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Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005 1.1

Introduction

Chapter 1


The Nature of Derivatives

A derivative is an instrument whose valuedepends on the values of other morebasic underlying variables


Examples of Derivatives

• Futures Contracts• Forward Contracts• Swaps• Options


Derivatives MarketsExchange traded

Traditionally exchanges have used the open-outcry system, but increasingly they are switchingto electronic tradingContracts are standard there is virtually no creditrisk

Over-the-counter (OTC)A computer- and telephone-linked network ofdealers at financial institutions, corporations, andfund managersContracts can be non-standard and there is somesmall amount of credit risk


Size of OTC and Exchange Markets(Figure 1.1, Page 3)

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

020406080

100120140160180200220240

Jun-98 Jun-99 Jun-00 Jun-01 Jun-02 Jun-03 Jun-04

Size of Market ($ trillion)

OTCExchange


Ways Derivatives are Used

To hedge risksTo speculate (take a view on thefuture direction of the market)To lock in an arbitrage profitTo change the nature of a liabilityTo change the nature of an investmentwithout incurring the costs of sellingone portfolio and buying another


Forward Contracts

Forward contracts are similar to futuresexcept that they trade in the over-the-counter marketForward contracts are particularly popularon currencies and interest rates


Foreign Exchange Quotes for GBPJune 3, 2003 (See page 4)

1.61001.60946-month forward



1.62851.6281SpotOfferBid


Forward Price

The forward price for a contract is thedelivery price that would be applicableto the contract if were negotiatedtoday (i.e., it is the delivery price thatwould make the contract worth exactlyzero)The forward price may be different forcontracts of different maturities


Terminology

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


Example (page 4)

On June 3, 2003 the treasurer of acorporation enters into a long forwardcontract to buy £1 million in six months atan exchange rate of 1.6100This obligates the corporation to pay$1,610,000 for £1 million on December 3,2003What are the possible outcomes?


Profit from aLong Forward Position

Profit

Price of Underlying at Maturity, STK


Profit from aShort Forward Position

Profit

Price of Underlying at Maturity, STK


Futures Contracts (page 6)

Agreement to buy or sell an asset for acertain price at a certain timeSimilar to forward contractWhereas a forward contract is traded OTC,a futures contract is traded on anexchange


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)


Examples of Futures Contracts

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


1. Gold: An ArbitrageOpportunity?

Suppose that:The spot price of gold is US$300The 1-year forward price of gold is

US$340The 1-year US$ interest rate is 5%

per annumIs there an arbitrage opportunity?


2. Gold: Another ArbitrageOpportunity?

Suppose that:- The spot price of gold is US$300- The 1-year forward price of gold

is US$300- The 1-year US$ interest rate is

5% per annumIs there an arbitrage opportunity?


The Forward Price of Gold

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

F = S (1+r )T

where r is the 1-year (domestic currency) risk-free rate of interest.In our examples, S = 300, T = 1, and r =0.05 sothat

F = 300(1+0.05) = 315


1. Oil: An ArbitrageOpportunity?

Suppose that:- The spot price of oil is US$19- The quoted 1-year futures price of

oil is US$25- The 1-year US$ interest rate is 5%

per annum- The storage costs of oil are 2%



2. Oil: Another ArbitrageOpportunity?

Suppose that:- The spot price of oil is US$19- The quoted 1-year futures price of

oil is US$16- The 1-year US$ interest rate is 5%

per annum- The storage costs of oil are 2%



Options

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


American vs European Options

An American option can be exercised atany time during its lifeA European option can be exercised onlyat maturity


Intel Option Prices (May 29, 2003;Stock Price=20.83); See Table 1.2 page 7

2.852.201.851.150.450.2022.50

1.500.850.452.401.601.2520.00

OctPut

JulyPut

JunePut

OctCall

JulyCall

JuneCall

StrikePrice


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 holderthe obligation to buy or sell at a certainpriceAn option gives the holder the right to buyor sell at a certain price


Types of Traders

• Hedgers

• Speculators

• Arbitrageurs

Some of the largest trading losses in derivatives haveoccurred because individuals who had a mandate to behedgers or arbitrageurs switched to being speculators(See for example Barings Bank, Business Snapshot 1.2,page 15)


Hedging Examples (pages 10-11)

A US company will pay £10 million forimports from Britain in 3 months anddecides to hedge using a long positionin a forward contractAn investor owns 1,000 Microsoftshares currently worth $28 per share. Atwo-month put with a strike price of$27.50 costs $1. The investor decidesto hedge by buying 10 contracts


Value of Microsoft Shares withand without Hedging (Fig 1.4, page 11)

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 feelsthat Amazon.com’s stock price willincrease over the next 2 months. Thecurrent stock price is $40 and the priceof a 2-month call option with a strike of45 is $2What are the alternative strategies?


Arbitrage Example

A stock price is quoted as £100 inLondon and $172 in New YorkThe current exchange rate is 1.7500What is the arbitrage opportunity?


Hedge Funds (see Business Snapshot 1.1, page 9)

Hedge funds are not subject to the same rules asmutual funds and cannot offer their securities publicly.Mutual funds must

disclose investment policies,makes shares redeemable at any time,limit use of leveragetake no short positions.

Hedge funds are not subject to these constraints.Hedge funds use complex trading strategies: big usersof derivatives for hedging, speculation and arbitrage


Mechanics of FuturesMarkets

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


Margins

A margin is cash or marketable securitiesdeposited by an investor with his or herbrokerThe balance in the margin account isadjusted to reflect daily settlementMargins minimize the possibility of a lossthrough a default on a contract


Example of a Futures Trade (page 27-28)

An investor takes a long position in 2December gold futures contracts onJune 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 OutcomeTable 2.1, Page 28

Daily Cumulative MarginFutures Gain Gain Account Margin

Price (Loss) (Loss) Balance CallDay (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

<


Other Key Points About Futures

They are settled dailyClosing out a futures positioninvolves entering into an offsettingtradeMost contracts are closed outbefore maturity


Collateralization in OTC Markets

It is becoming increasingly common forcontracts to be collateralized in OTCmarketsThey are then similar to futures contractsin that they are settled regularly (e.g. everyday or every week)


Futures Prices for Gold on Feb 4, 2004: PricesIncrease with Maturity (Figure 2.2, page 35)

(a) Gold

398399400401402403404405406407408

Feb-04 Apr-04 Jun-04 Aug-04 Oct-04 Dec-04

Contract Maturity Month

Futu

res

Pric

e ($

per

oz)


Futures Prices for Oil on February 4, 2004:Prices Decrease with Maturity (Figure 2.2, page 35)

(b) Brent Crude Oil

24

25

26

27

28

29

30

Mar-04 May-04 Jul-04 Sep-04 Nov-04 Jan-05

Contract Maturity Month

Futu

res

Pric

e ($

per

bar

rel)


Delivery

If a futures contract is not closed out beforematurity, it is usually settled by delivering theassets underlying the contract. When there arealternatives about what is delivered, where it isdelivered, and when it is delivered, the party withthe short position chooses. A few contracts (for example, those on stockindices and Eurodollars) are settled in cash


Some Terminology

Open interest: the total number of contractsoutstanding

equal to number of long positions ornumber of short positions

Settlement price: the price just before thefinal bell each day

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


Convergence of Futures to Spot(Figure 2.1, page 26)

Time Time

(a) (b)

FuturesPrice

FuturesPrice

Spot Price

Spot Price


Questions

When a new trade is completed whatare the possible effects on the openinterest?Can the volume of trading in a daybe greater than the open interest?


Regulation of Futures

Regulation is designed toprotect the public interestRegulators try to preventquestionable trading practicesby either individuals on the floorof the exchange or outsidegroups


Accounting & Tax

Ideally hedging profits (losses) should berecognized at the same time as the losses(profits) on the item being hedgedIdeally profits and losses from speculationshould be recognized on a mark-to-marketbasisRoughly speaking, this is what theaccounting and tax treatment of futures inthe U.S.and many other countries attemptsto achieve


Forward Contracts vs FuturesContracts

Private contract between 2 parties Exchange traded

Non-standard contract Standard contract

Usually 1 specified delivery date Range of delivery dates

Settled at end of contract Settled daily

Delivery or final cashsettlement usually occurs

Contract usually closed outprior to maturity

FORWARDS FUTURES

TABLE 2.3 (p. 41)

Some credit risk Virtually no credit risk


Foreign Exchange Quotes

Futures exchange rates are quoted as thenumber of USD per unit of the foreign currencyForward exchange rates are quoted in the sameway as spot exchange rates. This means thatGBP, EUR, AUD, and NZD are quoted as USDper unit of foreign currency. Other currencies(e.g., CAD and JPY) are quoted as units of theforeign currency per USD.

Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 3.1

Hedging Strategies UsingFutures

Chapter 3


Long & Short Hedges

A long futures hedge is appropriate whenyou know you will purchase an asset inthe future and want to lock in the priceA short futures hedge is appropriatewhen you know you will sell an asset inthe future & want to lock in the price


Arguments in Favor of Hedging

Companies should focus on the mainbusiness they are in and take steps tominimize risks arising from interest rates,exchange rates, and other marketvariables


Arguments against Hedging

Shareholders are usually well diversifiedand can make their own hedging decisionsIt may increase risk to hedge whencompetitors do notExplaining a situation where there is a losson the hedge and a gain on the underlyingcan be difficult


Convergence of Futures to Spot(Hedge initiated at time t1 and closed out at time t2)

Time

SpotPrice

FuturesPrice

t1 t2


Basis Risk

Basis is the difference betweenspot & futuresBasis risk arises because of theuncertainty about the basiswhen the hedge is closed out


Long Hedge

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

You hedge the future purchase of anasset by entering into a long futurescontractCost of Asset=S2 – (F2 – F1) = F1 + Basis


Short Hedge

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

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


Choice of Contract

Choose a delivery month that is as closeas possible to, but later than, the end ofthe life of the hedgeWhen there is no futures contract on theasset being hedged, choose the contractwhose futures price is most highlycorrelated with the asset price. This isknown as cross hedging.


Optimal Hedge Ratio

Proportion of the exposure that should optimally behedged is

whereσS is the standard deviation of ∆S, the change in thespot price during the hedging period,σF is the standard deviation of ∆F, the change in thefutures price during the hedging periodρ is the coefficient of correlation between ∆S and ∆F.

F

S

σσρ


Hedging Using Index Futures(Page 63)

To hedge the risk in a portfolio thenumber of contracts that should beshorted is

where P is the value of the portfolio,β is its beta, and A is the value of theassets underlying one futurescontract

βPA


Reasons for Hedging an EquityPortfolio

Desire to be out of the market for a shortperiod of time. (Hedging may be cheaperthan selling the portfolio and buying itback.)Desire to hedge systematic risk(Appropriate when you feel that you havepicked stocks that will outpeform themarket.)


Example

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

What position in futures contracts on theS&P 500 is necessary to hedge theportfolio?


Changing Beta

What position is necessary to reduce thebeta of the portfolio to 0.75?What position is necessary to increase thebeta of the portfolio to 2.0?


Hedging Price of an IndividualStock

Similar to hedging a portfolioDoes not work as well because only thesystematic risk is hedgedThe unsystematic risk that is unique to thestock is not hedged


Why Hedge Equity Returns

May want to be out of the market for a while.Hedging avoids the costs of selling andrepurchasing the portfolioSuppose stocks in your portfolio have anaverage beta of 1.0, but you feel they have beenchosen well and will outperform the market inboth good and bad times. Hedging ensures thatthe return you earn is the risk-free return plusthe excess return of your portfolio over themarket.


Rolling The Hedge Forward (page 67-68)

We can use a series of futurescontracts to increase the life of ahedgeEach time we switch from 1 futurescontract to another we incur a type ofbasis risk


Interest Rates

Chapter 4


Types of Rates

Treasury ratesLIBOR ratesRepo rates


Measuring Interest Rates

The compounding frequency usedfor an interest rate is the unit ofmeasurementThe difference between quarterlyand annual compounding isanalogous to the differencebetween miles and kilometers


Continuous Compounding(Page 79)

In the limit as we compound more and morefrequently we obtain continuously compoundedinterest rates$100 grows to $100eRT when invested at acontinuously compounded rate R for time T$100 received at time T discounts to $100e-RT attime zero when the continuously compoundeddiscount rate is R


Conversion Formulas(Page 79)

DefineRc : continuously compounded rateRm: same rate with compounding m times

per year

( )R m

Rm

R m e

cm

mR mc

= +⎛⎝⎜

⎞⎠⎟

= −

ln

/

1

1


Zero Rates

A zero rate (or spot rate), for maturity T isthe rate of interest earned on aninvestment that provides a payoff only attime T


Example (Table 4.2, page 81)

M aturity(years)

Zero Rate(% cont com p)

0.5 5.0

1.0 5.8

1.5 6.4

2.0 6.8


Bond Pricing

To calculate the cash price of a bond wediscount each cash flow at the appropriate zerorateIn our example, the theoretical price of a two-year bond providing a 6% coupon semiannuallyis

3 3 3103 98 39

0 05 0 5 0 058 1 0 0 064 1 5

0 068 2 0

e e ee

− × − × − ×

− ×

+ +

+ =

. . . . . .

. . .


Bond YieldThe bond yield is the discount rate thatmakes the present value of the cash flows onthe bond equal to the market price of thebondSuppose that the market price of the bond inour example equals its theoretical price of98.39The bond yield (continuously compounded) isgiven by solving

to get y=0.0676 or 6.76%.3 3 3 103 98 390 5 1 0 1 5 2 0e e e ey y y y− × − × − × − ×+ + + =. . . . .


Par YieldThe par yield for a certain maturity is thecoupon rate that causes the bond price toequal its face value.In our example we solve

g)compoundin s.a. (with get to 876

1002

100

2220.2068.0

5.1064.00.1058.05.005.0

.c=

ec

ececec

=⎟⎠⎞

⎜⎝⎛ ++

++

×−

×−×−×−


Par Yield continued

In general if m is the number of couponpayments per year, P is the present valueof $1 received at maturity and A is thepresent value of an annuity of $1 on eachcoupon date

cP m

A=

−( )100 100


Sample Data (Table 4.3, page 82)

Bond Time to Annual Bond CashPrincipal Maturity Coupon Price(dollars) (years) (dollars) (dollars)

100 0.25 0 97.5

100 0.50 0 94.9

100 1.00 0 90.0

100 1.50 8 96.0

100 2.00 12 101.6


The Bootstrap Method

An amount 2.5 can be earned on 97.5 during 3months.The 3-month rate is 4 times 2.5/97.5 or 10.256%with quarterly compoundingThis is 10.127% with continuous compoundingSimilarly the 6 month and 1 year rates are10.469% and 10.536% with continuouscompounding


The Bootstrap Method continued

To calculate the 1.5 year rate we solve

to get R = 0.10681 or 10.681%

Similarly the two-year rate is 10.808%

9610444 5.10.110536.05.010469.0 =++ ×−×−×− Reee


Zero Curve Calculated from theData (Figure 4.1, page 84)

9

10

11

12

0 0.5 1 1.5 2 2.5

ZeroRate (%)

Maturity (yrs)

10.127

10.469 10.53610.681 10.808


Forward Rates

The forward rate is the future zero rateimplied by today’s term structure of interestrates


Calculation of Forward RatesTable 4.5, page 85

Zero Rate for Forward Ratean n -year Investment for n th Year

Year (n ) (% per annum) (% per annum)

1 3.02 4.0 5.03 4.6 5.84 5.0 6.25 5.3 6.5


Formula for Forward Rates

Suppose that the zero rates for timeperiods T1 and T2 are R1 and R2 with bothrates continuously compounded.The forward rate for the period betweentimes T1 and T2 is

R T R TT T

2 2 1 1

2 1

−−


Instantaneous Forward Rate

The instantaneous forward rate for amaturity T is the forward rate that appliesfor a very short time period starting at T. Itis

where R is the T-year rate

R T RT

+∂∂


Upward vs Downward SlopingYield Curve

For an upward sloping yield curve:Fwd Rate > Zero Rate > Par Yield

For a downward sloping yield curvePar Yield > Zero Rate > Fwd Rate


Forward Rate Agreement

A forward rate agreement (FRA) is anagreement that a certain rate will apply toa certain principal during a certain futuretime period


Forward Rate Agreementcontinued

An FRA is equivalent to an agreementwhere interest at a predetermined rate, RKis exchanged for interest at the marketrateAn FRA can be valued by assuming thatthe forward interest rate is certain to berealized


Valuation Formulas (equations 4.9 and 4.10page 88)

Value of FRA where a fixed rate RK will bereceived on a principal L between times T1 andT2 isValue of FRA where a fixed rate is paid is

RF is the forward rate for the period and R2 is thezero rate for maturity T2

What compounding frequencies are used inthese formulas for RK, RM, and R2?

22))(( 12TR

FK eTTRRL −−−

22))(( 12TR

KF eTTRRL −−−


Duration of a bond that provides cash flow c i at time t i is

where B is its price and y is its yield (continuouslycompounded)This leads to

⎥⎦

⎤⎢⎣

⎡ −

=∑ B

ectiyt

in

ii

1

yDBB

∆−=∆

Duration (page 89)


Duration ContinuedWhen the yield y is expressed withcompounding m times per year

The expression

is referred to as the “modified duration”

myyBDB

+∆

−=∆1

Dy m1 +


Convexity

The convexity of a bond is defined as

2

1

2

2

2

)(21

thatso

1

yCyDBB

B

etc

yB

BC

n

i

ytii

i

∆+∆−=∆

=∂∂

=∑=

−


Theories of the Term Structure

Expectations Theory: forward rates equalexpected future zero ratesMarket Segmentation: short, medium andlong rates determined independently ofeach otherLiquidity Preference Theory: forwardrates higher than expected future zerorates


Determination of Forwardand Futures Prices

Chapter 5


Consumption vs Investment Assets

Investment assets are assets held bysignificant numbers of people purely forinvestment purposes (Examples: gold,silver)Consumption assets are assets heldprimarily for consumption (Examples:copper, oil)


Short Selling (Page 99-101)

Short selling involves sellingsecurities you do not ownYour broker borrows thesecurities from another client andsells them in the market in theusual way


Short Selling(continued)

At some stage you must buythe securities back so theycan be replaced in theaccount of the clientYou must pay dividends andother benefits the owner ofthe securities receives


Notation for Valuing Futures andForward Contracts

Risk-free interest rate formaturity T

r:

Time until delivery dateT:

Futures or forward price todayF0:

Spot price todayS0:


1. Gold: An ArbitrageOpportunity?

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

Is there an arbitrage opportunity?


2. Gold: Another ArbitrageOpportunity?

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



The Forward Price of Gold

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

F = S (1+r )T

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

F = 390(1+0.05) = 409.50


When Interest Rates are Measuredwith Continuous Compounding

F0 = S0erT

This equation relates the forward priceand the spot price for any investmentasset that provides no income and hasno storage costs


When an Investment AssetProvides a Known Dollar Income(page 105, equation 5.2)

F0 = (S0 – I )erT

where I is the present value of the incomeduring life of forward contract


When an Investment AssetProvides a Known Yield(Page 107, equation 5.3)

F0 = S0 e(r–q )T

where q is the average yield during thelife of the contract (expressed withcontinuous compounding)


Valuing a Forward ContractPage 108

Suppose that K is delivery price in a forward contract and 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 contractis

(K – F0 )e–rT


Forward vs Futures Prices

Forward and futures prices are usually assumedto be the same. When interest rates areuncertain they are, in theory, slightly different:A strong positive correlation between interestrates and the asset price implies the futuresprice is slightly higher than the forward priceA strong negative correlation implies thereverse


Stock Index (Page 110-112)

Can be viewed as an investment assetpaying a dividend yieldThe futures price and spot pricerelationship is therefore

F0 = S0 e(r–q )T where q is the average dividend yield on

the portfolio represented by the indexduring life of contract


Stock Index(continued)

For the formula to be true it isimportant that the index represent aninvestment assetIn other words, changes in the indexmust correspond to changes in thevalue of a tradable portfolioThe Nikkei index viewed as a dollarnumber does not represent aninvestment asset (See BusinessSnapshot 5.3, page 111)


Index Arbitrage

When F0 > S0e(r-q)T an arbitrageur buys thestocks underlying the index and sellsfuturesWhen F0 < S0e(r-q)T an arbitrageur buysfutures and shorts or sells the stocksunderlying the index


Index Arbitrage(continued)

Index arbitrage involves simultaneous trades infutures and many different stocksVery often a computer is used to generate thetradesOccasionally (e.g., on Black Monday)simultaneous trades are not possible and thetheoretical no-arbitrage relationship betweenF0 and S0 does not hold


A foreign currency is analogous to a securityproviding a dividend yieldThe continuous dividend yield is the foreignrisk-free interest rateIt follows that if rf is the foreign risk-free interestrate

Futures and Forwards onCurrencies (Page 112-115)

F S e r r Tf0 0= −( )


Why the Relation Must Be TrueFigure 5.1, page 113

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

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


Futures on Consumption Assets(Page 117-118)

F0 ≤ S0 e(r+u )T

where u is the storage cost per unittime as a percent of the asset value.

Alternatively, F0 ≤ (S0+U )erT

where U is the present value of thestorage costs.


The Cost of Carry (Page 118-119)

The cost of carry, c, is the storage cost plus theinterest costs less the income earnedFor an investment asset F0 = S0ecT

For a consumption asset F0 ≤ S0ecT

The convenience yield on the consumptionasset, y, is defined so that F0 = S0 e(c–y )T


Futures Prices & Expected FutureSpot Prices (Page 119-121)

Suppose k is the expected return required byinvestors on an assetWe can invest F0e–r T at the risk-free rate andenter into a long futures contract so that there isa cash inflow of ST at maturityThis shows that

TkrT

TkTrT

eSEF

SEeeF

)(0

0

)(

)()(

−

−

=

=or


Futures Prices & Future SpotPrices (continued)

If the asset hasno systematic risk, then k = r and F0 isan unbiased estimate of ST

positive systematic risk, then k > r andF0 < E (ST )negative systematic risk, then k < r andF0 > E (ST )


Interest Rate Futures

Chapter 6


Day Count Conventionsin the U.S. (Page 129)

Actual/360Money Market Instruments:

30/360Corporate Bonds:

Actual/Actual (in period)Treasury Bonds:


Treasury Bond Price Quotesin the U.S

Cash price = Quoted price + Accrued Interest


Treasury Bond FuturesPages 133-137

Cash price received by party with shortposition =

Most Recent Settlement Price ×Conversion factor + Accrued interest


Example

Settlement price of bond delivered = 90.00Conversion factor = 1.3800Accrued interest on bond =3.00Price received for bond is1.3800×9.00)+3.00 = $127.20per $100 of principal


Conversion Factor

The conversion factor for a bond isapproximately equal to the value of thebond on the assumption that the yieldcurve is flat at 6% with semiannualcompounding


CBOTT-Bonds & T-Notes

Factors that affect the futures price:Delivery can be made any timeduring the delivery monthAny of a range of eligible bondscan be deliveredThe wild card play


A Eurodollar is a dollar deposited in a bankoutside the United StatesEurodollar futures are futures on the 3-monthEurodollar deposit rate (same as 3-monthLIBOR rate)One contract is on the rate earned on $1 millionA change of one basis point or 0.01 in aEurodollar futures quote corresponds to acontract price change of $25

Eurodollar Futures (Page 137-142)


Eurodollar Futures continued

A Eurodollar futures contract is settled incashWhen it expires (on the third Wednesdayof the delivery month) the final settlementprice is 100 minus the actual three monthdeposit rate


Example

Suppose you buy (take a long position in)a contract on November 1The contract expires on December 21The prices are as shownHow much do you gain or lose a) on thefirst day, b) on the second day, c) over thewhole time until expiration?


Example

97.42Dec 21

………….

96.98Nov 3

97.23Nov 2

97.12Nov 1QuoteDate


Example continued

If on Nov. 1 you know that you will have$1 million to invest on for three months onDec 21, the contract locks in a rate of100 - 97.12 = 2.88%In the example you earn 100 – 97.42 =2.58% on $1 million for three months(=$6,450) and make a gain day by day onthe futures contract of 30×$25 =$750


Formula for Contract Value (page 138)

If Q is the quoted price of a Eurodollarfutures contract, the value of one contractis 10,000[100-0.25(100-Q)]


Forward Rates and EurodollarFutures (Page 139-142)

Eurodollar futures contracts last as long as10 yearsFor Eurodollar futures lasting beyond twoyears we cannot assume that the forwardrate equals the futures rate


There are Two Reasons

Futures is settled daily where forward issettled onceFutures is settled at the beginning of theunderlying three-month period; forward issettled at the end of the underlying three-month period


Forward Rates and EurodollarFutures continued

)012.0

21

1

2

1

212

about is (typically yearperchanges rate short the of deviation standard

the is and ) than later days (90contract futures the underlying rate the

ofmaturity the is contract, futures the ofmaturity to time the is where

rate Futures=rate Forward

is made often "adjustmentconvexity "A

σ

σ

σ−

t

tt

tt


Convexity Adjustment whenσ=0.012 (Table 6.3, page 141)

73.810

47.58

27.06

12.24

3.22

ConvexityAdjustment (bps)

Maturity ofFutures


Extending the LIBOR Zero Curve

LIBOR deposit rates define the LIBORzero curve out to one yearEurodollar futures can be used todetermine forward rates and the forwardrates can then be used to bootstrap thezero curve


Example

so that

If the 400 day LIBOR rate has been calculatedas 4.80% and the forward rate for the periodbetween 400 and 491 days is 5.30 the 491 daysrate is 4.893%

ii

iiiii TT

TRTRF−−

=+

++

1

11

1

11

)(

+

++

+−=

i

iiiiii T

TRTTFR


Duration Matching

This involves hedging against interestrate risk by matching the durations ofassets and liabilitiesIt provides protection against smallparallel shifts in the zero curve


Use of Eurodollar Futures

One contract locks in an interest rate on$1 million for a future 3-month periodHow many contracts are necessary to lockin an interest rate for a future six monthperiod?


Duration-Based Hedge Ratio

FC

P

DFPD

Duration of portfolio at hedge maturityDP

Value of portfolio being hedgedP

Duration of asset underlying futures atmaturity

DF

Contract price for interest rate futuresFC


ExampleIt is August. A fund manager has $10 million invested ina portfolio of government bonds with a duration of 6.80years and wants to hedge against interest rate movesbetween August and DecemberThe manager decides to use December T-bond futures.The futures price is 93-02 or 93.0625 and the duration ofthe cheapest to deliver bond is 9.2 yearsThe number of contracts that should be shorted is

7920.980.6

50.062,93000,000,10

=×


Limitations of Duration-BasedHedging

Assumes that only parallel shift in yieldcurve take placeAssumes that yield curve changes aresmall


GAP Management (Business Snapshot 6.3)

This is a more sophisticated approachused by banks to hedge interest rate. ItinvolvesBucketing the zero curveHedging exposure to situation where ratescorresponding to one bucket change andall other rates stay the same.


Swaps

Chapter 7


Nature of Swaps

A swap is an agreement toexchange cash flows at specifiedfuture times according to certainspecified rules


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 yearson 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, 2004 4.2%

Sept. 5, 2004 4.8% +2.10 –2.50 –0.40Mar.5, 2005 5.3% +2.40 –2.50 –0.10

Sept. 5, 2005 5.5% +2.65 –2.50 +0.15Mar.5, 2006 5.6% +2.75 –2.50 +0.25

Sept. 5, 2006 5.9% +2.80 –2.50 +0.30Mar.5, 2007 6.4% +2.95 –2.50 +0.45

Cash Flows to Microsoft(See Table 7.1, page 151)


Typical Uses of anInterest Rate Swap

Converting a liability fromfixed rate to floating ratefloating rate to fixed rate

Converting an investment fromfixed rate to floating ratefloating rate to fixed rate


Intel and Microsoft (MS)Transform a Liability(Figure 7.2, page 152)

Intel MS

LIBOR

5%

LIBOR+0.1%

5.2%


Financial Institution is Involved(Figure 7.4, page 153)

F.I.

LIBOR LIBORLIBOR+0.1%

4.985% 5.015%

5.2%Intel MS


Intel and Microsoft (MS)Transform an Asset(Figure 7.3, page 153)

Intel MS

LIBOR

5%

LIBOR-0.2%

4.7%


Financial Institution is Involved(See Figure 7.5, page 154)

Intel F.I. MS

LIBOR LIBOR

4.7%

5.015%4.985%

LIBOR-0.2%


Quotes By a Swap Market Maker(Table 7.3, page 155)

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 yearsSwap Rate (%)Offer (%)Bid (%)Maturity


The Comparative AdvantageArgument (Table 7.4, page 157)

AAACorp wants to borrow floatingBBBCorp wants to borrow fixed

Fixed Floating

AAACorp 4.0% 6-month LIBOR + 0.30%

BBBCorp 5.20% 6-month LIBOR + 1.00%


The Swap (Figure 7.6, page 158)

AAACorp BBBCorp

LIBOR

LIBOR+1%

3.95%

4%


The Swap when a FinancialInstitution is Involved(Figure 7.7, page 158)

AAACorp F.I. BBBCorp4%

LIBOR LIBOR

LIBOR+1%

3.93% 3.97%


Criticism of the ComparativeAdvantage Argument

The 4.0% and 5.2% rates available to AAACorpand BBBCorp in fixed rate markets are 5-yearratesThe LIBOR+0.3% and LIBOR+1% ratesavailable in the floating rate market are six-month ratesBBBCorp’s fixed rate depends on the spreadabove LIBOR it borrows at in the future


The Nature of Swap Rates

Six-month LIBOR is a short-term AA borrowingrateThe 5-year swap rate has a risk correspondingto the situation where 10 six-month loans aremade to AA borrowers at LIBORThis is because the lender can enter into aswap where income from the LIBOR loans isexchanged for the 5-year swap rate


Using Swap Rates to Bootstrap theLIBOR/Swap Zero Curve

Consider a new swap where the fixed rate is theswap rateWhen principals are added to both sides on thefinal payment date the swap is the exchange ofa fixed rate bond for a floating rate bondThe floating-rate rate bond is worth par. Theswap is worth zero. The fixed-rate bond musttherefore also be worth parThis shows that swap rates define par yieldbonds that can be used to bootstrap the LIBOR(or LIBOR/swap) zero curve


Valuation of an Interest RateSwap that is not New

Interest rate swaps can be valued asthe difference between the value of afixed-rate bond and the value of afloating-rate bondAlternatively, they can be valued as aportfolio of forward rate agreements(FRAs)


Valuation in Terms of Bonds

The fixed rate bond is valued in the usualwayThe floating rate bond is valued by notingthat it is worth par immediately after thenext payment date


Valuation in Terms of FRAs

Each exchange of payments in an interestrate swap is an FRAThe FRAs can be valued on theassumption that today’s forward rates arerealized


An Example of a Currency Swap

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


Exchange of Principal

In an interest rate swap theprincipal is not exchangedIn a currency swap the principal isusually exchanged at thebeginning and the end of theswap’s life


The Cash Flows (Table 7.7, page 166)

Year

Dollars Pounds$

------millions------2004 –15.00 +10.002005 +0.60 –0.70 2006 +0.60 –0.70 2007 +0.60 –0.702008 +0.60 –0.70 2009 +15.60 - 10.70

£


Typical Uses of aCurrency Swap

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

Conversion from an investment in onecurrency to an investment in anothercurrency


Comparative Advantage Argumentsfor Currency Swaps (Table 7.8, page 167)

General Motors wants to borrow AUDQantas wants to borrow USD

USD AUD

General Motors 5.0% 12.6%Qantas 7.0% 13.0%


Valuation of Currency Swaps

Like interest rate swaps, currencyswaps can be valued either as thedifference between 2 bonds or as aportfolio of forward contracts


Swaps & Forwards

A swap can be regarded as a convenient wayof packaging forward contractsThe “plain vanilla” interest rate swap in ourexample (slide 7.4) consisted of 6 FRAsThe “fixed for fixed” currency swap in ourexample (slide 7.22) consisted of a cashtransaction & 5 forward contracts


Swaps & Forwards(continued)

The value of the swap is the sum of thevalues of the forward contracts underlyingthe swapSwaps are normally “at the money” initially

This means that it costs nothing to enterinto a swapIt does not mean that each forwardcontract underlying a swap is “at themoney” initially


Credit Risk

A swap is worth zero to a companyinitiallyAt a future time its value is liable to beeither positive or negativeThe company has credit risk exposureonly when its value is positive


Other Types of Swaps

Floating-for-floating interest rate swaps,amortizing swaps, step up swaps, forwardswaps, constant maturity swaps,compounding swaps, LIBOR-in-arrearsswaps, accrual swaps, diff swaps, crosscurrency interest rate swaps, equity swaps,extendable swaps, puttable swaps,swaptions, commodity swaps, volatilityswaps……..


Mechanics of OptionsMarkets

Chapter 8


Review of Option Types

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


Option Positions

Long callLong putShort callShort put


Long Call on eBay(Figure 8.1, Page 182)

Profit from buying one eBay European call option: optionprice = $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(Figure 8.3, page 184)

Profit from writing one eBay European call option: optionprice = $5, strike price = $100

-30

-20

-10

05

70 80 90 100

110 120 130

Profit ($)



Long Put on IBM(Figure 8.2, page 183)

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

30

20

10

0-7

70605040 80 90 100

Profit ($)



Short Put on IBM(Figure 8.4, page 184)

Profit from writing an IBM European put option: optionprice = $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


Assets UnderlyingExchange-Traded OptionsPage 185-186

StocksForeign CurrencyStock IndicesFutures


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 Option series Intrinsic value Time value


Dividends & Stock Splits(Page 188-190)

Suppose you own N options with a strikeprice of K :

No adjustments are made to the optionterms 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 mannersimilar to stock splits


Dividends & Stock Splits(continued)

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

for a 2-for-1 stock split?for a 5% stock dividend?


Market Makers

Most exchanges use market makers tofacilitate options tradingA market maker quotes both bid and askprices when requestedThe market maker does not know whetherthe individual requesting the quotes wantsto buy or sell


Margins (Page 194-195)

Margins are required when options are soldWhen a naked option is written the margin is thegreater of:1 A total of 100% of the proceeds of the sale plus

20% of the underlying share price less theamount (if any) by which the option is out of themoney

2 A total of 100% of the proceeds of the sale plus10% of the underlying share price

For other trading strategies there are special rules


Warrants

Warrants are options that are issued by acorporation or a financial institutionThe number of warrants outstanding isdetermined by the size of the originalissue and changes only when they areexercised or when they expire


Warrants(continued)

The issuer settles up with the holderwhen a warrant is exercisedWhen call warrants are issued by acorporation on its own stock, exercisewill lead to new treasury stock beingissued


Executive Stock Options

Executive stock options are a form ofremuneration issued by a company to itsexecutivesThey are usually at the money whenissuedWhen options are exercised the companyissues more stock and sells it to the optionholder for the strike price


Executive Stock Options continued

They become vested after a period of time(usually 1 to 4 years)They cannot be soldThey often last for as long as 10 or 15yearsAccounting standards now require theexpensing of executive stock options


Convertible Bonds

Convertible bonds are regular bondsthat can be exchanged for equity atcertain times in the future according toa predetermined exchange ratioVery often a convertible is callableThe call provision is a way in which theissuer can force conversion at a timeearlier than the holder might otherwisechoose


Properties of StockOptions

Chapter 9


Notation c : European calloption price p : European putoption price S0 : Stock price today K : Strike price T : Life of option σ: Volatility of stockprice

C : American Call optionprice P : American Put optionprice ST :Stock price at optionmaturity D : Present value ofdividends during option’slife r : Risk-free rate formaturity T with cont comp


Effect of Variables on OptionPricing (Table 9.1, page 206)

c p C PVariableS0KTσrD

+ + –+

? ? + ++ + + ++ – + –

–– – +

– + – +


American vs European Options

An American option is worthat least as much as thecorresponding Europeanoption

C ≥ cP ≥ p


Calls: An Arbitrage Opportunity?

Suppose that c = 3 S0 = 20 T = 1 r = 10% K = 18 D = 0



Lower Bound for European CallOption Prices; No Dividends (Equation9.1, page 211)

c ≥ S0 –Ke -rT


Puts: An Arbitrage Opportunity?

Suppose that p = 1 S0 = 37T = 0.5 r =5%

K = 40 D = 0

Is there an arbitrageopportunity?


Lower Bound for European PutPrices; No Dividends(Equation 9.2, page 212)

p ≥ Ke -rT–S0


Put-Call Parity; No Dividends(Equation 9.3, page 212)

Consider the following 2 portfolios:Portfolio A: European call on a stock + PV of thestrike price in cashPortfolio C: European put on the stock + the stock

Both are worth max(ST , K ) at the maturity of theoptionsThey must therefore be worth the same today. Thismeans that

c + Ke -rT = p + S0


Arbitrage OpportunitiesSuppose that c = 3 S0 = 31T = 0.25 r = 10%K =30 D = 0What are the arbitragepossibilities when

p = 2.25 ? p = 1 ?


Early Exercise

Usually there is some chance that anAmerican option will be exercisedearlyAn exception is an American call on anon-dividend paying stockThis should never be exercised early


For an American call option:S0 = 100; T = 0.25; K = 60; D = 0

Should you exercise immediately?What should you do if

you want to hold the stock for the next 3months?you do not feel that the stock is worth holdingfor the next 3 months?

An Extreme Situation


Reasons For Not Exercising aCall Early (No Dividends)

No income is sacrificedPayment of the strike price isdelayedHolding the call provides insuranceagainst stock price falling belowstrike price


Should Puts Be ExercisedEarly ?

Are there any advantages toexercising an American put when

S0 = 60; T = 0.25; r=10% K = 100; D = 0


The Impact of Dividends onLower Bounds to Option Prices(Equations 9.5 and 9.6, pages 218-219)

rTKeDSc −−−≥ 0

0SKeDp rT −+≥ −


Extensions of Put-Call Parity

American options; D = 0S0 - K < C - P < S0 - Ke -rT

(Equation 9.4, p. 215)

European options; D > 0c + D + Ke -rT = p + S0(Equation 9.7, p. 219)

American options; D > 0S0 - D - K < C - P < S0 - Ke -rT

(Equation 9.8, p. 219)


Trading StrategiesInvolving Options

Chapter 10


Types of Strategies

Take a position in the option andthe underlyingTake a position in 2 or moreoptions of the same type (A spread)Combination: Take a position in amixture of calls & puts (Acombination)


Positions in an Option & theUnderlying (Figure 10.1, page 224)

Profit

STK

Profit

ST

K

Profit

ST

K

Profit

STK

(a) (b)

(c) (d)


Bull Spread Using Calls(Figure 10.2, page 225)

K1 K2

Profit

ST


Bull Spread Using PutsFigure 10.3, page 226

K1 K2

Profit

ST


Bear Spread Using PutsFigure 10.4, page 227

K1 K2

Profit

ST


Bear Spread Using CallsFigure 10.5, page 229

K1 K2

Profit

ST


Box Spread

A combination of a bull call spread and abear put spreadIf all options are European a box spread isworth the present value of the differencebetween the strike pricesIf they are American this is not necessarilyso. (See Business Snapshot 10.1)


Butterfly Spread Using CallsFigure 10.6, page 231

K1 K3

Profit

STK2


Butterfly Spread Using PutsFigure 10.7, page 232

K1 K3

Profit

STK2


Calendar Spread Using CallsFigure 10.8, page 232

Profit

ST

K


Calendar Spread Using PutsFigure 10.9, page 233

Profit

ST

K


A Straddle CombinationFigure 10.10, page 234

Profit

STK


Strip & StrapFigure 10.11, page 235

Profit

K ST

Profit

K ST

Strip Strap


A Strangle CombinationFigure 10.12, page 236

K1 K2

Profit

ST


Binomial Trees

Chapter 11


A Simple Binomial Model

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

Stock Price = $22

Stock Price = $18

Stock price = $20


Stock Price = $22Option Price = $1

Stock Price = $18Option Price = $0

Stock price = $20Option Price=?

A Call Option (Figure 11.1, page 242)

A 3-month call option on the stock has a strike price of21.


Consider the Portfolio: long ∆ sharesshort 1 call option

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

22∆ – 1

18∆

Setting Up a Riskless Portfolio


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 is22 × 0.25 – 1 = 4.50

The value of the portfolio today is 4.5e – 0.12×0.25 = 4.3670


Valuing the OptionThe portfolio that is

long 0.25 sharesshort 1 option

is worth 4.367The value of the shares is

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

0.633 (= 5.000 – 4.367 )


Generalization (Figure 11.2, page 243)

A derivative lasts for time T and isdependent on a stock

S0u ƒu

S0d ƒd

S0ƒ


Generalization(continued)

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

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

dSuSfdu

00 −−

=∆ƒ

S0u∆ – ƒu

S0d∆ – ƒd



Value of the portfolio at time T isS0u∆ – ƒu

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

Another expression for theportfolio value today is S0∆ – fHence

ƒ = S0∆ – (S0u∆ – ƒu )e–rT



Substituting for ∆ we obtain ƒ = [ pƒu + (1 – p)ƒd ]e–rT

where

p e du d

rT

=−

−


p as a Probability

It is natural to interpret p and 1-p as probabilities ofup and down movementsThe value of a derivative is then its expected payoffin a risk-neutral world discounted at the risk-free rate

S0u ƒu

S0d ƒd

S0ƒ

p

(1 – p )


Risk-neutral Valuation

When the probability of an up and downmovements are p and 1-p the expected stockprice at time T is S0erT

This shows that the stock price earns the risk-free rateBinomial trees illustrate the general result that tovalue a derivative we can assume that theexpected return on the underlying asset is therisk-free rate and discount at the risk-free rateThis is known as using risk-neutral valuation


Original Example Revisited

Since p is the probability that gives a return on thestock equal to the risk-free rate. We can find it from20e0.12 ×0.25 = 22p + 18(1 – p )which gives p = 0.6523Alternatively, we can use the formula

6523.09.01.1

9.00.250.12

=−

−=

−−

=×e

dudep

rT

S0u = 22 ƒu = 1

S0d = 18 ƒd = 0

S0 ƒ

p

(1 – p )


Valuing the Option Using Risk-Neutral Valuation

The value of the option is e–0.12×0.25 (0.6523×1 + 0.3477×0) = 0.633

S0u = 22 ƒu = 1

S0d = 18 ƒd = 0

S0ƒ

0.6523

0.3477


Irrelevance of Stock’s ExpectedReturn

When we are valuing an option in terms of thethe price of the underlying asset, theprobability of up and down movements in thereal world are irrelevantThis is an example of a more general resultstating that the expected return on theunderlying asset in the real world is irrelevant


A Two-Step ExampleFigure 11.3, page 246

Each time step is 3 monthsK=21, r=12%

20

22

18

24.2

19.8

16.2


Valuing a Call OptionFigure 11.4, page 247

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

201.2823

22

18

24.23.2

19.80.0

16.20.0

2.0257

0.0

A

B

C

D

E

F


A Put Option Example; K=52Figure 11.7, page 250

K = 52, time step = 1yrr = 5%

504.1923

60

40

720

484

3220

1.4147

9.4636

A

B

C

D

E

F


What Happens When an Option isAmerican (Figure 11.8, page 251)

505.0894

60

40

720

484

3220

1.4147

12.0

A

B

C

D

E

F


Delta

Delta (∆) is the ratio of the changein the price of a stock option to thechange in the price of theunderlying stockThe value of ∆ varies from node tonode


Choosing u and d

One way of matching the volatility is to set

where σ is the volatility and ∆t is the lengthof the time step. This is the approach usedby Cox, Ross, and Rubinstein

t

t

eud

eu∆σ−

∆σ

==

=

1


The Probability of an Up Move

contract futures a for rate free-risk

foreign the is herecurrency w a for

index the on yield dividend the is eindex wher stock a for

stock paying dnondividen a for

1

)(

)(

=

=

=

=

−−

=

∆−

∆−

∆

a

rea

qeaea

dudap

ftrr

tqr

tr

f

Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Wiener Processes andItô’s Lemma

Chapter 12


Types of Stochastic Processes

Discrete time; discrete variableDiscrete time; continuous variableContinuous time; discrete variableContinuous time; continuous variable


Modeling Stock Prices

We can use any of the four types ofstochastic processes to model stockpricesThe continuous time, continuousvariable process proves to be the mostuseful for the purposes of valuingderivatives


Markov Processes (See pages 263-64)

In a Markov process future movementsin a variable depend only on where weare, not the history of how we gotwhere we areWe assume that stock prices followMarkov processes


Weak-Form Market Efficiency

This asserts that it is impossible toproduce consistently superior returns witha trading rule based on the past history ofstock prices. In other words technicalanalysis does not work.A Markov process for stock prices isclearly consistent with weak-form marketefficiency


Example of a Discrete TimeContinuous Variable Model

A stock price is currently at $40At the end of 1 year it is considered that itwill have a probability distribution ofφ(40,10) where φ(µ,σ) is a normaldistribution with mean µ and standarddeviation σ.


QuestionsWhat is the probability distribution of thestock price at the end of 2 years?½ years?¼ years? ∆t years?

Taking limits we have defined acontinuous variable, continuous timeprocess


Variances & StandardDeviations

In Markov processes changes insuccessive periods of time areindependentThis means that variances are additiveStandard deviations are not additive


Variances & Standard Deviations(continued)

In our example it is correct to say thatthe variance is 100 per year.It is strictly speaking not correct to saythat the standard deviation is 10 peryear.


A Wiener Process (See pages 265-67)

We consider a variable z whose value changescontinuouslyThe change in a small interval of time ∆t is ∆zThe variable follows a Wiener process if

1.

2. The values of ∆z for any 2 different (non-overlapping) periods of time are independent

(0,1) is where φε∆ε=∆ tz


Properties of a Wiener Process

Mean of [z (T ) – z (0)] is 0Variance of [z (T ) – z (0)] is TStandard deviation of [z (T ) – z (0)] is

T


Taking Limits . . .

What does an expression involving dz and dtmean?It should be interpreted as meaning that thecorresponding expression involving ∆z and ∆t istrue in the limit as ∆t tends to zeroIn this respect, stochastic calculus is analogous toordinary calculus


Generalized Wiener Processes(See page 267-69)

A Wiener process has a drift rate (i.e.average change per unit time) of 0and a variance rate of 1In a generalized Wiener process thedrift rate and the variance rate can beset equal to any chosen constants


Generalized Wiener Processes(continued)

The variable x follows a generalizedWiener process with a drift rate of aand a variance rate of b2 if

dx=a dt+b dz


Generalized Wiener Processes(continued)

Mean change in x in time T is aTVariance of change in x in time T is b2TStandard deviation of change in x intime T is

tbtax ∆ε+∆=∆

b T


The Example Revisited

A stock price starts at 40 and has a probabilitydistribution of φ(40,10) at the end of the yearIf we assume the stochastic process is Markovwith no drift then the process is

dS = 10dzIf the stock price were expected to grow by $8on average during the year, so that the year-end distribution is φ(48,10), the process wouldbe

dS = 8dt + 10dz


Itô Process (See pages 269)

In an Itô process the drift rate and thevariance rate are functions of time

dx=a(x,t) dt+b(x,t) dzThe discrete time equivalent

is only true in the limit as ∆t tends to zero

ttxbttxax ∆ε+∆=∆ ),(),(


Why a Generalized Wiener Processis not Appropriate for Stocks

For a stock price we can conjecture that itsexpected percentage change in a short periodof time remains constant, not its expectedabsolute change in a short period of timeWe can also conjecture that our uncertainty asto the size of future stock price movements isproportional to the level of the stock price


An Ito Process for Stock Prices(See pages 269-71)

where µ is the expected return σ isthe volatility.

The discrete time equivalent is

dzSdtSdS σ+µ=

tStSS ∆εσ+∆µ=∆


Monte Carlo Simulation

We can sample random paths for thestock price by sampling values for εSuppose µ= 0.14, σ= 0.20, and ∆t = 0.01,then

ε+=∆ SSS 02.00014.0


Monte Carlo Simulation – One Path (See Table12.1, page 272)

Period

Stock Price at Start of Period

Random Sample for ε

Change in Stock Price, ∆S

0 20.000 0.52 0.236

1 20.236 1.44 0.611

2 20.847 -0.86 -0.329

3 20.518 1.46 0.628

4 21.146 -0.69 -0.262


Itô’s Lemma (See pages 273-274)

If we know the stochastic processfollowed by x, Itô’s lemma tells us thestochastic process followed by somefunction G (x, t )Since a derivative security is a function ofthe price of the underlying and time, Itô’slemma plays an important part in theanalysis of derivative securities


Taylor Series Expansion

A Taylor’s series expansion of G(x, t)gives

K+∆∂∂

+∆∆∂∂

∂+

∆∂∂

+∆∂∂

+∆∂∂

=∆

22

22

22

2

ttGtx

txG

xxGt

tGx

xGG

½

½


Ignoring Terms of Higher OrderThan ∆t

t

x

xxGt

tGx

xGG

ttGx

xGG

∆

∆

∆+∆+∆=∆

∆+∆=∆

½

22

2

order of

is whichcomponent a has because

becomes this calculus stochastic In

havewecalculusordinary In

∂∂

∂∂

∂∂

∂∂

∂∂


Substituting for ∆x

tbxGt

tGx

xGG

ttbtax

dztxbdttxadx

∆ε∂∂

+∆∂∂

+∆∂∂

=∆

∆∆ε∆∆

+=

222

2

½

order thanhigher of termsignoringThen + =

thatso),(),(

Suppose


The ε2∆t Term

tbxGt

tGx

xGG

ttttE

EEE

E

∆∂∂

+∆∂∂

+∆∂∂

=∆

∆∆

∆=∆ε

=ε

=ε−ε

=εφ≈ε

22

2

2

2

22

21

)(1)(

1)]([)(0)(,)1,0(

Hence ignored. be can and to alproportion is of variance The

that follows It

Since

2


Taking Limits

Lemma sIto' is This

½ obtain We

ngSubstituti

½ limits Taking

dzbxGdtb

xG

tGa

xGdG

dzbdtadx

dtbxGdt

tGdx

xGdG

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

=

+=∂∂

+∂∂

+∂∂

=

22

2

22

2


Application of Ito’s Lemmato a Stock Price Process

dzSSGdtS

SG

tGS

SGdG

tSGzdSdtSSd

½

and of function a For

is processprice stockThe

σ∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛σ

∂∂

+∂∂

+µ∂∂

=

σ+µ=

222

2


Examples

dzdtdG

SG

dzGdtGrdGeSG

TtTr

2.

time at maturing contract a for stock a of price forward The 1.

σ+⎟⎟⎠

⎞⎜⎜⎝

⎛ σ−µ=

=

σ+−µ== −

2

ln

)(

2

)(


The Black-Scholes-Merton Model

Chapter 13


The Stock Price Assumption

Consider a stock whose price is SIn a short period of time of length ∆t, thereturn on the stock is normally distributed:

where µ is expected return and σ is volatility

( )ttSS

∆∆≈∆ σµφ ,


The Lognormal Property(Equations 13.2 and 13.3, page 282)

It follows from this assumption that

Since the logarithm of ST is normal, ST islognormally distributed

ln ln ,

ln ln ,

S S T T

S S T T

T

T

− ≈ −⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

≈ + −⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

0

2

0

2

2

2

φ µσ

σ

φ µσ

σ

or


The Lognormal Distribution

E S S e

S S e eT

T

TT T

( )

( ) ( )

=

= −0

02 2 2

1

var

µ

µ σ


Continuously Compounded Return, x(Equations 13.6 and 13.7), page 283)

,2

or

ln1=

or

2

0

0

⎟⎟⎠

⎞⎜⎜⎝

⎛−≈

=

Tx

SS

Tx

eSS

T

xTT

σσµφ


The Expected Return

The expected value of the stock price is S0eµT

The expected return on the stock is µ – σ2/2 not µ

This is because

are not the same

)]/[ln()]/(ln[ 00 SSESSE TT and


µ and µ- σ2/2

Suppose we have daily data for a period ofseveral monthsµ is the average of the returns in each day[=E(∆S/S)]µ- σ2/2 is the expected return over thewhole period covered by the datameasured with continuous compounding(or daily compounding, which is almost thesame)


Mutual Fund Returns (See BusinessSnapshot 13.1 on page 285)

Suppose that returns in successive yearsare 15%, 20%, 30%, -20% and 25%The arithmetic mean of the returns is 14%The returned that would actually beearned over the five years (the geometricmean) is 12.4%


The Volatility

The volatility is the standard deviation of thecontinuously compounded rate of return in 1yearThe standard deviation of the return in time∆t isIf a stock price is $50 and its volatility is 25%per year what is the standard deviation ofthe price change in one day?

t∆σ


Estimating Volatility fromHistorical Data (page 286-88)

• Take observations S0, S1, . . . , Sn atintervals of τ years

• Calculate the continuously compoundedreturn in each interval as:

• Calculate the standard deviation, s , ofthe ui´s

• The historical volatility estimate is:

uS

Sii

i=

⎛⎝⎜

⎞⎠⎟

−

ln1

τ=σ

sˆ


Nature of Volatility

Volatility is usually much greater when themarket is open (i.e. the asset is trading)than when it is closedFor this reason time is usually measuredin “trading days” not calendar days whenoptions are valued


The Concepts Underlying Black-Scholes

The option price and the stock price dependon the same underlying source of uncertaintyWe can form a portfolio consisting of thestock and the option which eliminates thissource of uncertaintyThe portfolio is instantaneously riskless andmust instantaneously earn the risk-free rateThis leads to the Black-Scholes differentialequation


The Derivation of the Black-ScholesDifferential Equation

shares :ƒ+

derivative :1 of consisting portfolio a upset e W

ƒƒ½ƒƒƒ

222

2

S

zSS

tSSt

SS

zStSS

∂∂

−

∆σ∂∂

+∆⎟⎟⎠

⎞⎜⎜⎝

⎛σ

∂∂

+∂∂

+µ∂∂

=∆

∆σ+∆µ=∆


ƒƒ

bygiven is in time valueitsin change The

ƒƒ

bygiven is portfolio theof valueThe

SS

t

SS

∆∂∂

+∆−=∆Π

∆∂∂

+−=Π

Π

The Derivation of the Black-ScholesDifferential Equation continued


The Derivation of the Black-ScholesDifferential Equation continued

ƒƒ ½ƒƒ

:equation aldifferenti Scholes-Black get the toequations in these and ƒfor substitute We

Hence rate.

free-riskthebemust portfolioon thereturn The

2

222 r

SS

SrS

t

Str

=∂∂

σ+∂∂

+∂∂

∆∆Π∆=∆Π


The Differential Equation

Any security whose price is dependent on thestock price satisfies the differential equationThe particular security being valued is determinedby the boundary conditions of the differentialequationIn a forward contract the boundary condition is

ƒ = S – K when t =T The solution to the equation is

ƒ = S – K e–r (T – t )


The Black-Scholes Formulas(See pages 295-297)

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 normallydistributed variable with a mean of zeroand a standard deviation of 1 is less than xSee tables at the end of the book


Properties of Black-Scholes Formula

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

As S0 becomes very small c tends to zeroand 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 variablesaffected by risk preferenceThe solution to the differential equation istherefore the same in a risk-free world as itis in the real worldThis leads to the principle of risk-neutralvaluation


Applying Risk-Neutral Valuation(See appendix at the end of Chapter 13)

1. Assume that the expectedreturn from the stock price isthe risk-free rate

2. Calculate the expected payofffrom the option

3. Discount at the risk-free rate


Valuing a Forward Contract withRisk-Neutral Valuation

Payoff is ST – KExpected payoff in a risk-neutral world isSerT – KPresent value of expected payoff is

e-rT[SerT – K]=S – Ke-rT


Implied Volatility

The implied volatility of an option is thevolatility for which the Black-Scholes priceequals the market priceThe is a one-to-one correspondencebetween prices and implied volatilitiesTraders and brokers often quote impliedvolatilities rather than dollar prices


An Issue of Warrants & ExecutiveStock Options

When a regular call option is exercised the stockthat is delivered must be purchased in the openmarketWhen a warrant or executive stock option isexercised new Treasury stock is issued by thecompanyIf little or no benefits are foreseen by the marketthe stock price will reduce at the time the issue ofis announced.There is no further dilution (See BusinessSnapshot 13.3.)


The Impact of Dilution

After the options have been issued it is notnecessary to take account of dilution whenthey are valuedBefore they are issued we can calculatethe cost of each option as N/(N+M) timesthe price of a regular option with the sameterms where N is the number of existingshares and M is the number of new sharesthat will be created if exercise takes place


Dividends

European options on dividend-payingstocks are valued by substituting the stockprice less the present value of dividendsinto Black-ScholesOnly dividends with ex-dividend datesduring life of option should be includedThe “dividend” should be the expectedreduction in the stock price expected


American Calls

An American call on a non-dividend-payingstock should never be exercised earlyAn American call on a dividend-paying stockshould only ever be exercised immediatelyprior to an ex-dividend dateSuppose dividend dates are at times t1, t2,…tn. Early exercise is sometimes optimal attime ti if the dividend at that time is greaterthan

]1[ )( 1 ii ttreK −− +−


Black’s Approximation for Dealing withDividends in American Call Options

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


Options on Stock Indices,Currencies, and Futures

Chapter 14


European Options on StocksProviding a Dividend Yield

We get the same probabilitydistribution for the stock price at timeT in each of the following cases:1. The stock starts at price S0 andprovides a dividend yield = q2. The stock starts at price S0e–q T

and provides no income


European Options on StocksProviding Dividend Yieldcontinued

We can value European options byreducing the stock price to S0e–q T and thenbehaving as though there is no dividend


Extension of Chapter 9 Results(Equations 14.1 to 14.3)

rTqT KeeSc −− −≥ 0

Lower Bound for calls:

Lower Bound for puts

qTrT eSKep −− −≥ 0

Put Call Parity

qTrT eSpKec −− +=+ 0


Extension of Chapter 13 Results(Equations 14.4 and 14.5)

TTqrKSd

TTqrKSd

dNeSdNKep

dNKedNeScqTrT

rTqT

σσ−−+

=

σσ+−+

=

−−−=

−=−−

−−

)2/2()/ln(

)2/2()/ln(

)()(

)()(

02

01

102

210

where


The Binomial Model

S0u ƒu

S0d ƒd

S0 ƒ

p

(1 – p )

f=e-rT[pfu+(1-p)fd ]


The Binomial Modelcontinued

In a risk-neutral world the stock pricegrows at r-q rather than at r when thereis a dividend yield at rate qThe probability, p, of an up movementmust therefore satisfy

pS0u+(1-p)S0d=S0e (r-q)T

so that

p e du d

r q T

=−

−

−( )


Index Options (page 316-321)

The most popular underlying indices in theU.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 aresettled in cash; OEX is American and therest are European.


LEAPS

Leaps are options on stock indices thatlast up to 3 yearsThey have December expiration datesThey are on 10 times the indexLeaps also trade on some individualstocks


Index Option Example

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


Using Index Options for PortfolioInsurance

Suppose the value of the index is S0 and the strikeprice is KIf a portfolio has a β of 1.0, the portfolio insuranceis obtained by buying 1 put option contract on theindex for each 100S0 dollars heldIf the β is not 1.0, the portfolio manager buys β putoptions for each 100S0 dollars heldIn both cases, K is chosen to give the appropriateinsurance level


Example 1

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


Example 2

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


If index rises to 1040, it provides a40/1000 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 Leveland Portfolio Value in 3 months


Determining the Strike Price (Table14.2, page 320)

Value of Index in 3 months

Expected Portfolio Value in 3 months ($)

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

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 optionon a stock paying a dividend yieldSet S0 = current index levelSet q = average dividend yieldexpected during the life of the option


Currency OptionsCurrency options trade on the PhiladelphiaExchange (PHLX)There also exists an active over-the-counter(OTC) marketCurrency options are used by corporationsto buy insurance when they have an FXexposure


The Foreign Interest Rate

We denote the foreign interest rate by rf

When a U.S. company buys one unit ofthe foreign currency it has aninvestment of S0 dollarsThe return from investing at the foreignrate is rf S0 dollarsThis shows that the foreign currencyprovides a “dividend yield” at rate rf


Valuing European CurrencyOptions

A foreign currency is an asset thatprovides a “dividend yield” equal to rf

We can use the formula for an optionon a stock paying a dividend yield :

Set S0 = current exchange rate Set q = rƒ


Formulas for European CurrencyOptions(Equations 14.7 and 14.8, page 322)

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(Equations 14.9 and 14.10, page 322)

F S e r r Tf0 0= −( )

Using

TddT

TKFd

dNFdKNep

dKNdNFecrT

rT

σ−=

σσ+

=

−−−=

−=−

−

12

20

1

102

210

2/)/ln(

)]()([

)]()([


Mechanics of Call Futures Options

When a call futures option is exercisedthe holder acquires

1. A long position in the futures2. A cash amount equal to the excess of

the futures price over the strike price


Mechanics of Put Futures Option

When a put futures option is exercised theholder acquires

1. A short position in the futures2. A cash amount equal to the excess of

the strike price over the futures price


The Payoffs

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

where F0 is futures price at time ofexercise


Put-Call Parity for FuturesOptions (Equation 14.11, page 329)

Consider the following two portfolios:1. European call plus Ke-rT of cash

2. European put plus long futures pluscash equal to F0e-rT

They must be worth the same at time T sothat

c+Ke-rT=p+F0 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 of29.


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 futuresshort 1 call option

The value of the portfolio in 1 month is-1.6

The value of the portfolio today is -1.6e – 0.06/12 = -1.592


Valuing the Option

The portfolio that islong 0.8 futuresshort 1 option

is worth -1.592The value of the futures is zeroThe value of the option musttherefore be 1.592


Generalization of Binomial TreeExample (Figure 14.2, page 330)

A derivative lasts for time T and isdependent on a futures price

F0u ƒu

F0d ƒd

F0 ƒ



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

The portfolio is riskless when

∆ =−−

ƒ u dfF u F d0 0

F0u ∆ − F0 ∆ – ƒu

F0d ∆− F0∆ – ƒd



Value of the portfolio at time Tis 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

p du d

=−−

1


Valuing European FuturesOptions

We can use the formula for an optionon a stock paying a dividend yield

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

Setting q = r ensures that the expectedgrowth of F in a risk-neutral world iszero


Growth Rates For Futures Prices

A futures contract requires no initialinvestmentIn a risk-neutral world the expected returnshould be zeroThe expected growth rate of the futuresprice is therefore zeroThe futures price can therefore be treatedlike a stock paying a dividend yield of r


Black’s Formula(Equations 14.16 and 14.17, page 333)

The formulas for European options onfutures are known as Black’s formulas

[ ][ ]

TdT

TKFd

TTKFd

dNFdNKep

dNKdNFecrT

rT

σ−=σ

σ−=

σσ+

=

−−−=

−=−

−

10

2

01

102

210

2/2)/ln(

2/2)/ln(

)()(

)()(

where


Futures Option Prices vs SpotOption Prices

If futures prices are higher than spotprices (normal market), an Americancall on futures is worth more than asimilar American call on spot. AnAmerican put on futures is worth lessthan a similar American put on spotWhen futures prices are lower than spotprices (inverted market) the reverse istrue


Summary of Key Results

We can treat stock indices, currencies,and futures like a stock paying adividend yield of q

For stock indices, q = averagedividend yield on the index over theoption lifeFor currencies, q = rƒ

For futures, q = r


The Greek Letters

Chapter 15


Example

A bank has sold for $300,000 a European calloption on 100,000 shares of a nondividendpaying stock S0 = 49, K = 50, r = 5%, σ = 20%,T = 20 weeks, µ = 13%The Black-Scholes value of the option is$240,000How does the bank hedge its risk to lock in a$60,000 profit?


Naked & Covered Positions

Naked positionTake no action

Covered positionBuy 100,000 shares today

Both strategies leave the bankexposed to significant risk


Stop-Loss Strategy

This involves:Buying 100,000 shares as soon asprice reaches $50Selling 100,000 shares as soon asprice falls below $50This deceptively simple hedgingstrategy does not work well


Delta (See Figure 15.2, page 345)

Delta (∆) is the rate of change of theoption price with respect to the underlying

Optionprice

A

BSlope = ∆

Stock price


Delta Hedging

This involves maintaining a delta neutralportfolioThe delta of a European call on a stockpaying dividends at rate q is N (d 1)e– qT

The delta of a European put ise– qT [N (d 1) – 1]


Delta Hedgingcontinued

The hedge position must be frequentlyrebalancedDelta hedging a written option involves a“buy high, sell low” trading ruleSee Tables 15.2 (page 350) and 15.3(page 351) for examples of delta hedging


Using Futures for Delta Hedging

The delta of a futures contract is e(r-q)T

times the delta of a spot contractThe position required in futures for deltahedging is therefore e-(r-q)T times theposition required in the corresponding spotcontract


Theta

Theta (Θ) of a derivative (or portfolio ofderivatives) is the rate of change of the valuewith respect to the passage of timeThe theta of a call or put is usually negative.This means that, if time passes with the price ofthe underlying asset and its volatility remainingthe same, the value of the option declines


Gamma

Gamma (Γ) is the rate of change ofdelta (∆) with respect to the price of theunderlying assetGamma is greatest for options that areclose to the money (see Figure 15.9,page 358)


Gamma Addresses Delta HedgingErrors Caused By Curvature(Figure 15.7, page 355)

S

CStock price

S'

Callprice

C''C'


Interpretation of GammaFor a delta neutral portfolio,

∆Π ≈ Θ ∆t + ½Γ∆S 2

∆Π

∆S

Negative Gamma

∆Π

∆S

Positive Gamma


Relationship Between Delta,Gamma, and Theta

For a portfolio of derivatives on a stockpaying a continuous dividend yield atrate q

Θ ∆ Γ Π+ − + =( )r q S S r12

2 2σ


Vega

Vega (ν) is the rate of change of thevalue of a derivatives portfolio withrespect to volatilityVega tends to be greatest for optionsthat are close to the money (See Figure15.11, page 361)


Managing Delta, Gamma, & Vega

• ∆ can be changed by taking a position inthe underlying

To adjust Γ & ν it is necessary to take aposition in an option or other derivative


Rho

Rho is the rate of change of thevalue of a derivative with respectto the interest rate

For currency options there are 2rhos


Hedging in Practice

Traders usually ensure that their portfoliosare delta-neutral at least once a dayWhenever the opportunity arises, theyimprove gamma and vegaAs portfolio becomes larger hedgingbecomes less expensive


Scenario Analysis

A scenario analysis involves testing theeffect on the value of a portfolio of differentassumptions concerning asset prices andtheir volatilities


Hedging vs Creation of an OptionSynthetically

When we are hedging we takepositions that offset ∆, Γ, ν, etc.When we create an optionsynthetically we take positionsthat match ∆, Γ, & ν


Portfolio Insurance

In October of 1987 many portfoliomanagers attempted to create a putoption on a portfolio syntheticallyThis involves initially selling enough ofthe portfolio (or of index futures) tomatch the ∆ of the put option


Portfolio Insurancecontinued

As the value of the portfolio increases, the∆ of the put becomes less negative andsome of the original portfolio isrepurchasedAs the value of the portfolio decreases, the∆ of the put becomes more negative andmore of the portfolio must be sold


Portfolio Insurancecontinued

The strategy did not work well on October19, 1987...


Volatility Smiles

Chapter 16


Put-Call Parity Arguments

Put-call parity p +S0e-qT = c +K e–r T

holds regardless of the assumptionsmade about the stock price distributionIt follows thatpmkt-pbs=cmkt-cbs


Implied Volatilities

When pbs=pmkt, it must be true that cbs=cmkt

It follows that the implied volatilitycalculated from a European call optionshould be the same as that calculatedfrom a European put option when bothhave the same strike price and maturityThe same is approximately true ofAmerican options


Volatility Smile

A volatility smile shows the variation ofthe implied volatility with the strike priceThe volatility smile should be the samewhether calculated from call options orput options


The Volatility Smile for ForeignCurrency Options(Figure 16.1, page 377)

ImpliedVolatility

StrikePrice


Implied Distribution for ForeignCurrency Options (Figure 16.2, page 377)

Both tails are heavier than the lognormaldistributionIt is also “more peaked” than thelognormal distribution


The Volatility Smile for EquityOptions (Figure 16.3, page 380)

ImpliedVolatility

StrikePrice


Implied Distribution for EquityOptions (Figure 16.4, page 380)

The left tail is heavier and the right tail isless heavy than the lognormaldistribution


Other Volatility Smiles?

What is the volatility smile ifTrue distribution has a less heavy left tailand heavier right tailTrue distribution has both a less heavy lefttail and a less heavy right tail


Possible Causes of Volatility Smile

Asset price exhibiting jumps rather thancontinuous changeVolatility for asset price being stochastic

(One reason for a stochastic volatility inthe case of equities is the relationshipbetween volatility and leverage)


Volatility Term Structure

In addition to calculating a volatility smile,traders also calculate a volatility termstructureThis shows the variation of impliedvolatility with the time to maturity of theoption


Volatility Term Structure

The volatility term structure tends to bedownward sloping when volatility is highand upward sloping when it is low


Example of a Volatility Surface(Table 16.2, page 382)

S tr ik e P r ic e

0 .9 0 0 .9 5 1 .0 0 1 .0 5 1 .1 0

1 m n th 1 4 .2 1 3 .0 1 2 .0 1 3 .1 1 4 .5

3 m n th 1 4 .0 1 3 .0 1 2 .0 1 3 .1 1 4 .2

6 m n th 1 4 .1 1 3 .3 1 2 .5 1 3 .4 1 4 .3

1 y e a r 1 4 .7 1 4 .0 1 3 .5 1 4 .0 1 4 .8

2 y e a r 1 5 .0 1 4 .4 1 4 .0 1 4 .5 1 5 .1

5 y e a r 1 4 .8 1 4 .6 1 4 .4 1 4 .7 1 5 .0


Basic NumericalProcedures

Chapter 17


Tree Approaches to DerivativesValuation

TreesMonte Carlo simulationFinite difference methods


Binomial Trees

Binomial trees are frequently used toapproximate the movements in the price ofa stock or other assetIn each small interval of time the stockprice is assumed to move up by aproportional amount u or to move down bya proportional amount d


Movements in Time ∆t(Figure 17.1, page 392)

Su

SdS

p

1 – p


1. Tree Parameters for assetpaying a dividend yield of q

Parameters p, u, and d are chosen so that the treegives correct values for the mean & variance of thestock price changes in a risk-neutral world

Mean: e(r-q)∆t = pu + (1– p )dVariance: σ2∆t = pu2 + (1– p )d 2 – e2(r-q)∆t

A further condition often imposed is u = 1/ d


2. Tree Parameters for assetpaying a dividend yield of q(Equations 17.4 to 17.7)

When ∆t is small a solution to the equations is

tqr

t

t

eadudap

ed

eu

∆−

∆σ−

∆σ

=−−

=

=

=

)(


The Complete Tree(Figure 17.2, page 394)

S0u 2

S0u 4

S0d 2

S0d 4

S0

S0u

S0d S0 S0

S0u 2

S0d 2

S0u 3

S0u

S0d

S0d 3


Backwards Induction

We know the value of the option atthe final nodesWe work back through the treeusing risk-neutral valuation tocalculate the value of the option ateach node, testing for earlyexercise when appropriate


Example: Put Option(Example 17.1, page 394)

S0 = 50; K = 50; r =10%; σ = 40%;T = 5 months = 0.4167;∆t = 1 month = 0.0833

The parameters implyu = 1.1224; d = 0.8909;a = 1.0084; p = 0.5073


Example (continued)Figure 17.3, page 395

89.070.00

79.350.00

70.70 70.700.00 0.00

62.99 62.990.64 0.00

56.12 56.12 56.122.16 1.30 0.00

50.00 50.00 50.004.49 3.77 2.66

44.55 44.55 44.556.96 6.38 5.45

39.69 39.6910.36 10.31

35.36 35.3614.64 14.64

31.5018.50

28.0721.93


Calculation of Delta

Delta is calculated from the nodes at time∆t

Delta =−−

= −2 16 6 96

56 12 44 550 41. .

. ..


Calculation of Gamma

Gamma is calculated from the nodes attime 2∆t

∆ ∆

∆ ∆

1 2

2

064 37762 99 50

024 377 103650 39 69

064

1165003

=−

−= − =

−−

= −

−=

. ..

. ; . ..

.

..Gamma= 1


Calculation of Theta

Theta is calculated from the central nodesat times 0 and 2∆t

Theta= per year

or - . per calendar day

377 4 4901667

4 3

0012

. ..

.−= −


Calculation of Vega

We can proceed as followsConstruct a new tree with a volatility of 41%instead of 40%.Value of option is 4.62Vega is

4 62 4 49 0 13. . .− =per 1% change in volatility


Trees for Options on Indices,Currencies and Futures Contracts

As with Black-Scholes:For options on stock indices, q equals thedividend yield on the indexFor options on a foreign currency, q equals theforeign risk-free rateFor options on futures contracts q = r


Binomial Tree for DividendPaying Stock

Procedure:Draw the tree for the stock price less thepresent value of the dividendsCreate a new tree by adding the presentvalue of the dividends at each node

This ensures that the tree recombines andmakes assumptions similar to those when theBlack-Scholes model is used


Extensions of Tree Approach

Time dependent interest ratesThe control variate technique


Alternative Binomial Tree(Section 17.4, page 406)

Instead of setting u = 1/d we can seteach of the 2 probabilities to 0.5 and

ttqr

ttqr

ed

eu∆σ−∆σ−−

∆σ+∆σ−−

=

=)2/(

)2/(

2

2


Trinomial Tree (Page 409)

61

212

32

61

212

/1

2

2

2

2

3

+⎟⎟⎠

⎞⎜⎜⎝

⎛ σ−

σ∆

−=

=

+⎟⎟⎠

⎞⎜⎜⎝

⎛ σ−

σ∆

=

== ∆σ

rtp

p

rtp

udeu

d

m

u

t

S S

Sd

Su

pu

pm

pd


Time Dependent Parameters in aBinomial Tree (page 409)

Making r or q a function of time does notaffect the geometry of the tree. Theprobabilities on the tree become functionsof time.We can make σ a function of time bymaking the lengths of the time stepsinversely proportional to the variance rate.


Monte Carlo Simulation and π

How could you calculate π by randomlysampling points in the square?


Monte Carlo Simulation and Options When used to value European stock options, Monte

Carlo simulation involves the following steps:1. Simulate 1 path for the stock price in a risk neutral

world2. Calculate the payoff from the stock option3. Repeat steps 1 and 2 many times to get many sample

payoff4. Calculate mean payoff5. Discount mean payoff at risk free rate to get an

estimate of the value of the option


Sampling Stock Price Movements(Equations 17.13 and 17.14, page 411)

In a risk neutral world the process for a stockprice is

We can simulate a path by choosing timesteps of length ∆t and using the discreteversion of this

where ε is a random sample from φ(0,1)tStSS ∆εσ+∆µ=∆ ˆ

dS S dt S dz= +$µ σ


A More Accurate Approach(Equation 17.15, page 412)

( )

( )

( ) ttetSttS

tttSttS

dzdtSd

∆εσ+∆σ−µ=∆+

∆σε+∆σ−µ=−∆+

σ+σ−µ=

or

is this of version discrete The Use

2/ˆ

2

2

2

)()(

2/ˆ)(ln)(ln

2/ˆln


Extensions

When a derivative depends on severalunderlying variables we can simulatepaths for each of them in a risk-neutralworld to calculate the values for thederivative


Sampling from NormalDistribution (Page 414)

One simple way to obtain a samplefrom φ(0,1) is to generate 12 randomnumbers between 0.0 & 1.0, take thesum, and subtract 6.0In Excel =NORMSINV(RAND()) givesa random sample from φ(0,1)


To Obtain 2 Correlated NormalSamples

Obtain independent normal samples x1 andx2 and set

A procedure known as Cholesky’sdecomposition when samples are requiredfrom more than two normal variables

2212

11

1 ρ−+ρ=ε=ε

xxx


Standard Errors in Monte CarloSimulation

The standard error of the estimate of theoption price is the standard deviation ofthe discounted payoffs given by thesimulation trials divided by the square rootof the number of observations.


Application of Monte CarloSimulation

Monte Carlo simulation can deal withpath dependent options, optionsdependent on several underlyingstate variables, and options withcomplex payoffsIt cannot easily deal with American-style options


Determining Greek Letters

For ∆:1.Make a small change to asset price2.Carry out the simulation again using the same

random number streams3.Estimate ∆ as the change in the option price

divided by the change in the asset price

Proceed in a similar manner for other Greek letters


Variance Reduction Techniques

Antithetic variable techniqueControl variate techniqueImportance samplingStratified samplingMoment matchingUsing quasi-random sequences


Sampling Through the Tree

Instead of sampling from the stochasticprocess we can sample paths randomlythrough a binomial or trinomial tree tovalue a derivative


Finite Difference Methods

Finite difference methods aim torepresent the differential equation inthe form of a difference equationWe form a grid by consideringequally spaced time values andstock price valuesDefine ƒi,j as the value of ƒ at timei∆t when the stock price is j∆S


Finite Difference Methods(continued)

ƒ2ƒƒƒ

or ƒƒƒƒƒ

2

ƒƒƒset we

ƒƒ21ƒƒIn

2,1,1,

2

2

1,,,1,2

2

1,1,

2

222

SS

SSSS

SS

rS

SS

rSt

jijiji

jijijiji

jiji

∆

−+=

∂∂

∆⎟⎟⎠

⎞⎜⎜⎝

⎛∆−

−∆

−=

∂∂

∆

−=

∂∂

=∂∂

σ+∂∂

+∂∂

−+

−+

−+


Implicit Finite Difference Method(Equation 17.25, page 420)

If we also set ƒ ƒ ƒ

we obtain the implicit finite difference method.This involves solving simultaneous equations of the form: ƒ ƒ ƒ ƒ

, ,

, , , ,

∂∂ t t

a b c

i j i j

j i j j i j j i j i j

=−

+ + =

+

− + +

1

1 1 1

∆


Explicit Finite Difference Method(page 422-428)

ƒƒƒƒ :form the of equations solving involves This

method difference finite explicit the obtain wepoint )( the at arethey as point )( the at

same the be to assumed are and If

,,,,

2

11*

1*

11*

2

1

+++−+ ++=

+∂∂∂∂

jijjijjijji cba

i,j,jiSfSf


Implicit vs Explicit FiniteDifference Method

The explicit finite difference method isequivalent to the trinomial tree approachThe implicit finite difference method isequivalent to a multinomial tree approach


Implicit vs Explicit Finite Difference Methods(Figure 17.16, page 425)

ƒi , j ƒi +1, j

ƒi +1, j –1

ƒi +1, j +1

ƒi +1, jƒi , j

ƒi , j –1

ƒi , j +1

Implicit Method Explicit Method


Other Points on Finite DifferenceMethods

It is better to have ln S rather than S as theunderlying variableImprovements over the basic implicit andexplicit methods:

Hopscotch methodCrank-Nicolson method


Chapter 18

Value at Risk


The Question Being Asked in VaR

“What loss level is such that we are X%confident it will not be exceeded in Nbusiness days?”


VaR and Regulatory Capital(Business Snapshot 18.1, page 436)

Regulators base the capital they requirebanks to keep on VaRThe market-risk capital is k times the 10-day 99% VaR where k is at least 3.0


VaR vs. C-VaR(See Figures 18.1 and 18.2)

VaR is the loss level that will not beexceeded with a specified probabilityC-VaR (or expected shortfall) is theexpected loss given that the loss is greaterthan the VaR levelAlthough C-VaR is theoretically moreappealing, it is not widely used


Advantages of VaR

It captures an important aspect of riskin a single numberIt is easy to understandIt asks the simple question: “How bad canthings get?”


Time Horizon

Instead of calculating the 10-day, 99% VaRdirectly analysts usually calculate a 1-day 99%VaR and assume

This is exactly true when portfolio changes onsuccessive days come from independentidentically distributed normal distributions

day VaR1-day VaR-10 ×= 10


Historical Simulation(See Tables 18.1 and 18.2, page 438-439))

Create a database of the daily movements in allmarket variables.The first simulation trial assumes that thepercentage changes in all market variables areas on the first dayThe second simulation trial assumes that thepercentage changes in all market variables areas on the second dayand so on


Historical Simulation continued

Suppose we use m days of historical dataLet vi be the value of a variable on day iThere are m-1 simulation trialsThe ith trial assumes that the value of themarket variable tomorrow (i.e., on day m+1) is

1−i

im v

vv


The Model-Building Approach

The main alternative to historical simulation is tomake assumptions about the probabilitydistributions of return on the market variablesand calculate the probability distribution of thechange in the value of the portfolio analyticallyThis is known as the model building approach orthe variance-covariance approach


Daily Volatilities

In option pricing we measure volatility “peryear”In VaR calculations we measure volatility“per day”

252year

day

σ=σ


Daily Volatility continued

Strictly speaking we should define σday asthe standard deviation of the continuouslycompounded return in one dayIn practice we assume that it is thestandard deviation of the percentagechange in one day


Microsoft Example (page 440)

We have a position worth $10 million inMicrosoft sharesThe volatility of Microsoft is 2% per day(about 32% per year)We use N=10 and X=99


Microsoft Example continued

The standard deviation of the change inthe portfolio in 1 day is $200,000The standard deviation of the change in 10days is

200 000 10 456, $632,=


Microsoft Example continued

We assume that the expected change inthe value of the portfolio is zero (This isOK for short time periods)We assume that the change in the value ofthe portfolio is normally distributedSince N(–2.33)=0.01, the VaR is

2 33 632 456 473 621. , $1, ,× =


AT&T Example (page 441)

Consider a position of $5 million in AT&TThe daily volatility of AT&T is 1% (approx16% per year)The S.D per 10 days is

The VaR is50 000 10 144, $158,=

158114 233 405, . $368,× =


Portfolio

Now consider a portfolio consisting of bothMicrosoft and AT&TSuppose that the correlation between thereturns is 0.3


S.D. of Portfolio

A standard result in statistics states that

In this case σX = 200,000 and σY = 50,000and ρ = 0.3. The standard deviation of thechange in the portfolio value in one day istherefore 220,227

YXYXYX σρσ+σ+σ=σ + 222


VaR for Portfolio

The 10-day 99% VaR for the portfolio is

The benefits of diversification are(1,473,621+368,405)–1,622,657=$219,369What is the incremental effect of the AT&Tholding on VaR?

657,622,1$33.210220,227 =××


The Linear Model

We assumeThe daily change in the value of a portfoliois linearly related to the daily returns frommarket variablesThe returns from the market variables arenormally distributed


The General Linear Modelcontinued (equations 18.1 and 18.2)

deviation standard sportfolio' theis and variableofy volatilit theis where

21

222

1 1

2

1

P

i

n

iijjiji

jiiiP

n

i

n

jijjijiP

n

iii

i

xP

σσ

ρσσαασασ

ρσσαασ

α

∑ ∑

∑∑

∑

= <

= =

=

+=

=

∆=∆


Handling Interest Rates: CashFlow Mapping

We choose as market variables bondprices with standard maturities (1mth,3mth, 6mth, 1yr, 2yr, 5yr, 7yr, 10yr, 30yr)Suppose that the 5yr rate is 6% and the7yr rate is 7% and we will receive a cashflow of $10,000 in 6.5 years.The volatilities per day of the 5yr and 7yrbonds are 0.50% and 0.58% respectively


Example continued

We interpolate between the 5yr rate of 6%and the 7yr rate of 7% to get a 6.5yr rateof 6.75%The PV of the $10,000 cash flow is

540,60675.1

000,105.6 =


Example continued

We interpolate between the 0.5% volatilityfor the 5yr bond price and the 0.58%volatility for the 7yr bond price to get 0.56%as the volatility for the 6.5yr bondWe allocate α of the PV to the 5yr bondand (1- α) of the PV to the 7yr bond


Example continued

Suppose that the correlation betweenmovement in the 5yr and 7yr bond pricesis 0.6To match variances

This gives α=0.074

)1(58.05.06.02)1(58.05.056.0 22222 α−α××××+α−+α=


Example continuedThe value of 6,540 received in 6.5 years

in 5 years and by

in 7 years.This cash flow mapping preserves valueand variance

484$074.0540,6 =×

056,6$926.0540,6 =×


When Linear Model Can be Used

Portfolio of stocksPortfolio of bondsForward contract on foreign currencyInterest-rate swap


The Linear Model and Options

Consider a portfolio of options dependenton a single stock price, S. Define

andSP

∆∆

=δ

SSx ∆

=∆


Linear Model and Optionscontinued (equations 18.3 and 18.4)

As an approximation

Similarly when there are many underlyingmarket variables

where δi is the delta of the portfolio withrespect to the ith asset

xSSP ∆δ=∆δ=∆

∑ ∆δ=∆i

iii xSP


Example

Consider an investment in options on Microsoftand AT&T. Suppose the stock prices are 120and 30 respectively and the deltas of theportfolio with respect to the two stock prices are1,000 and 20,000 respectivelyAs an approximation

where ∆x1 and ∆x2 are the percentage changesin the two stock prices

21 000,2030000,1120 xxP ∆×+∆×=∆


Skewness(See Figures 18.3, 18.4 , and 18.5)

The linear model fails to capture skewnessin the probability distribution of theportfolio value.


Quadratic Model

For a portfolio dependent on a single stockprice it is approximately true that

this becomes

2)(21 SSP ∆γ+∆δ=∆

22 )(21 xSxSP ∆γ+∆δ=∆


Quadratic Model continued

With many market variables we get anexpression of the form

where

This is not as easy to work with as the linearmodel

∑ ∑= =

∆∆γ+∆δ=∆n

i

n

ijiijjiiii xxSSxSP

1 1 21

jiij

ii SS

PSP

∂∂

=γ∂∂

=δ2


Monte Carlo Simulation (page 448-449)

To calculate VaR using M.C. simulation weValue portfolio todaySample once from the multivariatedistributions of the ∆xi

Use the ∆xi to determine market variablesat end of one dayRevalue the portfolio at the end of day


Monte Carlo Simulation

Calculate ∆PRepeat many times to build up aprobability distribution for ∆PVaR is the appropriate fractile of thedistribution times square root of NFor example, with 1,000 trial the 1percentile is the 10th worst case.


Speeding Up Monte Carlo

Use the quadratic approximation tocalculate ∆P


Comparison of Approaches

Model building approach assumes normaldistributions for market variables. It tendsto give poor results for low delta portfoliosHistorical simulation lets historical datadetermine distributions, but iscomputationally slower


Stress Testing

This involves testing how well a portfolioperforms under some of the most extrememarket moves seen in the last 10 to 20years


Back-Testing

Tests how well VaR estimates would haveperformed in the pastWe could ask the question: How often wasthe actual 10-day loss greater than the99%/10 day VaR?


Principal Components Analysis forInterest Rates (Tables 18.3 and 18.4 on page 451)

The first factor is a roughly parallel shift(83.1% of variation explained)The second factor is a twist (10% ofvariation explained)The third factor is a bowing (2.8% ofvariation explained)


Using PCA to calculate VaR (page 453)

Example: Sensitivity of portfolio to rates ($m)

Sensitivity to first factor is from Table 18.3:10×0.32 + 4×0.35 – 8×0.36 – 7 ×0.36 +2 ×0.36 = – 0.08Similarly sensitivity to second factor = – 4.40

+2-7-8+4+105 yr4 yr3 yr2 yr1 yr


Using PCA to calculate VaR continued

As an approximation

The f1 and f2 are independentThe standard deviation of ∆P (from Table18.4) is

The 1 day 99% VaR is 26.66 × 2.33 =62.12

21 40.408.0 ffP −−=∆

66.2605.640.449.1708.0 2222 =×+×


Estimating Volatilitiesand Correlations

Chapter 19


Standard Approach to EstimatingVolatility (page 461)

Define σn as the volatility per day betweenday n-1 and day n, as estimated at end of dayn-1Define Si as the value of market variable atend of day iDefine ui= ln(Si/Si-1)

σ n n ii

m

n ii

m

mu u

um

u

2 2

1

1

11

1

=−

−

=

−=

−=

∑

∑

( )


Simplifications Usually Made (page462)

Define ui as (Si-Si-1)/Si-1

Assume that the mean value of ui is zeroReplace m-1 by m

This gives

σn n ii

m

mu2 2

1

1= −=∑


Weighting Scheme

Instead of assigning equal weights to theobservations we can set

σ α

α

n i n ii

m

ii

m

u2 21

11

=

=

−=

=

∑

∑

where


ARCH(m) Model (page 463)

In an ARCH(m) model we also assignsome weight to the long-run variance rate,VL:

∑

∑

=

= −

=α+γ

α+γ=σ

m

ii

m

i iniLn uV

1

122

1

where


EWMA Model

In an exponentially weighted movingaverage model, the weights assigned tothe u2 decline exponentially as we moveback through timeThis leads to

21

21

2 )1( −− λ−+λσ=σ nnn u


Attractions of EWMA

Relatively little data needs to be storedWe need only remember the currentestimate of the variance rate and the mostrecent observation on the market variableTracks volatility changesRiskMetrics uses λ = 0.94 for dailyvolatility forecasting


GARCH (1,1) page 465

In GARCH (1,1) we assign some weight tothe long-run average variance rate

Since weights must sum to 1γ + α + β =1

21

21

2−− βσ+α+γ=σ nnLn uV


GARCH (1,1) continued

Setting ω = γV the GARCH (1,1) model is

and

β−α−ω

=1LV

21

21

2−− βσ+α+ω=σ nnn u


Example (Example 19.2, page 465)

Suppose

The long-run variance rate is 0.0002 sothat the long-run volatility per day is 1.4%

σ σn n nu21

21

20 000002 013 0 86= + +− −. . .


Example continued

Suppose that the current estimate of thevolatility is 1.6% per day and the mostrecent percentage change in the marketvariable is 1%.The new variance rate is

The new volatility is 1.53% per day0 000002 013 0 0001 086 0 000256 0 00023336. . . . . .+ × + × =


GARCH (p,q)

σ ω α β σn i n i jj

q

i

p

n ju2 2

11

2= + +−==

−∑∑


Maximum Likelihood Methods

In maximum likelihood methods wechoose parameters that maximize thelikelihood of the observations occurring


Example 1

We observe that a certain event happensone time in ten trials. What is our estimateof the proportion of the time, p, that ithappens?The probability of the event happening onone particular trial and not on the others is

We maximize this to obtain a maximumlikelihood estimate. Result: p=0.1

9)1( pp −


Example 2

Estimate the variance of observations from anormal distribution with mean zero

∑

∑

∏

=

=

=

=

⎥⎦

⎤⎢⎣

⎡−−

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −π

m

ii

m

i

i

m

i

i

um

v

vuv

vu

v

1

2

1

2

1

2

1

)ln(

2exp

21

:Result

:maximizing to equivalent is this logarithms Taking

:Maximize


Application to GARCH

We choose parameters that maximize

∑

∏

=

=

⎥⎦

⎤⎢⎣

⎡−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−

π

m

i i

ii

i

im

i i

vuv

vu

v

1

2

2

1

)ln(

2exp

21

or


Excel Application (Table 19.1, page 469)

Start with trial values of ω, α, and βUpdate variancesCalculate

Use solver to search for values of ω, α, and βthat maximize this objective functionImportant note: set up spreadsheet so that youare searching for three numbers that are thesame order of magnitude (See page 470)

∑=

⎥⎦

⎤⎢⎣

⎡−−

m

i i

ii v

uv1

2

)ln(


Variance Targeting

One way of implementing GARCH(1,1)that increases stability is by using variancetargetingWe set the long-run average volatilityequal to the sample varianceOnly two other parameters then have to beestimated


How Good is the Model?

The Ljung-Box statistic tests forautocorrelationWe compare the autocorrelation of theui

2 with the autocorrelation of the ui2/σi

2


Forecasting Future Volatility(equation 19.3, page 472)

A few lines of algebra shows that

The variance rate for an option expiring onday m is

)()(][ 22Ln

kLkn VVE −σβ+α+=σ +

[ ]∑−

=+σ

1

0

21 m

kknE

m


Forecasting Future Volatilitycontinued (equation 19.4, page 473)

[ ]⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+

β+α=

−

L

aT

L VVaTeV

T

a

)0(1

1ln

252

is optionday - a for annum per volatility The

Define


Volatility Term Structures (Table 19.4)

The GARCH (1,1) suggests that, when calculating vega,we should shift the long maturity volatilities less thanthe short maturity volatilitiesImpact of 1% change in instantaneous volatility forJapanese yen example:

0.060.270.460.610.84Volatilityincrease (%)

500100503010Option Life(days)


Correlations and Covariances (page475-477)

Define xi=(Xi-Xi-1)/Xi-1 and yi=(Yi-Yi-1)/Yi-1

Alsoσx,n: daily vol of X calculated on day n-1σy,n: daily vol of Y calculated on day n-1covn: covariance calculated on day n-1The correlation is covn/(σu,n σv,n)


Updating Correlations

We can use similar models to those forvolatilitiesUnder EWMA

covn = λ covn-1+(1-λ)xn-1yn-1


Positive Finite Definite Condition

A variance-covariance matrix, Ω, isinternally consistent if the positive semi-definite condition

for all vectors w

w wT Ω ≥ 0


Example

The variance covariance matrix

is not internally consistent

1 0 0 90 1 0 9

0 9 0 9 1

.

.. .

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟


Credit Risk

Chapter 20


Credit Ratings

In the S&P rating system, AAA is the bestrating. After that comes AA, A, BBB, BB,B, and CCCThe corresponding Moody’s ratings areAaa, Aa, A, Baa, Ba, B, and CaaBonds with ratings of BBB (or Baa) andabove are considered to be “investmentgrade”


Historical Data

Historical data provided by rating agenciesare also used to estimate the probability ofdefault


Cumulative Ave Default Rates (%)(1970-2003, Moody’s, Table 20.1, page 482)

1 2 3 4 5 7 10 Aaa 0.00 0.00 0.00 0.04 0.12 0.29 0.62

Aa 0.02 0.03 0.06 0.15 0.24 0.43 0.68

A 0.02 0.09 0.23 0.38 0.54 0.91 1.59

Baa 0.20 0.57 1.03 1.62 2.16 3.24 5.10

Ba 1.26 3.48 6.00 8.59 11.17 15.44 21.01

B 6.21 13.76 20.65 26.66 31.99 40.79 50.02

Caa 23.65 37.20 48.02 55.56 60.83 69.36 77.91


Interpretation

The table shows the probability ofdefault for companies starting with aparticular credit ratingA company with an initial credit rating ofBaa has a probability of 0.20% ofdefaulting by the end of the first year,0.57% by the end of the second year,and so on


Do Default Probabilities Increasewith Time?

For a company that starts with a goodcredit rating default probabilities tend toincrease with timeFor a company that starts with a poorcredit rating default probabilities tend todecrease with time


Default Intensities vs UnconditionalDefault Probabilities (page 482-483)

The default intensity (also called hazard rate) isthe probability of default for a certain time periodconditional on no earlier defaultThe unconditional default probability is theprobability of default for a certain time period asseen at time zeroWhat are the default intensities andunconditional default probabilities for a Caa ratecompany in the third year?


Recovery Rate

The recovery rate for a bond is usuallydefined as the price of the bondimmediately after default as a percent ofits face value


Recovery Rates(Moody’s: 1982 to 2003, Table 20.2, page 483)

Class Mean(%)

Senior Secured 51.6

Senior Unsecured 36.1

Senior Subordinated 32.5

Subordinated 31.1

Junior Subordinated 24.5


Estimating Default Probabilities

Alternatives:Use Bond PricesUse CDS spreadsUse Historical DataUse Merton’s Model


Using Bond Prices (Equation 20.2, page 484)

Average default intensity over life of bond is approximately

where s is the spread of the bond’s yieldover the risk-free rate and R is the recoveryrate

Rs

−1


More Exact CalculationAssume that a five year corporate bond pays a couponof 6% per annum (semiannually). The yield is 7% withcontinuous compounding and the yield on a similar risk-free bond is 5% (with continuous compounding)Price of risk-free bond is 104.09; price of corporate bondis 95.34; expected loss from defaults is 8.75Suppose that the probability of default is Q per year andthat defaults always happen half way through a year(immediately before a coupon payment.


Calculations (Table 20.3, page 485)

288.48QTotal

50.67Q0.798563.46103.4640Q4.5

54.01Q0.839564.34104.3440Q3.5

57.52Q0.882565.17105.1740Q2.5

61.20Q0.927765.97105.9740Q1.5

65.08Q0.975366.73106.7340Q0.5

PV of ExpLoss

DiscountFactor

LGDRisk-freeValue

RecoveryAmount

DefProb

Time(yrs)


Calculations continued

We set 288.48Q = 8.75 to get Q = 3.03%This analysis can be extended to allowdefaults to take place more frequentlyWith several bonds we can use moreparameters to describe the defaultprobability distribution


The Risk-Free Rate

The risk-free rate when defaultprobabilities are estimated is usuallyassumed to be the LIBOR/swap zero rate(or sometimes 10 bps below theLIBOR/swap rate)To get direct estimates of the spread ofbond yields over swap rates we can lookat asset swaps


Real World vs Risk-NeutralDefault Probabilities

The default probabilities backed out ofbond prices or credit default swap spreadsare risk-neutral default probabilitiesThe default probabilities backed out ofhistorical data are real-world defaultprobabilities


A Comparison

Calculate 7-year default intensities fromthe Moody’s data (These are real worlddefault probabilities)Use Merrill Lynch data to estimateaverage 7-year default intensities frombond prices (these are risk-neutraldefault intensities)Assume a risk-free rate equal to the 7-year swap rate minus 10 basis point


Real World vs Risk NeutralDefault Probabilities, 7 yearaverages (Table 20.4, page 487)

Rating Real-world default probability per yr (bps)

Risk-neutral default probability per yr (bps)

Ratio Difference

Aaa 4 67 16.8 63 Aa 6 78 13.0 72 A 13 128 9.8 115 Baa 47 238 5.1 191 Ba 240 507 2.1 267 B 749 902 1.2 153 Caa 1690 2130 1.3 440


Risk Premiums Earned By BondTraders (Table 20.5, page 488)

Rating Bond Yield Spread over Treasuries

(bps)

Spread of risk-free rate used by market

over Treasuries (bps)

Spread to compensate for

default rate in the real world (bps)

Extra Risk Premium

(bps)

Aaa 83 43 2 38 Aa 90 43 4 43 A 120 43 8 69 Baa 186 43 28 115 Ba 347 43 144 160 B 585 43 449 93 Caa 1321 43 1014 264


Possible Reasons for These Results

Corporate bonds are relatively illiquidThe subjective default probabilities of bondtraders may be much higher than theestimates from Moody’s historical dataBonds do not default independently of eachother. This leads to systematic risk thatcannot be diversified away.Bond returns are highly skewed with limitedupside. The non-systematic risk is difficult todiversify away and may be priced by themarket


Which World Should We Use?

We should use risk-neutral estimates forvaluing credit derivatives and estimatingthe present value of the cost of defaultWe should use real world estimates forcalculating credit VaR and scenarioanalysis


Merton’s Model (page 489-491)

Merton’s model regards the equity as anoption on the assets of the firmIn a simple situation the equity value is

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


Equity vs. Assets

An option pricing model enables the valueof the firm’s equity today, E0, to be relatedto the value of its assets today, V0, and thevolatility of its assets, σV

E V N d De N d

dV D r T

Td d T

rT

V

V

V

0 0 1 2

10

2

2 12

= −

=+ +

= −

−( ) ( )

ln ( ) ( );

where

σ

σσ


Volatilities

σ∂∂

σ σE V VEEV

V N d V0 0 1 0= = ( )

This equation together with the option pricingrelationship enables V0 and σV to bedetermined from E0 and σE


Example

A company’s equity is $3 million and thevolatility of the equity is 80%The risk-free rate is 5%, the debt is $10million and time to debt maturity is 1 yearSolving the two equations yields V0=12.40and σv=21.23%


Example continued

The probability of default is N(-d2) or12.7%The market value of the debt is 9.40The present value of the promisedpayment is 9.51The expected loss is about 1.2%The recovery rate is 91%


The Implementation of Merton’sModel (e.g. Moody’s KMV)

Choose time horizonCalculate cumulative obligations to time horizon.This is termed by KMV the “default point”. Wedenote it by DUse Merton’s model to calculate a theoreticalprobability of defaultUse historical data or bond data to develop aone-to-one mapping of theoretical probabilityinto either real-world or risk-neutral probability ofdefault.


Credit Risk in DerivativesTransactions (page 491-493)

Three casesContract always an assetContract always a liabilityContract can be an asset or a liability


General Result

Assume that default probability is independent of thevalue of the derivativeConsider times t1, t2,…tn and default probability is qiat time ti. The value of the contract at time ti is fi andthe recovery rate is RThe loss from defaults at time ti is qi(1-R)E[max(fi,0)].Defining ui=qi(1-R) and vi as the value of a derivativethat provides a payoff of max(fi,0) at time ti, the costof defaults is

∑=

n

iiivu

1


Credit Risk Mitigation

NettingCollateralizationDowngrade triggers


Default Correlation

The credit default correlation between twocompanies is a measure of their tendency todefault at about the same timeDefault correlation is important in riskmanagement when analyzing the benefits ofcredit risk diversificationIt is also important in the valuation of somecredit derivatives, eg a first-to-default CDS andCDO tranches.


Measurement

There is no generally accepted measure ofdefault correlationDefault correlation is a more complexphenomenon than the correlation betweentwo random variables


Binomial Correlation Measure(page 499)

One common default correlation measure,between companies i and j is the correlationbetween

A variable that equals 1 if company i defaultsbetween time 0 and time T and zerootherwiseA variable that equals 1 if company j defaultsbetween time 0 and time T and zerootherwise

The value of this measure depends on T.Usually it increases at T increases.


Binomial Correlation continued

Denote Qi(T) as the probability that companyA will default between time zero and time T,and Pij(T) as the probability that both i and jwill default. The default correlation measureis

])()(][)()([

)()()()(

22 TQTQTQTQ

TQTQTPT

jjii

jiijij

−−

−=β


Survival Time Correlation

Define ti as the time to default for companyi and Qi(ti) as the probability distribution fortiThe default correlation betweencompanies i and j can be defined as thecorrelation between ti and tjBut this does not uniquely define the jointprobability distribution of default times


Gaussian Copula Model (page 496-499)

Define a one-to-one correspondence between the timeto default, ti, of company i and a variable xi by

Qi(ti ) = N(xi ) or xi = N-1[Q(ti)]where N is the cumulative normal distribution function.This is a “percentile to percentile” transformation. The ppercentile point of the Qi distribution is transformed to thep percentile point of the xi distribution. xi has a standardnormal distributionWe assume that the xi are multivariate normal. Thedefault correlation measure, ρij between companies i andj is the correlation between xi and xj


Binomial vs Gaussian CopulaMeasures (Equation 20.10, page 499)

The measures can be calculated fromeach other

function ondistributiy probabilitnormal bivariate cumulative the is where

that so

M

TQTQTQTQ

TQTQxxMT

xxMTP

jjii

jiijjiij

ijjiij

])()(][)()([

)()(];,[)(

];,[)(

22 −−

−ρ=β

ρ=


Comparison (Example 20.4, page 499)

The correlation number depends on thecorrelation metric usedSuppose T = 1, Qi(T) = Qj(T) = 0.01, avalue of ρij equal to 0.2 corresponds to avalue of βij(T) equal to 0.024.In general βij(T) < ρij and βij(T) is anincreasing function of T


Example of Use of Gaussian Copula(Example 20.3, page 498)

Suppose that we wish to simulate thedefaults for n companies . For eachcompany the cumulative probabilities ofdefault during the next 1, 2, 3, 4, and 5years are 1%, 3%, 6%, 10%, and 15%,respectively


Use of Gaussian Copula continued

We sample from a multivariate normaldistribution to get the xi

Critical values of xi areN -1(0.01) = -2.33, N -1(0.03) = -1.88,N -1(0.06) = -1.55, N -1(0.10) = -1.28,N -1(0.15) = -1.04


Use of Gaussian Copula continued

When sample for a company is less than-2.33, the company defaults in the first yearWhen sample is between -2.33 and -1.88, the companydefaults in the second yearWhen sample is between -1.88 and -1.55, the companydefaults in the third yearWhen sample is between -1,55 and -1.28, the companydefaults in the fourth yearWhen sample is between -1.28 and -1.04, the companydefaults during the fifth yearWhen sample is greater than -1.04, there is no defaultduring the first five years


A One-Factor Model for theCorrelation Structure (Equation 20.7, page 498)

The correlation between xi and xj is aiaj

The ith company defaults by time T when xi < N-1[Qi(T)]or

The probability of this is

2

1

1])([

i

iii

aMaTQNZ

−

−<

−

iiii ZaMax 21−+=

[ ]⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−

−=

−

2

1

1)()(

i

iii

aMaTQNNMTQ


Credit VaR (page 499-502)

Can be defined analogously to MarketRisk VaRA T-year credit VaR with an X%confidence is the loss level that we are X%confident will not be exceeded over Tyears


Calculation from a Factor-BasedGaussian Copula Model (equation 20.11, page500)

Consider a large portfolio of loans, each of whichhas a probability of Q(T) of defaulting by time T.Suppose that all pairwise copula correlations areρ so that all ai’s areWe are X% certain that M is less than N-1(1- X) =- N-1(X)It follows that the VaR is

[ ]⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

ρ−ρ+

=−−

1)()(

),(11 XNTQN

NTXV

ρ


CreditMetrics (page 500-502)

Calculates credit VaR by consideringpossible rating transitionsA Gaussian copula model is used to definethe correlation between the ratingstransitions of different companies


Credit Derivatives

Chapter 21


Credit Derivatives

Derivatives where the payoff depends onthe credit quality of a company or countryThe market started to grow fast in the late1990sBy 2003 notional principal totaled $3trillion


Credit Default Swaps

Buyer of the instrument acquires protection from theseller against a default by a particular company orcountry (the reference entity)Example: Buyer pays a premium of 90 bps per yearfor $100 million of 5-year protection against companyXPremium is known as the credit default spread. It ispaid for life of contract or until defaultIf there is a default, the buyer has the right to sellbonds with a face value of $100 million issued bycompany X for $100 million (Several bonds aretypically deliverable)


CDS Structure (Figure 21.1, page 508)

Default Protection Buyer, A

Default Protection Seller, B

90 bps per year

Payoff if there is a default byreference entity=100(1-R)

Recovery rate, R, is the ratio of the value of the bond issuedby reference entity immediately after default to the face valueof the bond


Other Details

Payments are usually made quarterly orsemiannually in arrearsIn the event of default there is a final accrualpayment by the buyerSettlement can be specified as delivery of thebonds or in cashSuppose payments are made quarterly in theexample just considered. What are the cashflows if there is a default after 3 years and 1month and recovery rate is 40%?


Attractions of the CDS Market

Allows credit risks to be traded in thesame way as market risksCan be used to transfer credit risks to athird partyCan be used to diversify credit risks


Using a CDS to Hedge a Bond

Portfolio consisting of a 5-year par yieldcorporate bond that provides a yield of 6% and along position in a 5-year CDS costing 100 basispoints per year is (approximately) a long positionin a riskless instrument paying 5% per year


Valuation Example (page 510-512)

Conditional on no earlier default a referenceentity has a (risk-neutral) probability of default of2% in each of the next 5 years. (This is a defaultintensity)Assume payments are made annually in arrears,that defaults always happen half way through ayear, and that the expected recovery rate is 40%Suppose that the breakeven CDS rate is s perdollar of notional principal


Unconditional Default andSurvival Probabilities (Table 21.1)

0.90390.01845

0.92240.01884

0.94120.01923

0.96040.01962

0.98000.02001

SurvivalProbability

DefaultProbability

Time(years)


Calculation of PV of PaymentsTable 21.2 (Principal=$1)

4.0704sTotal

0.7040s0.77880.9039s0.90395

0.7552s0.81870.9224s0.92244

0.8101s0.86070.9412s0.94123

0.8690s0.90480.9604s0.96042

0.9322s0.95120.9800s0.98001

PV of ExpPmt

DiscountFactor

ExpectedPaymt

SurvivalProb

Time (yrs)


Present Value of Expected Payoff(Table 21.3; Principal = $1)

0.0511Total

0.00880.79850.01110.40.01844.5

0.00950.83950.01130.40.01883.5

0.01020.88250.01150.40.01922.5

0.01090.92770.01180.40.01961.5

0.01170.97530.01200.40.02000.5

PV of Exp.Payoff

DiscountFactor

ExpectedPayoff

Rec.Rate

DefaultProbab.

Time(yrs)


PV of Accrual Payment Made in Event of aDefault. (Table 21.4; Principal=$1)

0.0426sTotal

0.0074s0.79850.0092s0.01844.5

0.0079s0.83950.0094s0.01883.5

0.0085s0.88250.0096s0.01922.5

0.0091s0.92770.0098s0.01961.5

0.0097s0.97530.0100s0.02000.5

PV of PmtDiscFactor

ExpectedAccr Pmt

DefaultProb

Time


Putting it all together

PV of expected payments is4.0704s+0.0426s=4.1130sThe breakeven CDS spread is given by4.1130s = 0.0511 or s = 0.0124 (124 bps)The value of a swap negotiated sometime ago with a CDS spread of 150bpswould be 4.1130×0.0150-0.0511 or0.0106 times the principal.


Implying Default Probabilitiesfrom CDS spreads

Suppose that the mid market spread for a 5 yearnewly issued CDS is 100bps per yearWe can reverse engineer our calculations toconclude that the default intensity is 1.61% peryear.If probabilities are implied from CDS spreadsand then used to value another CDS the result isnot sensitive to the recovery rate providing thesame recovery rate is used throughout


Other Credit Derivatives

Binary CDSFirst-to-default Basket CDSTotal return swapCredit default optionCollateralized debt obligation


Binary CDS (page 513)

The payoff in the event of default is a fixedcash amountIn our example the PV of the expectedpayoff for a binary swap is 0.0852 and thebreakeven binary CDS spread is 207 bps


CDS Forwards and Options (page 514-515)

Example: European option to buy 5 yearprotection on Ford for 280 bps starting in oneyear. If Ford defaults during the one-year life ofthe option, the option is knocked outDepends on the volatility of CDS spreads


Total Return Swap (page 515-516)

Agreement to exchange total return on acorporate bond for LIBOR plus a spreadAt the end there is a payment reflecting thechange in value of the bondUsually used as financing tools bycompanies that want an investment in thecorporate bond

Total ReturnPayer

Total Return Receiver

Total Return on Bond

LIBOR plus 25bps


First to Default Basket CDS(page 516)

Similar to a regular CDS except that severalreference entities are specified and there is apayoff when the first one defaultsThis depends on “default correlation”Second, third, and nth to default deals aredefined similarly


Collateralized Debt Obligation(Figure 21.3, page 517)

A pool of debt issues are put into a specialpurpose trustTrust issues claims against the debt in anumber of tranches

First tranche covers x% of notional and absorbs firstx% of default lossesSecond tranche covers y% of notional and absorbsnext y% of default lossesetc

A tranche earn a promised yield on remainingprincipal in the tranche


Bond 1Bond 2Bond 3

6

Bond n

Average Yield8.5%

Trust

Tranche 11st 5% of lossYield = 35%

Tranche 22nd 10% of loss

Yield = 15%

Tranche 33rd 10% of lossYield = 7.5%

Tranche 4Residual lossYield = 6%

CDO Structure


Synthetic CDO

Instead of buying the bonds thearranger of the CDO sells credit defaultswaps.


Single Tranche Trading (Table 21.6, page518)

This involves trading tranches of standard portfolios that are notfundedCDX IG (Aug 4, 2004):

iTraxx IG (Aug 4, 2004)

14.5bps47.5bps135.5bps

347bps41.8%Quote

15-30%10-15%7-10%3-7%0-3%Tranche

20bps43bps70bps168bps27.6%Quote

12-22%9-12%6-9%3-6%0-3%Tranche


Valuation of Correlation DependentCredit Derivatives (page 519-520)

A popular approach is to use a factor-based Gaussian copula model to definecorrelations between times to default thetime to defaultOften all pairwise correlations and all theunconditional default distributions areassumed to be the sameMarket likes to imply a pairwise correlationfrom market quotes.


Valuation of Correlation DependentCredit Derivatives continued

The probability of k defaults by time T conditional on Mis

This enables cash flows conditional on M to becalculated. By integrating over M the unconditionaldistributions are obtained

⎟⎟⎠

⎞⎜⎜⎝

⎛

ρ−ρ−

=−

1)]([

)(1 MTQN

NMTQ

( ) kNk MTQMTQkkN

N −−−

]1[)(!)!(

!


Convertible Bonds

Often valued with a tree where during a timeinterval ∆t there is

a probability pu of an up movementA probability pd of a down movementA probability 1-exp(-λt) that there will be a default

In the event of a default the stock price falls tozero and there is a recovery on the bond


The Probabilities

ud

eu

duauep

dudeap

t

t

d

t

u

1

)( 2

=

=

−−

=

−−

=

∆λ−σ

∆λ−

∆λ−


Node Calculations

Define:Q1: value of bond if neither converted nor

calledQ2: value of bond if calledQ3: value of bond if convertedValue at a node =max[min(Q1,Q2),Q3]


Example 21.1 (page 522)

9-month zero-coupon bond with face value of $100Convertible into 2 sharesCallable for $113 at any timeInitial stock price = $50,volatility = 30%,no dividendsRisk-free rates all 5%Default intensity, λ, is 1%Recovery rate=40%


The Tree (Figure 21.4, page 522)

G76.42

D 152.8566.34

B 132.69 H57.60 57.60

A 115.19 E 115.1950.00 50.00

106.93 C 106.36 I43.41 43.41

101.20 F 100.0037.6898.61 J

32.71100.00

Default Default Default0.00 0.00 0.00

40.00 40.00 40.00


Exotic Options

Chapter 22


Types of Exotics

PackageNonstandard AmericanoptionsForward start optionsCompound optionsChooser optionsBarrier options

Binary optionsLookback optionsShout optionsAsian optionsOptions to exchangeone asset for anotherOptions involvingseveral assets


Packages (page 529)

Portfolios of standard optionsExamples from Chapter 10: bull spreads,bear spreads, straddles, etcOften structured to have zero costOne popular package is a range forwardcontract


Non-Standard American Options(page 530)

Exercisable only on specific dates(Bermudans)Early exercise allowed during onlypart of life (initial “lock out” period)Strike price changes over the life(warrants, convertibles)


Forward Start Options (page 531)

Option starts at a future time, T1

Most common in employee stock optionplansOften structured so that strike priceequals asset price at time T1


Compound Option (page 531)

Option to buy or sell an optionCall on callPut on callCall on putPut on put

Can be valued analyticallyPrice is quite low compared with aregular option


Chooser Option “As You Like It”(page 532)

Option starts at time 0, matures at T2

At T1 (0 < T1 < T2) buyer chooses whether itis a put or callThis is a package!


Chooser Option as a Package

1

2

1))(()(

1

)(1

)(

1

),0max(

),max(

1212

1212

TT

SKeec

TeSKecp

pcT

TTqrTTq

TTqTTr

time at maturing put aplus time at maturing call a is This

therefore is time at value The

parity call-put From is value the time At

−+

−+=

−−−−−

−−−−


Barrier Options (page 535)

Option comes into existence only if stockprice hits barrier before option maturity

‘In’ optionsOption dies if stock price hits barrierbefore option maturity

‘Out’ options


Barrier Options (continued)

Stock price must hit barrier from below‘Up’ options

Stock price must hit barrier from above‘Down’ options

Option may be a put or a callEight possible combinations


Parity Relations

c = cui + cuo

c = cdi + cdo

p = pui + puo

p = pdi + pdo


Binary Options (page 535)

Cash-or-nothing: pays Q if ST > K,otherwise pays nothing.

Value = e–rT Q N(d2)Asset-or-nothing: pays ST if ST > K,otherwise pays nothing.

Value = S0 N(d1)


Decomposition of a Call Option

Long Asset-or-Nothing optionShort Cash-or-Nothing option where payoff

is K

Value = S0 N(d1) – e–rT KN(d2)


Lookback Options (page 536)

Lookback call pays ST – Smin at time TAllows buyer to buy stock at lowestobserved price in some interval of timeLookback put pays Smax– ST at time TAllows buyer to sell stock at highestobserved price in some interval of timeAnalytic solution


Shout Options (page 537)

Buyer can ‘shout’ once during option lifeFinal payoff is either

Usual option payoff, max(ST – K, 0), orIntrinsic value at time of shout, Sτ – K

Payoff: max(ST – Sτ , 0) + Sτ – KSimilar to lookback option but cheaperHow can a binomial tree be used tovalue a shout option?


Asian Options (page 538)

Payoff related to average stock priceAverage Price options pay:

Call: max(Save – K, 0)Put: max(K – Save , 0)

Average Strike options pay:Call: max(ST – Save , 0)Put: max(Save – ST , 0)


Asian Options

No analytic solutionCan be valued by assuming (as anapproximation) that the average stockprice is lognormally distributed


Exchange Options (page 540)

Option to exchange one asset foranotherFor example, an option to exchangeone unit of U for one unit of VPayoff is max(VT – UT, 0)


Basket Options (page 541)

A basket option is an option to buy or sella portfolio of assetsThis can be valued by calculating the firsttwo moments of the value of the basketand then assuming it is lognormal


How Difficult is it toHedge Exotic Options?

In some cases exotic options areeasier to hedge than thecorresponding vanilla options.(e.g., Asian options)In other cases they are more difficult tohedge (e.g., barrier options)


Static Options Replication(Section 22.13, page 541)

This involves approximately replicating an exoticoption with a portfolio of vanilla optionsUnderlying principle: if we match the value of anexotic option on some boundary , we havematched it at all interior points of the boundaryStatic options replication can be contrasted withdynamic options replication where we have totrade continuously to match the option


Example

A 9-month up-and-out call option an a non-dividend paying stock where S0 = 50, K = 50,the barrier is 60, r = 10%, and σ = 30%Any boundary can be chosen but the naturalone isc (S, 0.75) = MAX(S – 50, 0) when S < 60c (60, t ) = 0 when 0 ≤ t ≤ 0.75


Example (continued)

We might try to match the followingpoints on the boundaryc(S , 0.75) = MAX(S – 50, 0) for S < 60c(60, 0.50) = 0

c(60, 0.25) = 0 c(60, 0.00) = 0


Example continued(See Table 22.1, page 543)

We can do this as follows:

+1.00 call with maturity 0.75 & strike 50

–2.66 call with maturity 0.75 & strike 60




Example (continued)

This portfolio is worth 0.73 at time zerocompared with 0.31 for the up-and out optionAs we use more options the value of thereplicating portfolio converges to the value ofthe exotic optionFor example, with 18 points matched on thehorizontal boundary the value of the replicatingportfolio reduces to 0.38; with 100 points beingmatched it reduces to 0.32


Using Static OptionsReplication

To hedge an exotic option we shortthe portfolio that replicates theboundary conditionsThe portfolio must be unwound whenany part of the boundary is reached


Weather, Energy, andInsurance Derivatives

Chapter 23


Pricing Issues (page 551)

To a good approximation many underlyingvariables in insurance, weather, andenergy derivatives contracts can beassumed to have zero systematic risk.This means that we can calculateexpected payoff in the real world anddiscount at the risk-free rate


Weather Derivatives: Definitions(page 552)

Heating degree days (HDD): For each daythis is max(0, 65 – A) where A is theaverage of the highest and lowesttemperature in ºF.Cooling Degree Days (CDD): For eachday this is max(0, A – 65)Contracts specify the weather station to beused


Weather Derivatives: Products

A typical product is a forward contract oran option on the cumulative CDD or HDDduring a monthWeather derivatives are often used byenergy companies to hedge the volume ofenergy required for heating or coolingduring a particular monthHow would you value an option on AugustCDD at a particular weather station?


Energy Derivatives (page 553-556)

Main energy sources:OilGasElectricity


Oil Derivatives (page 553)

Virtually all derivatives available on stocks andstock indices are also available in the OTCmarket with oil as the underlying assetFutures and futures options traded on the NewYork Mercantile Exchange (NYMEX) and theInternational Petroleum Exchange (IPE) are alsopopular


Natural Gas Derivatives (page 554)

A typical OTC contract is for the delivery ofa specified amount of natural gas at aroughly uniform rate to specified locationduring a month.NYMEX and IPE trade contracts thatrequire delivery of 10,000 million Britishthermal units of natural gas to a specifiedlocation


Electricity Derivatives (page 555)

Electricity is an unusual commodity in thatit cannot be storedThe U.S is divided into about 140 controlareas and a market for electricity iscreated by trading between control areas.


Electricity Derivatives continued

A typical contract allows one side toreceive a specified number of megawatthours for a specified price at a specifiedlocation during a particular monthTypes of contracts:

5x8, 5x16, 7x24, daily or monthly exercise, swing options


How an Energy Producer HedgesRisks

Estimate a relationship of the formY=a+bP+cT+ε

where Y is the monthly profit, P is theaverage energy prices, T is temperature,and ε is an error termTake a position of –b in energy forwardsand –c in weather forwards.


Modeling Energy Prices (equation 23.1,page 555)

For oil a is about 0.5 and σ is about 20%;for natural gas these parameters are about1.0 and 40%; for electricity they are about15 and 150%.

dzdtSatSd σ+−θ= ]ln)([ln


Insurance Derivatives (page 556-557)

CAT bonds are an alternative to traditionalreinsuranceThis is a bond issued by a subsidiary of aninsurance company that pays a higher-than-normal interest rate.If claims of a certain type are above a certainlevel the interest and possibly the principal onthe bond are used to meet claims


More on Models andNumerical Procedures

Chapter 24


Three Alternatives to GeometricBrownian Motion

Constant elasticity of variance(CEV)Mixed Jump diffusionVariance Gamma


CEV Model (page 562 to 563))

When α = 1 the model is Black-ScholescaseWhen α > 1 volatility rises as stock pricerisesWhen α < 1 volatility falls as stock pricerisesEuropean option can be value analyticallyin terms of the cumulative non-central chisquare distribution

dzSSdtqrdS ασ+−= )(


CEV Models Implied Volatilities

σimp

K

α < 1

α > 1


Mixed Jump Diffusion Model (page563 to 564)

Merton produced a pricing formula when the stockprice follows a diffusion process overlaid with randomjumps

dp is the random jump k is the expected size of the jumpλ dt is the probability that a jump occurs in the nextinterval of length dt

dpdzdtkSdS +σ+λ−µ= )(/


Jumps and the Smile

Jumps have a big effect on the impliedvolatility of short term optionsThey have a much smaller effect on theimplied volatility of long term options


The Variance-Gamma Model (page564 to 566)

g is change over time T in a variable that followsa gamma process. This is a process where smalljumps occur frequently and there are occasionallarge jumpsConditional on g, ln ST is normal. Its varianceproportional to gThere are 3 parameters

v, the variance rate of the gamma processσ2, the average variance rate of ln S per unit timeθ, a parameter defining skewness


Understanding the Variance-Gamma Model

g defines the rate at which informationarrives during time T (g is sometimesreferred to as measuring economic time)If g is large the the change in ln S has arelatively large mean and varianceIf g is small relatively little informationarrives and the change in ln S has arelatively small mean and variance


Time Varying Volatility

Suppose the volatility is σ1 for the first yearand σ2 for the second and thirdTotal accumulated variance at the end ofthree years is σ1

2 + 2σ22

The 3-year average volatility is

2 22 2 2 1 2

1 223 2 ;

3σ + σ

σ = σ + σ σ =


Stochastic Volatility Models (equations24.2 and 24.3, page 567)

When V and S are uncorrelated aEuropean option price is the Black-Scholes price integrated over thedistribution of the average variance

VL

S

dzVdtVVadV

dzVdtqrS

dS

αξ+−=

+−=

)(

)(


Stochastic Volatility Models continued

When V and S are negatively correlatedwe obtain a downward sloping volatilityskew similar to that observed in the marketfor equitiesWhen V and S are positively correlated theskew is upward sloping


The IVF Model (page 568)

SdztSdttqtrdS

SdzSdtqrdS

),()]()([by replaced is)(

modelmotion Brownian geomeric usual The prices.

option observed matchesexactly that priceasset for the process a create todesigned

is modelfunction y volatilitimplied The

σ+−=

σ+−=


The Volatility Function (equation 24.4)

The volatility function that leads to themodel matching all European option pricesis

)()]()([)(2)],([ 222

2

KcKKctqtrKctqtctK

mkt

mktmktmkt

∂∂∂∂−++∂∂

=σ


Strengths and Weaknesses of theIVF Model

The model matches the probabilitydistribution of stock prices assumed by themarket at each future timeThe models does not necessarily get thejoint probability distribution of stock pricesat two or more times correct


Numerical Procedures

Topics:Path dependent options using treeBarrier optionsOptions where there are two stochasticvariablesAmerican options using Monte Carlo


Path Dependence:The Traditional View

Backwards induction works well forAmerican options. It cannot be used forpath-dependent optionsMonte Carlo simulation works well forpath-dependent options; it cannot be usedfor American options


Extension of BackwardsInductionBackwards induction can be used for some path-dependent optionsWe will first illustrate the methodology usinglookback options and then show how it can beused for Asian options


Lookback Example (Page 570)

Consider an American lookback put on a stock whereS = 50, σ = 40%, r = 10%, ∆t = 1 month & the life ofthe option is 3 monthsPayoff is Smax-ST

We can value the deal by considering all possiblevalues of the maximum stock price at each node

(This example is presented to illustrate the methodology. It is not the mostefficient way of handling American lookbacks (See Technical Note 13)


Example: An American Lookback PutOption (Figure 24.2, page 570)

S0 = 50, σ = 40%, r = 10%, ∆t = 1 month,

56.12

56.124.68

44.55

50.00

6.38

62.99

62.993.36

50.00

56.12 50.006.12 2.66

36.69

50.00

10.31

70.7070.70

0.00

62.99 56.126.87 0.00

56.12

56.12 50.0011.57 5.45

44.55

35.3650.0014.64

50.005.47 A


Why the Approach Works

This approach works for lookback options becauseThe payoff depends on just 1 function of the path followedby the stock price. (We will refer to this as a “pathfunction”)The value of the path function at a node can be calculatedfrom the stock price at the node & from the value of thefunction at the immediately preceding nodeThe number of different values of the path function at anode does not grow too fast as we increase the number oftime steps on the tree


Extensions of the Approach

The approach can be extended so that thereare no limits on the number of alternativevalues of the path function at a nodeThe basic idea is that it is not necessary toconsider every possible value of the pathfunctionIt is sufficient to consider a relatively smallnumber of representative values of thefunction at each node


Working ForwardFirst work forward through the treecalculating the max and min values ofthe “path function” at each nodeNext choose representative values ofthe path function that span the rangebetween the min and the max

Simplest approach: choose the min, themax, and N equally spaced values betweenthe min and max


Backwards Induction

We work backwards through the tree in theusual way carrying out calculations for each ofthe alternative values of the path function thatare considered at a nodeWhen we require the value of the derivative ata node for a value of the path function that isnot explicitly considered at that node, we uselinear or quadratic interpolation


Part of Tree to Calculate Valueof an Option on the ArithmeticAverage(Figure 24.3, page 572)

S = 50.00

Average S46.6549.0451.4453.83

Option Price5.6425.9236.2066.492

S = 45.72

Average S43.8846.7549.6152.48

Option Price 3.430 3.750 4.079 4.416

S = 54.68

Average S47.9951.1254.2657.39

Option Price 7.575 8.101 8.635 9.178

X

Y

Z

0.5056

0.4944

S=50, X=50, σ=40%, r=10%, T=1yr,∆t=0.05yr. We are at time 4∆t


Part of Tree to Calculate Value of anOption on the Arithmetic Average(continued)

Consider Node X when the average of 5observations is 51.44

Node Y: If this is reached, the average becomes51.98. The option price is interpolated as 8.247

Node Z: If this is reached, the average becomes50.49. The option price is interpolated as 4.182

Node X: value is(0.5056×8.247 + 0.4944×4.182)e–0.1×0.05 = 6.206


Using Trees with Barriers(Section 24.5, page 573)

When trees are used to valueoptions with barriers, convergencetends to be slowThe slow convergence arises fromthe fact that the barrier isinaccurately specified by the tree


True Barrier vs Tree Barrier for aKnockout Option: The Binomial Tree Case

Barrier assumed by treeTrue barrier


Inner and Outer Barriers for Trinomial Trees(Figure 24.4, page 574)

Outer barrierTrue barrier

Inner Barrier


Alternative Solutionsto Valuing Barrier Options

Interpolate between value when innerbarrier is assumed and value whenouter barrier is assumedEnsure that nodes always lie on thebarriersUse adaptive mesh methodology

In all cases a trinomial tree ispreferable to a binomial tree


Modeling Two CorrelatedVariables (Section 24.6, page 576)

APPROACHES:1.Transform variables so that they are not

correlated & build the tree in the transformedvariables

2.Take the correlation into account by adjustingthe position of the nodes

3.Take the correlation into account by adjustingthe probabilities


Monte Carlo Simulation andAmerican Options

Two approaches:The least squares approachThe exercise boundary parameterizationapproach

Consider a 3-year put option where theinitial asset price is 1.00, the strike price is1.10, the risk-free rate is 6%, and there isno income


Sampled Paths

1.341.220.881.0081.010.840.921.0070.900.770.761.0061.521.561.111.0050.920.970.931.0041.031.071.221.0031.541.261.161.0021.341.081.091.001t=3t=2t=1t=0Path


The Least Squares Approach (page579)

We work back from the end using a leastsquares approach to calculate thecontinuation value at each timeConsider year 2. The option is in themoney for five paths. These giveobservations on S of 1.08, 1.07, 0.97, 0.77,and 0.84. The continuation values are 0.00,0.07e-0.06, 0.18e-0.06, 0.20e-0.06, and 0.09e-0.06


The Least Squares Approachcontinued

Fitting a model of the form V=a+bS+cS2 weget a best fit relation

V=-1.070+2.983S-1.813S2

for the continuation value VThis defines the early exercise decision att=2. We carry out a similar analysis at t=1


The Least Squares Approachcontinued

In practice more complex functional formscan be used for the continuation value andmany more paths are sampled


The Early Exercise BoundaryParametrization Approach (page 582)

We assume that the early exercise boundarycan be parameterized in some wayWe carry out a first Monte Carlo simulation andwork back from the end calculating the optimalparameter valuesWe then discard the paths from the first MonteCarlo simulation and carry out a new MonteCarlo simulation using the early exerciseboundary defined by the parameter values.


Application to Example

We parameterize the early exerciseboundary by specifying a critical assetprice, S*, below which the option isexercised.At t=1 the optimal S* for the eight paths is0.88. At t=2 the optimal S* is 0.84In practice we would use many more pathsto calculate the S*


Martingales andMeasures

Chapter 25


Derivatives Dependent on a SingleUnderlying Variable

dzσ dtµƒ

dƒ

dzσ dtµƒ

d ƒ

ƒƒ

dzsdtmd

222

2

111

1

.

+=

+=

θ

+=θθ

θ

Suppose and prices withon dependent sderivative two Imagine

process the follows that security) traded a of

price they necessaril (not , variable, a Consider

21


Forming a Riskless Portfolio

tƒƒσµƒƒσµ=

ƒƒσƒƒσ

ƒσƒσ

∆−∆Π

−=Π

−

Π

)(

)()(

21122121

211122

11

22

derivative 2nd the of and derivative 1st the of +

of consisting , portfolio riskless a up set can We


or

:gives This

:riskless is portfolio the Since

2

2

1

1

121221

σrµ

σrµ

r σr σσµσµ

t=r

−=

−

−=−

Π∆∆Π

Market Price of Risk (Page 590)

This shows that (µ – r )/σ is the same for allderivatives dependent on the sameunderlying variable, θWe refer to (µ – r )/σ as the market price ofrisk for θ and denote it by λ


Extension of the Analysisto Several Underlying Variables(Equations 25.12 and 25.13, page 593)

then

withvariables underlying several on depends If

σλrµ

dzσµ dtƒ

dƒ

f

n

iii

n

iii

∑

∑

=

=

=−

+=

1

1


Martingales (Page 594)

A martingale is a stochastic process withzero driftA variable following a martingale has theproperty that its expected future valueequals its value today


Alternative Worlds

dzfdtfrdf

dzσfdtrfdf

σ+λσ+=

λ

+=

)(

is risk of price market the where worlda In

worldneutral-risk ltraditiona the In


The Equivalent MartingaleMeasure Result (Page 595)

ion)considerat under period the during income no

provide to assumed are and pricessecurity derivative all

for martingale a is that shows lemma sIto' then ,security a

of volatility the to equal set weIf

gff

gfg

(

λ


Forward Risk Neutrality

We refer to a world where the market priceof risk is the volatility of g as a world that isforward risk neutral with respect to g.If Eg denotes a world that is FRN wrt g

fg

E fgg

T

T

0

0

=⎛⎝⎜

⎞⎠⎟


Alternative Choices for theNumeraire Security g

Money Market AccountZero-coupon bond priceAnnuity factor


Money Market Accountas the Numeraire

The money market account is an accountthat starts at $1 and is always invested atthe short-term risk-free interest rateThe process for the value of the account is

dg=rg dtThis has zero volatility. Using the moneymarket account as the numeraire leads tothe traditional risk-neutral world


Money Market Accountcontinued

worldneutral-risk ltraditiona the in nsexpectatio denotes where

becomes

equation the ,= and 1= Since

E

feEf

gfE

gf

egg

T

rdt

T

Tg

rdt

T

T

T

ˆ

ˆ 0

0

0

0

0

0

⎥⎦

⎤⎢⎣

⎡ ∫=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

∫

−


Zero-Coupon Bond Maturing attime T as Numeraire

price bond the wrtFRN is that worlda in nsexpectatio denotes and price bond coupon-zero the is ),( where

becomes

equation The

T

TT

T

Tg

ETP

fETPf

gfE

gf

0

][),0(0

0

0

=

⎟⎟⎠

⎞⎜⎜⎝

⎛=


Forward Prices

In a world that is FRN wrt P(0,T), theexpected value of a security at time T is itsforward price


Interest Rates

In a world that is FRN wrt P(0,T2) theexpected value of an interest ratelasting between times T1 and T2 is theforward interest rate


Annuity Factor as the Numeraire

⎥⎦

⎤⎢⎣

⎡=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

)()0(0

0

0

TAfEAf

gfE

gf

TA

T

Tg

becomes

equation The


Annuity Factors and Swap Rates

Suppose that s(t) is the swap ratecorresponding to the annuity factor A.Then:

s(t)=EA[s(T)]


Extension to Several IndependentFactors (Page 599)

∑∑

∑∑

∑

∑

==

==

=

=

σ+⎥⎦

⎤⎢⎣

⎡σλ+=

σ+⎥⎦

⎤⎢⎣

⎡σλ+=

σ+=

σ+=

m

iiig

m

iigi

m

iiif

m

iifi

m

iiig

m

iiif

dztgtdttgttrtdg

dztftdttfttrtdf

dztgtdttgtrtdg

dztftdttftrtdf

1,

1,

1,

1,

1,

1,

)()()()()()(

)()()()()()(

)()()()()(

)()()()()(

consistent internally are that worldsother For

worldneutral-riskltraditiona the In


Extension to Several IndependentFactors continued

hold. results the of rest the and martingalea is case, factor-one the in As

where worldas wrtFRN is that worlda define We

gf

σλg

igi ,=


Applications(Section 25.6, page 600)

Valuation of a European call option wheninterest rates are stochasticValuation of an option to exchange oneasset for another


Change of Numeraire(Section 25.7, page 602)

wvwghwv

vhg

q

vqv

and between ncorrelatio the is and of volatility the is of

volatility the is whereby increases variable a of drift the , to from

security numeraire the change weWhen

ρσ=σσρσ

,,,


Interest Rate Derivatives:The Standard Market Models

Chapter 26


The Complications in ValuingInterest Rate Derivatives (page 611)

We need a whole term structure to define thelevel of interest rates at any timeThe stochastic process for an interest rate ismore complicated than that for a stock priceVolatilities of different points on the termstructure are differentInterest rates are used for discounting the payoffas well as for defining the payoff


Approaches to PricingInterest Rate Options

Use a variant of Black’smodelUse a no-arbitrage (yieldcurve based) model


Black’s Model

Similar to the model proposed byFischer Black for valuing options onfuturesAssumes that the value of an interestrate, a bond price, or some othervariable at a particular time T in thefuture has a lognormal distribution


Black’s Model (continued)

The mean of the probability distribution isthe forward value of the variableThe standard deviation of the probabilitydistribution of the log of the variable is

where σ is the volatilityThe expected payoff is discounted at theT-maturity rate observed today

σ T


Black’s Model (Eqn 26.1 and 26.2, page 611-612)

TddT

TKFd

dNFdKNTPpdKNdNFTPc

σ−=σ

σ+=

−−−=−=

12

20

1

102

210

;2/)/ln(

)]()()[,0()]()()[,0(

• K : strike price• F0 : forward value of

variable today

• T : option maturity• σ : volatility of F


Black’s Model: Delayed Payoff

• K : strike price• F0 : forward value of

variable• σ : volatility of F

• T : time whenvariable is observed

• T * : time of payoff

TddT

TKFd

dNFdKNTPp

dKNdNFTPc

σ−=σ

σ+=

−−−=

−=

12

20

1

102*

210*

;2/)/ln(

)]()()[,0(

)]()()[,0(


Validity of Black’s Model

Two assumptions: 1. The expected value of the underlying

variable is its forward price 2. We can discount expected payoffs at

rate observed in the market todayIt turns out that these assumptions offseteach other in the applications of Black’smodel that we will consider


Black’s Model for European BondOptions

Assume that the future bond price is lognormal

Both the bond price and the strike price shouldbe cash prices not quoted prices

TddT

TKFd

dNFdKNTPpdKNdNFTPc

BB

BB

B

B

σ−=σ

σ+=

−−−=−=

12

2

1

12

21

;2/)/ln(

)]()()[,0()]()()[,0(


Forward Bond and Forward Yield

Approximate duration relation between forwardbond price, FB, and forward bond yield, yF

where D is the (modified) duration of theforward bond at option maturity

F

FF

B

BF

B

B

yyDy

FFyD

FF ∆

−≈∆

∆−≈∆ or


Yield Vols vs Price Vols (Equation 26.8,page 617)

This relationship implies the followingapproximation

where σy is the forward yield volatility, σB isthe forward price volatility, and y0 istoday’s forward yield Often σy is quoted with the understandingthat this relationship will be used tocalculate σB

yB Dy σ=σ 0


Theoretical Justification for BondOption Model

model sBlack' to leads This

Also

is price option the time at maturing bond coupon-zero a wrtFRN is that worlda in Working

0][

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

FBE

KBETPT

TT

TT

=

−


Caps and Floors

A cap is a portfolio of call options on LIBOR. Ithas the effect of guaranteeing that the interestrate in each of a number of future periods willnot rise above a certain levelPayoff at time tk+1 is Lδk max(Rk-RK, 0) where L isthe principal, δk =tk+1-tk , RK is the cap rate, and Rkis the rate at time tk for the period between tk andtk+1A floor is similarly a portfolio of put options onLIBOR. Payoff at time tk+1 is Lδk max(RK -Rk , 0)


Caplets

A cap is a portfolio of “caplets”Each caplet is a call option on a futureLIBOR rate with the payoff occurring inarrearsWhen using Black’s model we assumethat the interest rate underlying eachcaplet is lognormal


Black’s Model for Caps(Equation 26.13, p. 621)

The value of a caplet, for period (tk, tk+1) is

• Fk : forward interest rate for (tk, tk+1)

• σk : forward rate volatility

• L: principal• RK : cap rate• δk=tk+1-tk

-= and where

kkk

kkKk

Kkkk

tddt

tRFd

dNRdNFtPL

σσ

σ+=

−δ +

12

2

1

211

2/)/ln(

)]()()[,0(


When Applying Black’s ModelTo Caps We Must ...

EITHERUse spot volatilitiesVolatility different for each caplet

ORUse flat volatilitiesVolatility same for each caplet within aparticular cap but varies according tolife of cap


Theoretical Justification for CapModel

modelsBlack'toleads This][

Also)]0,[max(),0(

is priceoption the at time maturing bondcoupon -zero

a wrt FRN is that worldain Working

1

11

1

kkk

Kkkk

k

FRE

RREtPt

=

−

+

++

+


Swaptions

A swaption or swap option gives theholder the right to enter into an interestrate swap in the futureTwo kinds

The right to pay a specified fixed rate andreceive LIBORThe right to receive a specified fixed rate andpay LIBOR


Black’s Model for EuropeanSwaptionsWhen valuing European swap options it isusual to assume that the swap rate islognormalConsider a swaption which gives the right topay sK on an n -year swap starting at time T .The payoff on each swap payment date is

where L is principal, m is payment frequencyand sT is market swap rate at time T

max )0,( KT ssmL

−


Black’s Model for EuropeanSwaptions continued (Equation 26.15, page 627)

The value of the swaption is

s0 is the forward swap rate; σ is the swap ratevolatility; ti is the time from today until the i th swappayment; and

)]()( [ 210 dNsdNsLA K−

Am

P tii

m n

==

∑1 01

( , )

TddT

Tssd K σ−=σ

σ+= 12

20

1 ;2/)/ln(where


Theoretical Justification for SwapOption Model

modelsBlack'toleads This][

Also)]0,[max(

is priceoption the swap, theunderlyingannuity the

wrt FRN is that worldain Working

0ssE

ssLAE

TA

KTA

=

−


Relationship Between Swaptionsand Bond Options

An interest rate swap can be regarded asthe exchange of a fixed-rate bond for afloating-rate bondA swaption or swap option is therefore anoption to exchange a fixed-rate bond for afloating-rate bond


Relationship Between Swaptionsand Bond Options (continued)

At the start of the swap the floating-ratebond is worth par so that the swaption canbe viewed as an option to exchange a fixed-rate bond for parAn option on a swap where fixed is paid andfloating is received is a put option on thebond with a strike price of parWhen floating is paid and fixed is received, itis a call option on the bond with a strikeprice of par


Deltas of Interest RateDerivatives

Alternatives:Calculate a DV01 (the impact of a 1bps parallelshift in the zero curve)Calculate impact of small change in the quote foreach instrument used to calculate the zero curveDivide zero curve (or forward curve) into bucketsand calculate the impact of a shift in each bucketCarry out a principal components analysis forchanges in the zero curve. Calculate delta withrespect to each of the first two or three factors


Convexity, Timing, andQuanto Adjustments

Chapter 27


Forward Yields and ForwardPricesWe define the forward yield on a bond as the yieldcalculated from the forward bond priceThere is a non-linear relation between bond yields and bondpricesIt follows that when the forward bond price equals theexpected future bond price, the forward yield does notnecessarily equal the expected future yield


Relationship Between Bond Yieldsand Prices (Figure 27.1, page 636)

BondPrice

YieldY3

B 1

Y1Y2

B 3B 2


Convexity Adjustment for BondYields (Eqn 27.1, p. 637)

Suppose a derivative provides a payoff at time Tdependent on a bond yield, yT observed at time T .Define:G(yT) : price of the bond as a function of its yield

y0 : forward bond yield at time zeroσy : forward yield volatilityThe expected bond price in a world that is FRN wrtP(0,T) is the forward bond priceThe expected bond yield in a world that is FRN wrtP(0,T) is

)()(

21Yield Bond Forward

0

0220 yG

yGTy y ′′′

σ−


Convexity Adjustment for SwapRate

The expected value of the swap rate for the period T toT+τ in a world that is FRN wrt P(0,T) is

where G(y) defines the relationship between price andyield for a bond lasting between T and T+τ that pays acoupon equal to the forward swap rate

)()(

21Rate Swap Forward

0

0220 yG

yGTy y ′′′

σ−



An instrument provides a payoff in 3 years equalto the 1-year zero-coupon rate multiplied by$1000Volatility is 20%Yield curve is flat at 10% (with annualcompounding)The convexity adjustment is 10.9 bps so that thevalue of the instrument is 101.09/1.13 = 75.95


Example 27.2 (Page 638-639)

An instrument provides a payoff in 3 years =to the 3-year swap rate multiplied by $100Payments are made annually on the swapVolatility is 22%Yield curve is flat at 12% (with annualcompounding)The convexity adjustment is 36 bps so thatthe value of the instrument is 12.36/1.123 =8.80


Timing Adjustments (Equation 27.4, page640)

The expected value of a variable, V, in a world that isFRN wrt P(0,T*) is the expected value of the variablein a world that is FRN wrt P(0,T) multiplied by

where R is the forward interest rate between T and T*expressed with a compounding frequency of m, σR isthe volatility of R, R0 is the value of R today, σV is thevolatility of F, and ρ is the correlation between R andV

⎥⎦

⎤⎢⎣

⎡+

−σσρ− T

mRTTRRVVR

/1)(exp

0

*0



A derivative provides a payoff 6 years equal to the valueof a stock index in 5 years. The interest rate is 8% withannual compounding1200 is the 5-year forward value of the stock indexThis is the expected value in a world that is FRN wrtP(0,5)To get the value in a world that is FRN wrt P(0,6) wemultiply by 1.00535The value of the derivative is 1200×1.00535/(1.086)


Quantos(Section 27.3, page 641)

Quantos are derivatives where thepayoff is defined using variablesmeasured in one currency and paid inanother currencyExample: contract providing a payoff ofST – K dollars ($) where S is the Nikkeistock index (a yen number)


Diff Swap

Diff swaps are a type of quantoA floating rate is observed in one currencyand applied to a principal in anothercurrency


Quanto Adjustment (page 642)

The expected value of a variable, V, in aworld that is FRN wrt PX(0,T) is itsexpected value in a world that is FRN wrtPY(0,T) multiplied by exp(ρVWσVσWT) whereW is the forward exchange rate (units of Yper unit of X) and ρVW is the correlationbetween V and W.



Current value of Nikkei index is 15,000This gives one-year forward as 15,150.75Suppose the volatility of the Nikkei is 20%,the volatility of the dollar-yen exchangerate is 12% and the correlation betweenthe two is 0.3The one-year forward value of the Nikkeifor a contract settled in dollars is15,150.75e0.3 ×0.2×0.12×1 or 15,260.23


Quantos continued

twothe between ncorrelatio of tcoefficien the is

and ), of unit per of (units rate exchange the of volatility the is , of volatility the is where

by increases variable a of rate growth the ,currency in worldneutral risk ltraditiona the to currency in worldneutral

risk ltraditiona the from move weWhen

ρ

σσσρσ

XYV

VX

Y

SV

SV


Siegel’s Paradox

is?explain thyou Can . ofdrift a have to1

for process expect the we, of ratedrift a has for process theGiven that

)/1()/1]([)/1( thatlemma sIto' from implies This

][process neutral-risk thefollows )currency

ofuntit per currency of (units rate exchangeAn

2

YX

XY

SSYX

SXY

rrSrr

SdzSdtSrrSd

SdzSdtrrdSX

YS

−−

σ−σ+−=

σ+−=


When is a Convexity, Timing, orQuanto Adjustment Necessary

A convexity or timing adjustment isnecessary when interest rates are used ina nonstandard way for the purposes ofdefining a payoffNo adjustment is necessary for a vanillaswap, a cap, or a swap option


Interest Rate Derivatives:Models of the Short Rate

Chapter 28


Term Structure Models

Black’s model is concerned withdescribing the probability distribution ofa single variable at a single point intimeA term structure model describes theevolution of the whole yield curve


The Zero Curve

The process for the instantaneous short rate, r,in the traditional risk-neutral world defines theprocess for the whole zero curve in this worldIf P(t, T ) is the price at time t of a zero-couponbond maturing at time T

where is the average r between times t and T[ ] P t T E e r T t( , ) $ ( )= − −

r


Equilibrium Models

Rendleman & Bartter: Vasicek: Cox, Ingersoll, & Ross (CIR):

dr r dt r dz

dr a b r dt dz

dr a b r dt r dz

= +

= − +

= − +

µ σ

σ

σ

( )

( )


Mean Reversion(Figure 28.1, page 651)

Interestrate

HIGH interest rate has negative trend

LOW interest rate has positive trend

ReversionLevel


Alternative Term Structuresin Vasicek & CIR(Figure 28.2, page 652)

Zero Rate

Maturity

Zero Rate

Maturity

Zero Rate

Maturity


Equilibrium vs No-ArbitrageModels

In an equilibrium model today’sterm structure is an outputIn a no-arbitrage model today’sterm structure is an input


Developing No-ArbitrageModel for r

A model for r can be made to fitthe initial term structure byincluding a function of time inthe drift


Ho and Lee

dr = θ(t )dt + σdzMany analytic results for bond pricesand option pricesInterest rates normally distributedOne volatility parameter, σAll forward rates have the samestandard deviation


Initial ForwardCurve

ShortRate

r

r

r

rTime

Diagrammatic Representation ofHo and Lee (Figure 28.3, page 655)


Hull and White Model

dr = [θ(t ) – ar ]dt + σdzMany analytic results for bond prices andoption pricesTwo volatility parameters, a and σInterest rates normally distributedStandard deviation of a forward rate is adeclining function of its maturity


Diagrammatic Representation of Hulland White (Figure 28.4, page 656)

ShortRate

r

r

r

rTime

Forward RateCurve


Black-Karasinski Model (equation 28.18)

Future value of r is lognormalVery little analytic tractability

[ ] dztdtrrtatrd )()ln()()()ln( σ+−θ=


Options on Zero-Coupon Bonds(equation 28.20, page 658)

In Vasicek and Hull-White model, price of call maturingat T on a bond lasting to s is

LP(0,s)N(h)-KP(0,T)N(h-σP)Price of put is

KP(0,T)N(-h+σP)-LP(0,s)N(h)where

[ ]

TTsσ

KLa

eeaKTP

sLPh

P

aTTsa

PP

P

)( Lee-HoFor

price. strike theis and principal theis 2

112),0(

),0(ln1 2)(

−σ=

−−

σ=σ

σ+

σ=

−−−


Options on Coupon BearingBonds

In a one-factor model a European option on acoupon-bearing bond can be expressed as aportfolio of options on zero-coupon bonds.We first calculate the critical interest rate at theoption maturity for which the coupon-bearingbond price equals the strike price at maturityThe strike price for each zero-coupon bond is setequal to its value when the interest rate equalsthis critical value


Interest Rate Trees vs Stock PriceTrees

The variable at each node in aninterest rate tree is the ∆t-periodrateInterest rate trees work similarly tostock price trees except that thediscount rate used varies fromnode to node


Two-Step Tree Example(Figure 28.6, page 660))

Payoff after 2 years is MAX[100(r – 0.11), 0]pu=0.25; pm=0.5; pd=0.25; Time step=1yr

10%0.35**

12%1.11*

10%0.23

8%0.00

14%3

12%1

10%0

8%0

6%0 *: (0.25×3 + 0.50×1 + 0.25×0)e–0.12×1

**: (0.25×1.11 + 0.50×0.23 +0.25×0)e–0.10×1


Alternative Branching Processesin a Trinomial Tree(Figure 28.7, page 661)

(a) (b) (c)


Procedure for Building Tree

dr = [θ(t ) – ar ]dt + σdz

1.Assume θ(t ) = 0 and r (0) = 02.Draw a trinomial tree for r to match the mean

and standard deviation of the process for r3.Determine θ(t ) one step at a time so that the

tree matches the initial term structure


Example (page 662 to 667)

σ = 0.01 a = 0.1 ∆t = 1 year The zero curve is as shown in

Table 28.1 on page 665


Building the First Tree for the ∆trate R

Set vertical spacingChange branching when jmax nodes frommiddle where jmax is smallest integergreater than 0.184/(a∆t)Choose probabilities on branches so thatmean change in R is -aR∆t and S.D. ofchange is

tR ∆σ=∆ 3

t∆σ


The Initial Tree(Figure 28.8, page 663)

A

B

C

D

E

F

G

H

I

Node A B C D E F G H I

R 0.000% 1.732% 0.000% -1.732% 3.464% 1.732% 0.000% -1.732% -3.464%

p u 0.1667 0.1217 0.1667 0.2217 0.8867 0.1217 0.1667 0.2217 0.0867

p m 0.6666 0.6566 0.6666 0.6566 0.0266 0.6566 0.6666 0.6566 0.0266

p d 0.1667 0.2217 0.1667 0.1217 0.0867 0.2217 0.1667 0.1217 0.8867


Shifting Nodes

Work forward through treeRemember Qij the value of a derivativeproviding a $1 payoff at node j at time i∆tShift nodes at time i∆t by αi so that the(i+1)∆t bond is correctly priced


The Final Tree(Figure 28.9, Page 665)

A

B

C

D

E

F

G

H

I

Node A B C D E F G H I

R 3.824% 6.937% 5.205% 3.473% 9.716% 7.984% 6.252% 4.520% 2.788%

p u 0.1667 0.1217 0.1667 0.2217 0.8867 0.1217 0.1667 0.2217 0.0867

p m 0.6666 0.6566 0.6666 0.6566 0.0266 0.6566 0.6666 0.6566 0.0266

p d 0.1667 0.2217 0.1667 0.1217 0.0867 0.2217 0.1667 0.1217 0.8867


Extensions

The tree building procedure can beextended to cover more general models ofthe form:

dƒ(r ) = [θ(t ) – a ƒ(r )]dt + σdzWe set x=f(r) and proceed similarly tobefore


Calibration to determine a and σ

The volatility parameters a and σ are chosen so that themodel fits the prices of actively traded instruments suchas caps and European swap options as closely aspossibleWe minimize a function of the form

where Ui is the market price of the ith calibratinginstrument, Vi is the model price of the ith calibratinginstrument and P is a function that penalizes big changesor curvature in a and σ

∑=

+−n

iii PVU

1

2)(


Interest Rate Derivatives:HJM and LMM

Chapter 29


HJM Model: Notation

P(t,T ): price at time t of a discount bondwith principal of $1 maturing at T

Ωt : vector of past and present values ofinterest rates and bond prices at time tthat are relevant for determining bondprice volatilities at that time

v(t,T,Ωt ): volatility of P(t,T)


Notation continued

ƒ(t,T1,T2): forward rate as seen at t for theperiod between T1 and T2

F(t,T): instantaneous forward rate as seen att for a contract maturing at T

r(t): short-term risk-free interest rate at tdz(t): Wiener process driving term structure

movements


Modeling Bond Prices (Equation 29.1,page 680)

) for process a get we approach Letting for

process the determine to lemma sIto' use can we

Because all for

providing function any choose can We

t,TFTT),Tf(t,T

TTTtPTtP),Tf(t,T

tttvv

tdzTtPTtvdtTtPtrTtdP

t

t

(.

)],(ln[)],(ln[

0),,(

)(),(),,(),()(),(

1221

12

2121 −

−=

=Ω

Ω+=


The process for F(t,T)Equation 29.4 and 29.5, page 681)

factor one than more is there whenhold results Similar

have must we(),(

write weif that means result This

dτs(t,)ΩT,s(t,)ΩT, m(t,

)dzΩT,s(t,dtΩT,t,mTtdF

tdzTtvdtTtvTtvTtdF

T

t ttt

tt

tTtTt

∫ Ωτ=

+=

Ω−ΩΩ=

),

)

)(),,(),,(),,(),(


Tree Evolution of Term Structureis Non-Recombining

Tree for the short rate r is non-Markov(see Figure 29.1, page 682)


The LIBOR Market Model

The LIBOR market model is a modelconstructed in terms of the forward ratesunderlying caplet prices


Notation

t kF t t tm t t

t F t tv t P t t t

t t

k

k k k

k k

k k

k k k

: th reset date forward rate between times and

: index for next reset date at time volatility of at time volatility of ( , at time

( ):( )( ): ( )( ): )

:

+

+ −

1

1

ς

δ


Volatility Structure

We assume a stationary volatility structurewhere the volatility of depends only on the number of accrual periods between the next reset date and [i.e., it is a function onlyof ]

F t

tk m t

k

k

( )

( )−


In Theory the Λ’s can bedetermined from Cap Prices

yinductivel determined be to s the allows This

have must weprices cap to fit perfect a provides

model the If caplet. the for volatility the is If when of volatility the as Define i

'

),()()(

11

22

1

Λ

δΛ=σ

σ=−Λ

∑=

−−

+

k

iiikkk

kkk

k

t

ttitmktF


Example 29.1 (Page 684)

If Black volatilities for the first threecaplets are 24%, 22%, and 20%, then

Λ0=24.00%Λ1=19.80%Λ2=15.23%


Example 29.2 (Page 684)

n 1 2 3 4 5

σn(%) 15.50 18.25 17.91 17.74 17.27

Λn-1(%) 15.50 20.64 17.21 17.22 15.25

n 6 7 8 9 10

σn(%) 16.79 16.30 16.01 15.76 15.54

Λn-1(%) 14.15 12.98 13.81 13.60 13.40


The Process for Fk in a One-Factor LIBOR Market Model

dF F dz

P t t

k k m t k

i

= + −

+

K Λ ( )

( , ),

The drift depends on the world chosenIn a world that is forward risk - neutralwith respect to the drift is zero1


Rolling Forward Risk-Neutrality(Equation 29.12, page 685)

It is often convenient to choose a worldthat is always FRN wrt a bond maturing atthe next reset date. In this case, we candiscount from ti+1 to ti at the δi rateobserved at time ti. The process for Fk is

dFF

FF

dt dzk

k

i i i m t k m t

i ij m t

i

k m t=+

+− −

=−∑

δδ

Λ ΛΛ( ) ( )

( )( )1


The LIBOR Market Model andHJM

In the limit as the time between resetstends to zero, the LIBOR market modelwith rolling forward risk neutrality becomesthe HJM model in the traditional risk-neutral world


Monte Carlo Implementation ofLMM Model(Equation 29.14, page 685)

We assume no change to the drift betweenreset dates so that

F t F tF t

Lk j k ji i j i j k j

j j

k j

j k

i

k k j j( ) ( )exp( )

+− − −

=−=

+−

⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

⎡

⎣⎢⎢

⎤

⎦⎥⎥

∑1

2

1 2δ

δδ ε δ

Λ Λ ΛΛ


Multifactor Versions of LMM

LMM can be extended so that there areseveral components to the volatilityA factor analysis can be used to determinehow the volatility of Fk is split intocomponents


Ratchet Caps, Sticky Caps, andFlexi Caps

A plain vanilla cap depends only on oneforward rate. Its price is not dependent onthe number of factors.Ratchet caps, sticky caps, and flexi capsdepend on the joint distribution of two ormore forward rates. Their prices tend toincrease with the number of factors


Valuing European Options in theLIBOR Market Model

There is an analytic approximation thatcan be used to value European swapoptions in the LIBOR market model. Seeequations 29.18 and 29.19 on page 689


Calibrating the LIBOR MarketModel

In theory the LMM can be exactly calibrated tocap prices as described earlierIn practice we proceed as for short rate modelsto minimize a function of the form

where Ui is the market price of the ith calibratinginstrument, Vi is the model price of the ithcalibrating instrument and P is a function thatpenalizes big changes or curvature in a and σ

∑=

+−n

iii PVU

1

2)(


Types of Mortgage-BackedSecurities (MBSs)

Pass-ThroughCollateralized MortgageObligation (CMO)Interest Only (IO)Principal Only (PO)


Option-Adjusted Spread(OAS)

To calculate the OAS for aninterest rate derivative we value itassuming that the initial yield curveis the Treasury curve + a spreadWe use an iterative procedure tocalculate the spread that makesthe derivative’s model price =market price. This is the OAS.


Chapter 30

Swaps Revisited


Valuation of Swaps

The standard approach is to assume thatforward rates will be realizedThis works for plain vanilla interest rateand plain vanilla currency swaps, but doesnot necessarily work for non-standardswaps


Variations on Vanilla InterestRate Swaps

Principal different on two sidesPayment frequency different on two sidesCan be floating-for-floating instead of floating-for-fixedIt is still correct to assume that forward rates arerealizedHow should a swap exchanging the 3-monthLIBOR for 3-month CP rate be valued?


Compounding Swaps (Business Snapshot30.2, page 699)

Interest is compounded instead of being paidExample: the fixed side is 6% compoundedforward at 6.3% while the floating side isLIBOR plus 20 bps compounded forward atLIBOR plus 10 bps.This type of compounding swap can bevalued using the “assume forward rates arerealized” rule. This is because we can enterinto a series of forward contracts that havethe effect of exchanging cash flows for theirvalues when forward rates are realized.


Currency Swaps

Standard currency swaps can be valuedusing the “assume forward LIBOR rate arerealized” rule.Sometimes banks make a smalladjustment because LIBOR in currency Ais exchanged for LIBOR plus a spread incurrency B


More Complex Swaps

LIBOR-in-arrears swapsCMS and CMT swapsDifferential swaps

These cannot be accurately valued byassuming that forward rates will berealized


LIBOR-in Arrears Swap (Equation 30.1,page 701)

Rate is observed at time ti and paid at time ti rather thantime ti+1

It is necessary to make a convexity adjustment to eachforward rate underlying the swapSuppose that Fi is the forward rate between time ti andti+1 and σi is its volatilityWe should increase Fi by

when valuing a LIBOR-in-arrears swap

ii

iiiii

FtttF

τ+−σ +

1)( 1

22


CMS swaps

Swap rate observed at time ti is paid attime ti+1We must

make a convexity adjustment becausepayments are swap rates (= yield on a paryield bond)Make a timing adjustment because paymentsare made at time ti+1 not ti

See equation 30.2 on page 702


Differential Swaps

Rate is observed in currency Y andapplied to a principal in currency XWe must make a quanto adjustment to therateSee equation 30.3 on page 704.


Equity Swaps (page 704-705)

Total return on an equity index isexchanged periodically for a fixed orfloating returnWhen the return on an equity index isexchanged for LIBOR the value of theswap is always zero immediately after apayment. This can be used to value theswap at other times.


Swaps with Embedded Options(page 705-708)

Accrual swapsCancelable swapsCancelable compounding swaps


Other Swaps (page 708-709)

Indexed principal swapCommodity swapVolatility swapBizarre deals (for example, the P&G 5/30swap in Business Snapshot 30.4 on page709)


Real Options

Chapter 31


An Alternative to the NPV Rulefor Capital Investments

Define stochastic processes for the keyunderlying variables and use risk-neutralvaluationThis approach (known as the real optionsapproach) is likely to do a better job atvaluing growth options, abandonmentoptions, etc than NPV


The Problem with using NPV toValue Options

Consider the example from Chapter 11: risk-free rate=12%; strike price = $21

Suppose that the expected return required by investorsin the real world on the stock is 16%. What discount rateshould we use to value an option with strike price $21?

Stock Price = $22

Stock price = $20

Stock Price=$18


Correct Discount Rates areCounter-Intuitive

Correct discount rate for a call option is42.6%Correct discount rate for a put option is–52.5%


General Approach to Valuation

We can value any asset dependent on avariable θ by

Reducing the expected growth rate of θ by λswhere λ is the market price of θ-risk and s isthe volatility of θAssuming that all investors are risk-neutral


Extension to Many UnderlyingVariables

When there are several underlyingvariables θi we reduce the growth rate ofeach one by its market price of risk timesits volatility and then behave as though theworld is risk-neutralNote that the variables do not have to beprices of traded securities


Estimating the Market Price ofRisk Using CAPM (equation 31.2, page 716)

rate free-riskterm-short the is and market; the on return

expected the is return; smarket' the of volatility the is market; the on returns and variable the in changes percentage between

ncorrelatio ousinstantane the is where

by given is variable a of risk of price market The

r

r

m

m

mm

µσ

ρ

−µσρ

=λ )(


Example of Application of Real OptionsApproach to Valuing Amazon.com (BusinessSnapshot 31.1; Schwartz and Moon)

Estimate stochastic processes for the company’s salesrevenue and its average growth rate.Estimated the market price of risk and other keyparameters (cost of goods sold as a percent of sales,variable expenses as a percent of sales, fixed expenses,etc.)Use Monte Carlo simulation to generate differentscenarios in a risk-neutral world.The stock price is the average of the present values ofthe net cash flows discounted at the risk-free rate.


Commodity Prices

Futures prices can be used to define theprocess followed by a commodity price ina risk-neutral world.We can build in mean reversion and use aprocess for constructing trinomial treesthat is analogous to that used for interestrates in Chapter 28


Example (page 671)

A company has to decide whether to invest $15million to obtain 6 million barrels of oil at the rateof 2 million barrels per year for three years. Thefixed operating costs are $6 million per year andthe variable costs are $17 per barrel. The spotprice of oil $20 per barrel and 1, 2, and 3-yearfutures prices are $22, $23, and $24,respectively. The risk-free rate is 10% perannum for all maturities.


The Process for Oil

We assume that this is d ln(S)=[θ(t)-aln(S)] dt+σ dzWhere a=0.1 and σ=0.2


Tree Assuming θ(t)=0; Fig 31.1

E J0.6928 0.6928

B F K0.3464 0.3464 0.3464

A C G L0.0000 0.0000 0.0000 0.0000

D H M-0.3464 -0.3464 -0.3464

I N-0.6928 -0.6928

0.88670.12170.16670.22170.08670.12170.16670.22170.1667pd

0.02660.65660.66660.65660.02660.65660.66660.65660.6666pm

0.08670.22170.16670.12170.88670.22170.16670.12170.1667pu

IHGFEDCBANode


Final Tree for Oil Prices; Fig 31.2

E J44.35 45.68

B F K30.49 31.37 32.30

A C G L20.00 21.56 22.18 22.85

D H M15.25 15.69 16.16

I N11.10 11.43

0.88670.12170.16670.22170.08670.12170.16670.22170.1667pd

0.02660.65660.66660.65660.02660.65660.66660.65660.6666pm

0.08670.22170.16670.12170.88670.22170.16670.12170.1667pu

IHGFEDCBANode


Valuation of Base Project; Fig 31.3

E J42.24 0.00

B F K38.32 21.42 0.00

A C G L14.46 10.80 5.99 0.00

D H M-9.65 -5.31 0.00

I N-13.49 0.00

0.88670.12170.16670.22170.08670.12170.16670.22170.1667pd

0.02660.65660.66660.65660.02660.65660.66660.65660.6666pm

0.08670.22170.16670.12170.88670.22170.16670.12170.1667pu

IHGFEDCBANode


Valuation of Option to Abandon; Fig 31.4

(No Salvage Value; No Further Payments)

E J0.00 0.00

B F K0.00 0.00 0.00

A C G L1.94 0.80 0.00 0.00

D H M9.65 5.31 0.00

I N13.49 0.00

0.88670.12170.16670.22170.08670.12170.16670.22170.1667pd

0.02660.65660.66660.65660.02660.65660.66660.65660.6666pm

0.08670.22170.16670.12170.88670.22170.16670.12170.1667pu

IHGFEDCBANode


Value of Expansion Option; Fig 31.5 (CompanyCan Increase Scale of Project by 20% for $2million)

E J6.45 0.00

B F K5.66 2.28 0.00

A C G L1.06 0.34 0.00 0.00

D H M0.00 0.00 0.00

I N0.00 0.00

0.88670.12170.16670.22170.08670.12170.16670.22170.1667pd

0.02660.65660.66660.65660.02660.65660.66660.65660.6666pm

0.08670.22170.16670.12170.88670.22170.16670.12170.1667pu

IHGFEDCBANode


Derivatives Mishaps andWhat We Can Learn

from Them

Chapter 32


Big Losses by FinancialInstitutions

Allied Irish Bank ($700 million)Barings ($1 billion)Daiwa ($1 billion)Kidder Peabody ($350 million)LTCM ($4 billion)Midland Bank ($500 million)National Westminster Bank ($130 million)


Big Losses by Non-FinancialCorporations

Allied Lyons ($150 million)Gibsons Greetings ($20 million)Hammersmith and Fulham ($600 million)Metallgesellschaft ($1.8 billion)Orange County ($2 billion)Procter and Gamble ($90 million)Shell ($1 billion)Sumitomo ($2 billion)


Lessons for All Users ofDerivatives

Risk must be quantified and risk limits setExceeding risk limits not acceptable evenwhen profits resultDo not assume assume that a trader witha good track record will always be rightBe diversifiedScenario analysis and stress testing isimportant


Lessons for Financial Institutions

Do not give too much independence to startradersSeparate the front middle and back officeModels can be wrongBe conservative in recognizing inceptionprofitsDo not sell clients inappropriate productsLiquidity risk is importantThere are dangers when many are followingthe same strategy


Lessons for Non-FinancialCorporations

It is important to fully understand theproducts you tradeBeware of hedgers becoming speculatorsIt can be dangerous to make theTreasurer’s department a profit center

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