


The Comparative Advantage Argument

AAACorp wants to borrow floatingBBBCorp wants to borrow fixed

Fixed Floating

AAACorp 10.00% 6-month LIBOR + 0.30%

BBBCorp 11.20% 6-month LIBOR + 1.00%


The Swap

AAA BBB

LIBOR

LIBOR+1%

9.95%

10%


The Swap when a Financial Institution is Involved

AAA F.I. BBB10%

LIBOR LIBOR

LIBOR+1%

9.93% 9.97%


Reason for the Comparative Advantage

The 10.0% and 11.2% rates available to AAACorpand BBBCorp in fixed rate markets are 5-year ratesThe LIBOR+0.3% and LIBOR+1% rates available in the floating rate market are six-month ratesThe spread reflects the probability of default

2006-4-14

北航 21


Example 1Companies A and B have been offered the following rates per annum on a $20 million five-year loan:

fixed rate floating rate Company A 12.0% LIBOR+0.1%Company B 13.4% LIBOR+0.6%

Company A requires a floating-rate loan; company B requires a fixed-rate loan. Design a swap that will net a bank, acting as intermediary, 0.1% per annum and that will appear equally attractive to both companies.


Valuation of an Interest Rate Swap

A swap is worth zero to a company initially. This means that it costs nothing to enter into a swapAt a future time its value is liable to be either positive or negative


Valuation of an Interest Rate Swap (cont.)

Interest rate swaps can be valued as the difference between the value of a fixed-rate bond and the value of a floating-rate bondAlternatively, they can be valued as a portfolio of forward rate agreements (FRAs)


Valuation in Terms of Bonds

Vswap=Bfl-Bfix (or, Bfix-Bfl)The fixed rate bond is valued in the usual wayThe floating rate bond is valued by noting that it is worth par immediately after the next payment date


Example 2

Suppose that a financial institution pays 6-month LIBOR and receives 8% per annum (with semiannual compounding) on a swap with a notional principle of $100 and the remaining payment dates are in 3, 9 and 15 months. The swap has a remaining life of 15months. The LIBOR rates with continuous compounding for 3-month, 9-month and 15-month maturities are 10%, 10.5% and 11%, respectively. The 6-month LIBOR rate at the last payment date was 10.2% (with semiannual compounding).


2. An Example of a fixed-for-fixed Currency Swap

An agreement to pay 11% on a sterlingprincipal of £10,000,000 & receive 8% on a US$ principal of $15,000,000 every year for 5 years

2006-4-14

北航 22


Exchange of Principal

In an interest rate swap the principal is not exchangedIn a currency swap the principal is exchanged at the beginning and the end of the swap


The Cash Flows

Year

Dollars Pounds$

------millions------2001 –15.00 +10.002002 +1.20 –1.102003 +1.20 –1.102004 +1.20 –1.102005 +1.20 –1.102006 +16.20 -11.10

£


Typical Uses of a Currency Swap

Conversion from a liability in one currency to a liability in another currency

Conversion from an investment in one currency to an investment in another currency


Comparative Advantage Arguments for Currency Swaps

General Motors wants to borrow AUDQantas wants to borrow USD

USD AUD

General Motors 5.0% 12.6%Qantas 7.0% 13.0%


Company X wishes to borrow U.S. dollars at a fixed rate of interest and company Y wishes to borrow Japanese Yen at a fixed rate of interest. The companies have been quoted the following interest rates.

Yen DollarsCompany X 5.0% 9.6%Company Y 6.5% 10.0%

Design a swap that will net a bank, acting as intermediary, 50bp per annum and make the swap equally attractive to the two companies.

Example 3


Valuation of Currency Swaps

Like interest rate swaps, currency swaps can be valued either as the difference between 2 bonds or as a portfolio of forward contractsValuation in Terms of Bonds:

Vswap=BD-S0 BF (or, S0BF -BD )

2006-4-14

北航 23


Example 4

Suppose that the term structure of interest rates is flat in both Japan and United States. The Japanese rate is 4% per annum and the U.S. rate is 9% per annum (both with continuous compounding). A financial institution enters into a currency swap in which it receives 5% per annum in yen and pays 8% per annum in dollars once a year. The principles in the two currencies are $10 million and 1,200 million yen. The swap will last for another three years and the current exchange rate is 110yen=$1.


Swaps & Forwards

A swap can be regarded as a convenient way of packaging forward contractsThe “fixed for fixed” currency swap in our example consisted of a cash transaction & 5 forward contractsThe value of the swap is the sum of the values of the forward contracts underlying the swap


Credit Risk

A swap is worth zero to a company initiallyAt a future time its value is liable to be either positive or negativeSwaps are private agreements between two partiesThe company has credit risk exposure only when its value is positive


Chapter 7

Mechanics of Options Markets


Types of Options

A call is an option to buyA put is an option to sellA European option can be exercised only at the end of its lifeAn American option can be exercised at any time


Option Positions

Long callLong putShort callShort put

2006-4-14

北航 24


Long Call on eBayProfit from buying one eBay European call option: option

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

30

20

10

0-5

70 80 90 100

110 120 130

Profit ($)

Terminalstock price ($)


Short Call on eBay Profit from writing one eBay European call option: option

price = $5, strike price = $100

-30

-20

-10

05

70 80 90 100

110 120 130

Profit ($)



Long Put on IBM Profit from buying an Oracle European put option: option

price = $7, strike price = $70

30

20

10

0-7

70605040 80 90 100

Profit ($)



Short Put on IBM

Profit from writing an IBM European put option: option price = $7, strike price = $70

-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

Payoff Payoff

ST STKK

Payoff Payoff

ST STKK


Types of options (cont.)

Stock options: Exchange-traded options in U.S. are American

Index options：traded on CBOEAn option is to buy or sell 100 times the index valueOptions on S&P500 are European Options on S&P100 are AmericanSettled in cash

2006-4-14

北航 25


Futures option 期货到期日比期权到期日稍晚

和期货合约在同一交易所交易

When a call is exercised, the holder get a long position in the underlying futures plus a cash amount equal to the excess of the futures price over the strike price



Foreign currency optionTraded on Philadelphia Stock Exchange以外币的本币价格表示：如英镑买入期权的价格为$ 0.035/£The contract size depends on the currency, such as £, it is £31250



Specification ofExchange-Traded Options

Expiration dateStrike priceEuropean or AmericanCall or Put (option class)


Terminology

Moneyness :At-the-money optionIn-the-money optionOut-of-the-money option


Terminology (continued)

Option class (call or put)Option series: options of a given class with the same expiration date and strike price

Intrinsic valueTime value


Dividends & Stock Splits

Suppose you own N options with a strike price of K :

No adjustments are made to the option terms for cash dividendsWhen there is an n-for-m stock split,

the strike price is reduced to mK/nthe no. of options is increased to nN/m

Stock dividends are handled in a manner similar to stock splits

2006-4-14

北航 26


Dividends & Stock Splits(continued)

Consider a call option to buy 100 shares for $20/shareHow should terms be adjusted:

for a 2-for-1 stock split?for a 25% stock dividend?

200 share, $10/share125 share, $16/share


Position limits and Exercise limits

Position limits: the maximum number of option contracts that an investor can hold on one side of the market (long call and short put are considered to be on the same side of the market)Exercise limits: the maximum number of contracts that can be exercised by any investor in any period of five consecutive trading days


Newspaper quotes Market Makers

Most exchanges use market makers to facilitate options tradingA market maker quotes both bid and ask prices when requestedThe market maker does not know whether the individual requesting the quotes wants to buy or sell


Margins

Margins are required when options are soldWhen a naked call (put) option is written the margin is the greater of:1 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

2 A total of 100% of the proceeds of the sale plus 10% of the underlying share price (exercise price)

When writing covered calls, no margin is required


Margins (cont.)

Example: An investor writes four naked call options

on a stock. The option price is $5, the strike price is $40, and the stock price is $38. What is the margin requirement?

$42402006年春北航金融系李平 156

Warrants, Executive Stock Options and convertible bonds

They are call options that are written by a company on its own stockWhen they are exercised, the company issues more of its own stock and sells them to the option holder for the strike priceThe exercise leads to an increase in the number of the company’s stock outstanding

2006-4-14

北航 27


Warrants

Warrants are call options coming into existence as a result of a bond issueThey are added to the bond issue to make the bond more attractive to investorsOnce they are created, they sometimes trade separately from the bonds宝钢权证


Executive Stock Options

Call option issued by a company to executives to motivate them to act in the best interests of the company’s shareholdersUsually at-the-money when issuedCan’t be tradedOften last for 10 or 15 years


Convertible Bonds

Convertible bonds are regular bonds that can be exchanged for equity at certain times in the future according to a predetermined exchange ratioIs a bond with an embedded call option on the company’s stock


Chapter 8

Properties ofStock Option Prices


Outline

The relationship between the option price and the underlying stock price (by arbitrage argument)Whether an American option should be exercised early


Notation

c : European call option pricep : European put option priceS0 : Stock price todayK : Strike priceT : Life of option σ: Volatility of stock price

C : American Call option priceP : American Put option priceST :Stock price at option maturityD : Present value of dividends during option’s lifer : Risk-free rate for maturity T with cont. comp.

2006-4-14

北航 28


Effect of Variables on Option Pricing

c p C PVariableS0KTσrD

+ + –+

? ? + ++ + + ++ – + –

–– – +

– + – +2006年春北航金融系李平 164

American vs European Options

An American option is worth at least as much as the corresponding European option

C ≥ cP ≥ p

) ,max() ,max(

SKpPKScC

−=−=


Upper bounds for option prices

,

, 00

KPKep

SCScrT ≤≤

≤≤−


Lower bound for European calls on non-dividend-paying stocks

Portfolio A: one European call & an amount of cash equal to Ke-rT

Portfolio B: one share

c(t) ≥ max(S(t) –Ke –rT, 0)


Calls: An Arbitrage opportunity?

Suppose that c(t) = 3 S(t) = 20 T = 1 r = 10% K = 18 D = 0Is there an arbitrage opportunity?

S(t) –Ke –rT=3.71>3=c，buy call, short stock. If the inflow ($17) is invested for one year at 10% per Annum, it will be $18.79.


Lower bound for European puts on non-dividend-paying stocks

Portfolio A: one European put & one share Portfolio B: an amount of cash equal to Ke-rt

p(t) ≥max( Ke-rT–S(t),0)

2006-4-14

北航 29


Puts: An Arbitrage opportunity?Suppose that

p(t)= 1 S(t) = 37 T = 0.5 r =5% K = 40 D = 0


Ke-rT–S(t)=2.01>1=p, 借$38，为期6个月，用借款购买卖权和股票， 6个月后借款为$38.96。


Put-call parity for non-dividend-paying stocks

Portfolio A: One European call on a stock + an amount of cash equal to Ke-rT

Portfolio B: One European put on the stock + one shareBoth are worth MAX(ST , K ) at the maturity of the optionsThey must therefore be worth the same today

c(t) + Ke -rT = p(t) + S(t)


Arbitrage Opportunities

Suppose that c(t)= 3 S(t)= 31 T = 0.25 (3-m) r = 10% K =30 D = 0What are the arbitrage possibilities when

p(t) = 2.25 ? p(t) = 1 ?


When p(t)=2.25

c(t)+Ke-rt=32.26, p(t)+S(t)=33.25Portfolio B is overpriced. The arbitrage strategy: buy the call, short both the stock and the put.Generating a positive cash flow of 2.25+31-3=30.25After three months, this amount grows to 31.02


If S(T)>K, exercise the callIf S(T)≤ K, the put is exercisedIn either case, the investor ends up buying one share for $30 to close the short position.The net profit: 31.02-30 continue

Continued


When p(t)=1

c(t)+Ke-rt=32.26, p(t)+S(t)=32Portfolio A is overpriced. The arbitrage strategy: short the call, buy both the stock and the put. Initial investment:

1+31-3=$29

2006-4-14

北航 30


The initial investment is financed at 10%. A repayment of $29.73 is required at the end of three months.Either the call or put is exercised, the stock will be sold for $30.The net profit: 30-29.73

Continued


Early Exercise for American options

Usually there is some chance that an American option will be exercised earlyAn exception is an American call on a non-dividend paying stockThis should never be exercised early


For an American call option:S(t) = 50; T = 1m; K = 40; D = 0

Should you exercise immediately?What should you do if

1. You want to hold the stock for one month? 2. You do not feel that the stock is worth

holding for the next 1 month?

An Extreme Situation


Reasons For Not Exercising a Call Early--No Dividends

Case 1: should keep the option and exercise it at the end of the month.

We delay paying the strike price, earn the interestNo income is sacrificed (no dividend)Holding the call provides insuranceagainst stock price falling below strike price


Reasons For Not Exercising a Call Early--No Dividends (cont.)

Case 2: Better action: sell the optionThe option will be bought by another investor who does want to hold the stock.Such investors must exist, otherwise the current stock price would not be $50.The price obtained for the option will be greater than its intrinsic value of $10.


More formal argument

C ≥ c ≥ S0–Ke –rT>S0–KIf it is optimal to exercise early,

C=S0–K

2006-4-14

北航 31


Should Puts Be Exercised Early?

A put option should be exercised early if it is deep in the money.An extreme case:

S(t)= 0; K = $10; D = 0The profit of exercise now: $10, and can also get interest.If wait, the profit will be less than 10.


The Impact of Dividends on Lower Bounds

c S D Kep D Ke S

rT

rT

≥ − −

≥ + −

−

−

0

0

Portfolio A: one European call & an amount of cash equal to Ke-rT+DPortfolio B: one share


Impact on Put-Call Parity

European options; D > 0c + D + Ke -rT = p + S0

American options; D = 0

American options; D > 0

rTKeSPCKS −−≤−≤− 00

rTKeSPCKDS −−≤−≤−− 00


Chapter 9

Trading Strategies Involving Options


Three Alternative Strategies

Take a position in the option and the underlyingTake a position in 2 or more options of the same type (A spread) Combination: Take a position in a mixture of calls & puts (A combination)


Positions in an Option & the Underlying

Long a stock & short a call = writing a covered call (a)Short a stock & long a call = reverse of a covered callLong a stock & long a put = protective put (b)Short a stock & short a put= reverse of a protective put

2006-4-14

北航 32


Profit

STK

Profit

ST

K

Profit

ST

K

Profit

STK

(a) (b)

(c) (d)


Bull Spread Using Calls

K1 K2

ST

Profit

Buy lower & sell higher call


Continue

Bull spread created from calls requires an initial investmentProfit from a bull spreadExample:

K1=30, c1=3, K2=35, c2=1Construct a bull and give the profit.


Bull Spread Using Puts

Buy lower & sell higher put

K1 K2

Profit

ST


Continue

Bull spread created from puts brings a cash inflow to investorsA bull spread strategy limits the upside potential as well as the downside risk


Bear Spread Using Calls

Profit

K1 K2 ST

Buy higher & sell lower call

2006-4-14

北航 33


Bear Spread Using Puts

K1 K2

Profit

ST

Buy higher & sell lower put


Butterfly Spread Using Calls

K1 K3

Profit

STK2

Buy 1 high & 1 low, sell 2 middle


Butterfly Spread Strategy

Generally K2 is close to the current stock priceWhen it is appropriate?PayoffExample:

K1=55, c1=10, K2=60, c2=7, K2=65, c2=5, S0=61


Butterfly Spread Using Puts

K1 K3

Profit

STK2


Calendar Spread Using Calls

Profit

ST

K

Buy longer & sell shorter (maturity)2006年春北航金融系李平 198

Calendar Spread Using Puts

Profit

ST

K

2006-4-14

北航 34


A Straddle Combination

Profit

STK

Buy a call & a put2006年春北航金融系李平 200

Continue

Payoff structureWhen it is appropriate?

expect a significant move in the stock price, but does not know the direction

Example: S0=69, expect a significant move in the future, K=70, c=4, p=3


Strip & Strap

Profit

K ST

Strip Strap

Buy 1 call & 2 puts Buy 2 calls & 1 put

K ST

Profit


When it is appropriate?

Strip: expect a big stock price move and considers a decrease to be more likely than an increaseStrap: expect a big stock price move and considers increase a to be more likely than an decrease


A Strangle Combination

Buy 1 call with higher strike & 1 put with lower strike

K1 K2

Profit

ST


Difference from straddle

The strike priceThe stock price has to move farther in a strangle than in a straddle for the investor to make profit

2006-4-14

北航 35


Example 1

Suppose that put options with strike prices $30 and $35 cost $4 and $7, respectively. How can these two options can be used to create

(a) a bull spread(b) a bear spread? Show the profit for both spreads.


Example 2

Call options on a stock are available with strike prices of $15, $17.5 and $20 and expiration dates in three months. Their prices are $4, $2 and $0.5 respectively. Explain how these options can be used to create a butterfly spread. What is the pattern of profits from this spread?


Example 3

A call with a strike price of $50 costs $2. A put with a strike price of $45 costs $3.

Explain how a strangle can be created from these two options. What is the pattern of profits from the strangle?


Example 4

An investor believes that there will be a big jump in a stock price, but is uncertain to the direction. Identify six differentstrategies the investor can follow and explain the differences between them.


Chapter 10

Binomial Model


A Simple Example

A stock price is currently $20In three months it will be either $22 or $18

Stock Price = $18

Stock Price = $22Stock price = $20

2006-4-14

北航 36


Stock Price = $22Option Price = $1

Stock Price = $18Option Price = $0

Stock price = $20Option Price=?

A Call Option

A 3-month call option on the stock has a strike price of 21.


Consider the Portfolio: long ∆ sharesshort 1 call option

Portfolio is riskless when 22 ∆ – 1 = 18 ∆ or ∆ = 0.25

18∆

Setting Up a Riskless Portfolio

22∆ – 1

20∆-c


Valuing the Portfolio(Risk-Free Rate is 12%)

The riskless portfolio is: long 0.25 sharesshort 1 call option

The value of the portfolio in 3 months is 22′0.25 – 1 = 4.50

The value of the portfolio today is (no-arbitrage argument)

4.5e – 0.12′0.25 = 4.36702006年春北航金融系李平 214

Valuing the Option

The portfolio that is long 0.25 sharesshort 1 option

is worth 4.367 today.The value of the shares today is

5.000 (= 0.25′20 )The value of the option is therefore

0.633 (= 5.000 – 4.367 )


Generalization

A derivative lasts for time T and is dependent on a stock

S0u ƒu

S0dƒd

S0ƒ


Generalization (continued)

Consider the portfolio that is long ∆ shares and short 1 derivative

The portfolio is riskless when S0u ∆ – ƒu = S0d ∆ – ƒd or

∆ =−−

ƒ u dfS u S d0 0

S0 u∆ – ƒu

S0d∆ – ƒd

S0∆– f

2006-4-14

北航 37



Value of the portfolio at time T is S0u ∆ – ƒu

Value of the portfolio today is (S0u ∆ – ƒu )e–rT

Another expression for the portfolio value today is S0 ∆ – fHence

ƒ = S0 ∆ – (S0u ∆ – ƒu )e–rT



Substituting for ∆ we obtainƒ = [ p ƒu + (1 – p )ƒd ]e–rT

where

p e du d

r T

=−

−


Risk-Neutral Valuation

ƒ = [ p ƒu + (1 – p )ƒd ]e-rT

The variables p and (1 – p ) can be interpreted as the risk-neutral probabilities of up and down movementsThe value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate

S0uƒu

S0dƒd

S0ƒ

p

(1 – p )


Irrelevance of Stock’s Expected Return

When we are valuing an option in terms of the underlying stock the expected return on the stock is irrelevant


Original Example Revisited

One way is to use the formula

Alternatively, since p is a risk-neutral probability20e0.12 ′0.25 = 22p + 18(1 – p ); p = 0.6523

6523.09.01.1

9.00.250.12

=−

−=

−−

=×e

dudep

rT

S0d = 18ƒd = 0

p

S0u = 22ƒu = 1

S0ƒ

(1 – p )


Valuing the Option

The value of the option is e–0.12′0.25 [0.6523′1 + 0.3477′0]

= 0.633

S0u = 22ƒu = 1

S0d = 18ƒd = 0

S0ƒ

0.6523

0.3477

2006-4-14

北航 38


A Two-Step Example

Each time step is 3 months

20

22

18

24.2

19.8

16.2


Valuing a Call Option

Value at node B = e–0.12′0.25(0.6523′3.2 + 0.3477′0) = 2.0257

Value at node A = e–0.12′0.25(0.6523′2.0257 + 0.3477′0)= 1.2823

24.23.2

201.2823

22

18

19.80.0

16.20.0

2.0257

0.0

A

B

C

D

E

F


A Put Option Example; K=52

504.1923

60

40

720

484

3220

1.4147

9.4636

A

B

C

D

E

F


What Happens When an Option is American

Procedure: work back through the tree from the end to the beginning, testing at each node to see whether early exercise is optimal.The value at the final nodes is the same as for the European.At earlier nodes the value is the greater of

The value given as an European;The payoff from early exercise.


An American Put Option Example; K=52

505.0894

60

40

720

484

3220

1.4147

12.0

A

B

C

D

E

F


Delta

Delta (∆) is the ratio of the change in the price of a stock option to the change in the price of the underlying stockThe value of ∆ varies from node to node

2006-4-14

北航 39


Choosing u and d

One way of matching the volatility is to set

where σ is the volatility and ∆t is the length of the time step. This is the approach used by Cox, Ross, and Rubinstein

t

t

eud

eu∆σ−

∆σ

==

=

1


Examples

1. S0=$40, T=1m, ST=$42 or $38, r=8% per annum (cont comp), what is the

value of a 1-m European call with K=$39?Use both of the no-arbitrage argument and

the risk-neutral argument.

1.69


Examples

2. S0=$100. Over each of the next two six-month periods it is expected to go up by 10%, or go down by 10%,

r=8% per annum (cont comp), what are the value of a one-year European call and a one-year European put with K=$100? Verify the put-call parity.

1.92, 9.61


Examples

3. S0=$25, T=2m, ST=$23 or $27, r=10% per annum (cont comp), what is the value of a derivative that pays

off at the end of two months?2TS

639.3


Valuing Stock Options: The Black-Scholes Model

Chapter 12


The Black-Scholes Random Walk Assumption

Consider a stock whose price is SIn a short period of time of length ∆t the change in the stock price is assumed to be normal with mean µS∆t and standard deviation

• µ is expected return and σ is volatilitytS ∆σ

2006-4-14

北航 40


The Lognormal Property

These assumptions imply ln ST is normally distributed with mean:

and standard deviation:

Because the logarithm of ST is normal, STis lognormally distributed

TS )2/(ln 20 σ−µ+

Tσ


The Lognormal Property(continued)

where φ [m,s] is a normal distribution with mean m and standard deviation s

[ ]

[ ]TTSS

TTSS

T

T

σσ−µφ=

σσ−µ+φ≈

,)2(ln

or,)2(lnln

2

0

20


The Lognormal Distribution

E S S e

S S e eT

T

TT T

( )

( ) ( )

=

= −0

02 2 2

1

var

µ

µ σ


The Expected Return

The expected value of the stock price is S0eµT

The expected return on the stock with continuous compounding is µ – σ2/2The arithmetic mean of the returns over short periods of length ∆t is µThe geometric mean of these returns is µ –σ2/2


The Volatility

The volatility is the standard deviation of the continuously compounded rate of return in 1 yearThe standard deviation of the return in time ∆t is

If a stock price is $50 and its volatility is 25% per year what is the standard deviation of the price change in one day?

t∆σ


Estimating Volatility from Historical Data

1. Take observations S0, S1, . . . , Sn at intervals of τyears

2. Define the continuously compounded return as:

3. Calculate the standard deviation, s , of the ui ′s4. The historical volatility estimate is:

uS

Sii

i=

−

ln1

τ=σ

sˆ

2006-4-14

北航 41


Nature of Volatility

Volatility is usually much greater when the market is open (i.e. the asset is trading) than when it is closedFor this reason time is usually measured in “trading days” rather than calendar days when options are valued


The Concepts Underlying Black-Scholes

The option price and the stock price depend on the same underlying source of uncertaintyWe can form a portfolio consisting of the stock and the option which eliminates this source of uncertaintyThe portfolio is instantaneously riskless and must instantaneously earn the risk-free rate


The Black-Scholes Formulas

TdT

TrKSd

TTrKSd

dNSdNeKp

dNeKdNScrT

rT

σ−=σ

σ−+=

σσ++

=

−−−=

−=−

−

10

2

01

102

210

)2/2()/ln(

)2/2()/ln(

)()(

)()(

where


The N(x) Function

N(x) is the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than xSee tables at the end of the book


Properties of Black-ScholesFormula

As S0 becomes very large c tends to S – Ke-rT and p tends to zero

As S0 becomes very small c tends to zero and p tends to Ke-rT – S


Risk-Neutral Valuation

The variable µ does not appear in the Black-Scholes equationThe equation is independent of all variables affected by risk preferenceThis is consistent with the risk-neutral valuation principle

2006-4-14

北航 42


Applying Risk-Neutral Valuation

1. Assume that the expected return from an asset is the risk-free rate

2. Calculate the expected payoff from the derivative

3. Discount at the risk-free rate


Valuing a Forward Contract with Risk-Neutral Valuation

Payoff is ST – KExpected payoff in a risk-neutral world is SerT

– KPresent value of expected payoff is

e-rT[SerT – K]=S – Ke-rT


Implied Volatility

The implied volatility of an option is the volatility for which the Black-Scholes price equals the market priceThe is a one-to-one correspondence between prices and implied volatilitiesTraders and brokers often quote implied volatilities rather than dollar prices


Dividends

European options on dividend-paying stocks are valued by substituting the stock price less the present value of dividendsinto the Black-Scholes formulaOnly dividends with ex-dividend dates during life of option should be included The “dividend” should be the expected reduction in the stock price


American Calls

An American call on a non-dividend-paying stock should never be exercised earlyAn American call on a dividend-paying stock should only ever be exercised immediately prior to an ex-dividend date


Black’s Approach to Dealing with Dividends in American Call Options

Set the American price equal to the maximum of two European prices:1. The 1st European price is for an option maturing at the same time as the American option2. The 2nd European price is for an option maturing just before the final ex-dividend date

2006-4-14

北航 43


Options on Stock Indices and Currencies

Chapter 13


European Options on StocksPaying Dividend Yields

We get the same probability distribution for the stock price at time T in each of the following cases:1. The stock starts at price S0 and provides a dividend yield = q2. The stock starts at price S0e–q T and provides no income


European Options on StocksPaying Dividend Yield (continued)

We can value European options by reducing the stock price to S0e–q T and then behaving as though there is no dividend


Extension of Chapter 8 Results

rTqT KeeSc −− −≥ 0

Lower Bound for calls:

Lower Bound for puts

qTrT eSKep −− −≥ 0

Put Call Parity

qTrT eSpKec −− +=+ 0


Extension of Chapter 12 Results

TTqrKSd

TTqrKSd

dNeSdNKep

dNKedNeScqTrT

rTqT

σσ−−+

=

σσ+−+

=

−−−=

−=−−

−−

)2/2()/ln(

)2/2()/ln(

)()(

)()(

02

01

102

210

where


The Binomial Model

S0uƒu

S0dƒd

S0ƒ

p

(1 – p )

F = e-rT[pfu+(1– p)fd ]

2006-4-14

北航 44


The Binomial Model (continued)In a risk-neutral world the stock price grows at r – q rather than at r when there is a dividend yield at rate qThe probability, p, of an up movement must therefore satisfy

pS0u+(1 – p)S0d=S0e (r-q)T

so that

p e du d

r q T

=−

−

−( )


Index OptionsThe most popular underlying indices in the U.S. are

The Dow Jones Index times 0.01 (DJX)The Nasdaq 100 Index (NDX)The Russell 2000 Index (RUT)The S&P 100 Index (OEX)The S&P 500 Index (SPX)

Contracts are on 100 times index; they are settled in cash; OEX is American and the rest are European.


Index Option Example

Consider a call option on an index with a strike price of 560Suppose 1 contract is exercised when the index level is 580What is the payoff?


Using Index Options for Portfolio InsuranceSuppose the value of the index is S0 and the strike price is KIf a portfolio has a β of 1.0, the portfolio insurance is obtained by buying 1 put option contract on the index for each 100S0 dollars heldIf the β is not 1.0, the portfolio manager buys β put options for each 100S0 dollars heldIn both cases, K is chosen to give the appropriate insurance level


Example 1

Portfolio has a beta of 1.0It is currently worth $500,000The index currently stands at 1000What trade is necessary to provide insurance against the portfolio value falling below $450,000?


Example 2

Portfolio has a beta of 2.0It is currently worth $500,000 and index stands at 1000The risk-free rate is 12% per annumThe dividend yield on both the portfolio and the index is 4%How many put option contracts should be purchased for portfolio insurance?

2006-4-14

北航 45


If index rises to 1040, it provides a 40/1000 or 4% return in 3 monthsTotal return (incl. dividends)=5%Excess return over risk-free rate=2%Excess return for portfolio=4%Increase in Portfolio Value=4+3–1=6%Portfolio value=$530,000

Calculating Relation Between Index Level and Portfolio Value in 3 months


Determining the Strike Price

Value of Index in 3months

Expected Portfolio Valuein 3 months ($)

1,080 570,0001,040 530,0001,000 490,000 960 450,000 920 410,000 880 370,000

An option with a strike price of 960 will provide protection against a 10% decline in the portfolio value


Valuing European Index Options

We can use the formula for an option on a stock paying a continuous dividend yieldSet S0 = current index levelSet q = average dividend yield expected during the life of the option


Currency Options

Currency options trade on the Philadelphia Exchange (PHLX)There also exists an active over-the-counter (OTC) marketCurrency options are used by corporations to buy insurance when they have an FX exposure


The Foreign Interest Rate

We denote the foreign interest rate by rf

When a U.S. company buys one unit of the foreign currency it has an investment of S0dollarsThe return from investing at the foreign rate is rf S0 dollarsThis shows that the foreign currency provides a “dividend yield” at rate rf


Valuing European Currency Options

A foreign currency is an asset that provides a continuous “dividend yield”equal to rfWe can use the formula for an option on a stock paying a continuous dividend yield :

Set S0 = current exchange rateSet q = rƒ

2006-4-14

北航 46


Formulas for European Currency Options

T

TfrrKSd

T

TfrrKSd

dNeSdNKep

dNKedNeScTrrT

rTTr

f

f

σ

σ−−+=

σ

σ+−+=

−−−=

−=−−

−−

)2/2()/ln(

)2/2()/ln(

)()(

)()(

0

2

0

1

102

210

where


Alternative Formulas

F S e r r Tf0 0= −( )Using

TddT

TKFd

dNFdKNep

dKNdNFecrT

rT

σ−=

σσ+

=

−−−=

−=−

−

12

20

1

102

210

2/)/ln(

)]()([

)]()([


Chapter 14

Futures Options


Mechanics of Call Futures Options

When a call futures option is exercised the holder acquires

1. A long position in the futures 2. A cash amount equal to the excess of

the futures price over the strike price at previous settlement


Mechanics of Put Futures Option

When a put futures option is exercised the holder acquires

1. A short position in the futures 2. A cash amount equal to the excess of

the strike price over the futures price at previous settlement


The Payoffs

If the futures position is closed out immediately:Payoff from call = F0 – KPayoff from put = K – F0

where F0 is futures price at time of exercise

2006-4-14

北航 47


Potential Advantages of FuturesOptions over Spot Options

Futures contract may be easier to trade than underlying assetExercise of the option does not lead to delivery of the underlying asset Futures options and futures usually trade in adjacent pits at exchangeFutures options may entail lower transactions costs


Put-Call Parity for Futures Options

Consider the following two portfolios:1. European call plus Ke-rT of cash2. European put plus long futures plus

cash equal to F0e-rT

They must be worth the same at time T so that

c+Ke-rT=p+F0 e-rT


Other Relations

Fe-rT – K < C – P < F – Ke-rT

c > (F – K)e-rT

p > (F – K)e-rT


Futures Price = $33Option Price = $4

Futures Price = $28Option Price = $0

Futures price = $30Option Price=?

Binomial Tree Example

A 1-month call option on futures has a strike price of 29.


Consider the Portfolio: long ∆ futuresshort 1 call option

Portfolio is riskless when 3∆ – 4 = –2∆ or ∆ = 0.8

3∆ – 4

-2∆

Setting Up a Riskless Portfolio


Valuing the Portfolio( Risk-Free Rate is 6% )

The riskless portfolio is: long 0.8 futures

short 1 call optionThe value of the portfolio in 1 month is

–1.6The value of the portfolio today is

–1.6e – 0.06/12 = –1.592

2006-4-14

北航 48


Valuing the Option

The portfolio that is long 0.8 futuresshort 1 option

is worth –1.592The value of the futures is zeroThe value of the option must therefore be 1.592


Generalization of Binomial Tree Example

A derivative lasts for time T and is dependent on a futures

F0dƒd

F0uƒuF0

ƒ



Consider the portfolio that is long ∆ futures and short 1 derivative

The portfolio is riskless when

∆ =−−

ƒ u dfF u F d0 0

F0u ∆ − F0 ∆ – ƒu

F0d ∆− F0∆ – ƒd



Value of the portfolio at time T is F0u ∆ –F0∆ – ƒu

Value of portfolio today is – ƒHence

ƒ = – [F0u ∆ –F0∆ – ƒu]e-rT



Substituting for ∆ we obtainƒ = [ p ƒu + (1 – p )ƒd ]e–rT

where

dudp

−−

=1


Valuing European Futures Options

We can use the formula for an option on a stock paying a continuous dividend yield

Set S0 = current futures price (F0)Set q = domestic risk-free rate (r )

Setting q = r ensures that the expected growth of F in a risk-neutral world is zero

2006-4-14

北航 49


Growth Rates For Futures Prices

A futures contract requires no initial investmentIn a risk-neutral world the expected return should be zeroThe expected growth rate of the futures price is therefore zeroThe futures price can therefore be treated like a stock paying a dividend yield of r


Black’s Model The formulas for European options on futures are known as Black’s model

[ ][ ]

TdT

TKFd

TTKF

d

dNFdNKep

dNKdNFecrT

rT

σ−=σ

σ−=

σ

σ+=

−−−=

−=−

−

10

2

01

102

210

2/2)/ln(

2/2)/ln(

)()(

)()(

where


Futures Option Prices vs Spot Option Prices

If futures prices are higher than spot prices (normal market), an American call on futures is worth more than a similar American call on spot. An American put on futures is worth less than a similar American put on spotWhen futures prices are lower than spot prices (inverted market) the reverse is true


Put-Call Parity Results

:Futures

:exchange Foreign

:Indices

rTrT

TrrT

qTrT

eFpeKc

eSpeKc

eSpeKc

f

−−

−−

−−

+=+

+=+

+=+


Summary of Key Results from Chapters 13 and 14

We can treat stock indices, currencies, & futures like a stock paying a continuous dividend yield of q

For stock indices, q = average dividend yield on the index over the option lifeFor currencies, q = rƒ

For futures, q = r


Chapter 15

The Greek Letters

2006-4-14

北航 50


The problem to the option writer: managing the riskEach Greek letter measures a different dimension to the risk in an option positionThe aim of a trader: manage the Greek letters so that all risks are acceptable


A bank has sold for $300,000 a European call option on 100,000 shares of a nondividend paying stockS0 = 49, K = 50, r = 5%, σ = 20%,

T = 20 weeks (0.3846y), µ = 13%The Black-Scholes value of the option is $240,000The bank get $60,000 more than the theoretical value, but it is faced the problem of hedging the

risk.

An Example


Naked & Covered Positions

Naked position: Take no action, works well when ST <50, otherwise (e.g. ST=60), lose (60-50)* 100,000 Covered positionBuy 100,000 shares todayworks well when exercised (ST >50), otherwise (e.g. ST=40), lose (59-40)* 100,000 Neither strategy provides a satisfactory hedge, most traders employ Greek letters.


Delta (∆)

Delta is the rate of change of the option price with respect to the underlying

Optionprice

A

BSlope = ∆

Stock price


Example

S=100, c=10, ∆ =0.6 An investor sold 20 calls, this position could be hedged by buying 0.6*2000=1200 sharesThe gain (lose) on the option position will be offset by the lose (gain) on the stock positionDelta of a call on a stock (0.6)delta of the short option position (-2000*0.6)delta of the long share position


Delta Hedging

This involves maintaining a delta neutralportfolio--- ∆ =0In Black-Scholes model,

-1: option+ ∆ : shares

set up a delta neutral portfolio

2006-4-14

北航 51


Delta Hedging (cont.)

The delta of a European call on a non-dividend-paying stock:

∆ =N (d 1)>0Short position in a call should be hedged by a long position on sharesThe delta of a European put is

∆ = - N (-d 1) =N (d 1) – 1<0Short position in a put should be hedged by a short position on shares


Delta Hedging (cont.)

The variation of delta w.r.t the stock priceThe hedge position must be frequently rebalancedIn the example, when S increase from $100 to $110, the delta will increase from 0.6 to 0.65, then an extra 0.05*2000=100 shares should be purchased to maintain the hedge


Delta for other European options

Call on asset paying yield q∆ =e-qt N (d 1)

For put∆ = e-qt [N (d 1) -1]

For index option, foreign currency options and futures optionsDelta of a portfolio

ii

iwS

∆=∂Π∂

=∆ ∑


Gamma (Γ)

Gamma is always positive (for buyer), negative for writerIf gamma is large, delta is highly sensitive to the stock price, then it will be quite risky to leave a delta-neutral portfolio unchanged.

Sc

S 2

2

∂∂

=∂∆∂

=Γ


Gamma Addresses Delta Hedging Errors Caused By Curvature

S

CStock price

S’

Callprice

C’C’’


Making a portfolio gamma neutral

The gamma of the underlying asset is 0, so it can’t be used to change the gamma of a portfolio. What is required is an instrument such as an option which is not linearly dependenton the underlying asset.

2006-4-14

北航 52


Gamma hedging

Suppose the gamma of a delta-neutral portfolio is Γ, the gamma of a traded option is ΓT, then the gamma of a new portfolio with the number of wT options added is

wT ΓT + ΓIn order that the new portfolio is gamma neutral, the number of the options should be

wT= - Γ/ΓT


Gamma hedging (cont.)

Including the traded option will change the delta of the portfolio, so the position in the underlying asset has to be changed to maintain delta neutral. The portfolio is gamma neutral only for a short period of time. As time passes, gamma neutrality can be maintained only when the position in the option is adjusted so that it is always equal to - Γ/ΓT.


An example

Suppose that a portfolio is delta neutraland has a gamma of –3000. The delta and gamma of a particular traded call are 0.62 and 1.5. The portfolio can be made gamma neutral by including in the portfolio a long position of 2000(=-[-3000/1.5]).


Example (cont.)

The delta of the new portfolio will change from 0 to 2000*0.62=1240.A quantity of 1240 of the underlying assetmust be sold from the portfolio to keep it delta neutral.


Theta (Θ)

Theta of a derivative is the rate of change of the value with respect to the passage of time with all else remain the same, often referred to as the time decay of the optionIn practice, when theta is quoted, time is measured in days so that theta is the change in the option value when one day passes.Theta is usually negative for an option, since as time passes, the option tends to be less valuable.


Vega (ν)

If ν is the vega of a portfolio and νT is the vega of a traded option, a position of –ν/νTin the traded option makes the portfolio vega neutral.If a hedger requires a portfolio to be both gamma and vega neutral, at least two traded derivatives dependent on the underlying asset must be used.

σν

∂∂

=c

2006-4-14

北航 53


Rho (ρ)

For currency options there are 2 rhosFor a European call

rc

∂∂

=ρ

)(

)(

1

2

dSNTe

dNKTeTr

r

rTr

f

f

−

−

−=

=

ρ

ρ


Hedging in Practice

Traders usually ensure that their portfolios are delta-neutral at least once a day.Zero gamma and zero vega are less easy to achieve because of the difficulty of finding suitable options.Whenever the opportunity arises, they improve gamma and vegaAs portfolio becomes larger hedging becomes less expensive

financial derivatives how do futures and options markets

Documents