Swaps and other derivatives

Wiley Finance SeriesSecurities Operations: A Guide to Trade and Position Management Building and Using Dynamic Interest Rate Models Ken Kortanek and Vladimir Medvedev Structured Equity D

Swaps and Other Derivatives

Wiley Finance Series

Securities Operations: A Guide to Trade and Position Management

Building and Using Dynamic Interest Rate Models

Ken Kortanek and Vladimir Medvedev

Structured Equity Derivatives: The Definitive Guide to Exotic Options and Structured Notes

Harry Kat

Advanced Modelling in Finance using Excel and VBA

Mary Jackson and Mike Staunton

Operational Risk: Measurement and Modelling

Jack King

Advanced Credit Risk Analysis: Financial Approach and Mathematical Models to Assess, Price and Manage Credit Risk

Didier Cossin and Hugues Pirotte

Dictionary of Financial Engineering

John F Marshall

Pricing Financial Derivatives: The Finite Difference Method

Domingo A Tavella and Curt Randall

Interest Rate Modelling

Jessica James and Nick Webber

Handbook of Hybrid Instruments: Convertible Bonds, Preferred Shares, Lyons, ELKS, DECS and Other Mandatory Convertible Notes

Izzy Nelken (ed.)

Options on Foreign Exchange, Revised Edition

David F DeRosa

The Handbook of Equity Derivatives, Revised Edition

Jack Francis, William Toy and J Gregg Whittaker

Volatility and Correlation in the Pricing of Equity, FX and Interest-rate Options

Riccardo Rebonato

Risk Management and Analysis vol 1: Measuring and Modelling Financial Risk

Carol Alexander (ed.)

Risk Management and Analysis vol 2: New Markets and Products

Carol Alexander (ed.)

Implementing Value at Risk

Philip Best

Credit Derivatives: A Guide to Instruments and Applications

Janet Tavakoli

Implementing Derivatives Models

Les Clewlow and Chris Strickland

Interest-rate Option Models: Understanding, Analysing and Using Models for Exotic Interest-rate Options (second edition)

Riccardo Rebonato

Swaps and Other Derivatives

Richard Flavell

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Preface and Acknowledgements ix

1 Introduction 1

1.1 Introduction 11.2 Applications of swaps 31.3 An overview of the swap market 61.4 The evolution of a swap market 81.5 Conclusion 10

2 Short-term interest rate swaps 11

Objective 112.1 Discounting, the time value of money and other matters 112.2 Forward rate agreements and interest rate futures 162.3 Short-term swaps 202.4 Future valuing a swap 31

3 Generic interest rate swaps 35

Objective 353.1 Generic interest rate swaps 353.2 Pricing through comparative advantage 383.3 The relative pricing of generic IRS 413.4 The relationship between the bond and swap markets 433.5 Implying a discount function 503.6 Building a blended curve 58

4 The pricing and valuation of non-generic swaps 65

Objective 654.1 The pricing of simple non-generic swaps 654.2 Rollercoasters 724.3 A more complex example 754.4 An alternative to discounting 854.5 Swap valuation 85

5 More complex swaps 95

Objective 955.1 Asset packaging 95

5.2 Credit swaps

5.3 Credit-adjusted swap pricing

5.4 Simple mismatch swaps

5.5 Average rate swaps

5.6 Overnight indexed swaps

5.7 Basis swaps

5.8 Yield curve swaps

5.9 Convexity effects of swaps

6.1 Floating-floating cross-currency swaps

6.2 Pricing and hedging of CCBS

6.3 CCBS and discounting

6.4 Fixed-floating cross-currency swaps

6.5 Floating-floating swaps continued

6.6 Fixed-fixed cross-currency swaps

6.7 Cross-currency swaps valuation

6.8 Dual currency swaps

6.9 Cross-currency equity swaps

6.10 Conclusion

Appendix Adjustments to the pricing of a quanto diff swap

7 Interest rate OTC options

Objective

7.1 Introduction

7.2 The Black option pricing model

7.3 Interest rate volatility

7.4 Par and forward volatilities

7.5 Caps, floors and collars

8.2 Interest rate risk management

8.3 Gridpoint risk management — market rates

106121128129131137142152156165175184

205205205207211224229234241247259262262

267267267268271277288299300307309316320326

333333333336337

mtents vii

8.4 Equivalent portfolios

8.5 Gridpoint risk management — forward rates

8.6 Gridpoint risk management — zero coupon rates

8.7 Yield curve risk management

8.8 Swap futures

8.9 Theta risk

8.10 Risk management of IR option portfolios

8.11 Hedging of inflation swaps

Appendix Analysis of swap curves

9 Imperfect risk management

Objective

9.1 Introduction

9.2 A very simple example

9.3 A very simple example extended

9.4 Multifactor delta VaR

9.5 Choice of risk factors and cashflow mapping

9.6 Estimation of volatility and correlations

9.7 A running example

9.8 Simulation methods

9.9 Shortcomings and extensions to simulation methods

9.10 Delta-gamma and other methods

Index 447

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Preface and Acknowledgements

This book is designed for financial professionals to understand how the vast bulk of OTCderivatives are structured, priced and hedged, and ultimately how to use such derivativesthemselves A wide range of books already exist that describe in conceptual terms how andwhy such derivatives are used, and it is not the ambition of this book to supplant them.There are also a number of books which describe the pricing and hedging of derivatives,especially exotic ones, primarily in mathematical terms Whilst exotics are an importantand growing segment of the market, by far the majority of derivatives are still very muchfirst generation, and as such relatively straightforward

For example, interest rate swaps constitute over half of the $100 trillion OTC derivativemarket, and yet there have been few books published in the last decade that describe howthey are created and valued in practical detail So how do many of the professionals gaintheir knowledge? One popular way is "learning on the job", reinforced by the odd trainingcourse But swap structures can be quite complex, requiring more than just superficialknowledge, and probably every professional uses a computer-based system, certainly forthe booking and regular valuation of trades, and most likely for their initial pricing andrisk management These systems are complex, having to deal with real-world situations,and their practical inner details bear little resemblance to the idealized world of mostbooks So often practitioners tend to treat the systems as black boxes, relying on someinitial and frequently inadequate range of tests and hoping their intuition will guide them.The greatest sources of comfort are often the existing customer list of the system (theycan't all be wrong!) and, if the system is replacing an old one, comparative valuations.The objective of this book is to describe how the pricing, valuation and riskmanagement of generic OTC derivatives may be performed, in sufficient detail and withvarious alternatives, so that the approaches may be applied in practice It is based uponsome 15 years of varying experience as a financial engineer for ANZ Merchant Bank inLondon, as a trainer and consultant to banks worldwide, and as Director of FinancialEngineering at Lombard Risk Systems responsible for all the mathematics in the variouspricing and risk management systems

The audience for the book is firstly traders, sales people and front-line risk managers.But increasingly such knowledge needs to be more widely spread within financialinstitutions, such as internal audit, risk control and IT Then there are the counterpartiessuch as organizations using derivatives for risk management, who have frequentlyidentified the need for transparent pricing This need has been exacerbated in recent years

as many developed countries now require that these organizations demonstrate theeffectiveness of risk management, and also perform regular (usually annual) mark-to-

Preface and Acknowledgements

market Similarly, organizations using complex funding structures want to understand

how the structures are created and priced Turning to the other side, many fund managers

and in particular hedge funds are also using derivatives to manage their risk profile, and

then to report using one of the value-at-risk techniques This has been particularly true

since the collapse of Long Term Capital Management, despite the fact that most

implementations of VaR would not have recognized the risk Other potential readers are

the auditors, consultants and regulators of the banks and their client organizations

Institutions offer derivatives with a wide range of maturities, ranging from a few hours

(used to provide risk management over the announcement of an economic figure) to

perpetuals (i.e no upfront maturity defined) There is however a golden rule when pricing

derivatives, namely, always price them off the market that will be used to hedge them This

leads to the first separation in the interest rate swap market between the following

Chapter 2 The short end of the curve, which uses cash, futures and occasionally FRAs

to hedge swaps This chapter first discusses the derivation of discount factors from cash

rates, and concentrates on the range of alternative approaches that may be used It then

looks at the derivation of forward interest rates, and how FRAs may be priced using cash

and futures The convexity effect is highlighted for future discussion Finally an approach

is introduced that does not require discounting, but permits the introduction of a funding

cost

Chapter 3 The medium to long end of the curve The highly liquid inter-bank market

typically trades plain swaps (usually known as "generic" or "vanilla"), very often between

market makers and intermediaries These are hedged in other financial markets, typically

futures for the shorter exposures and bonds for the longer ones This chapter concentrates

initially on the relationship between the bond and swap markets, and how generic swap

prices may be implied It concludes by developing various techniques for the estimation of

discount factors from a generic swap curve

Chapter 4 The end-user market provides customers with tailored (i.e non-generic)

swaps designed to meet their specific requirements Such swaps are not traded as such, but

created as one-off structures This chapter describes a range of simple non-generic swaps,

and discusses various techniques for pricing them, including one that requires no

discounting Finally, two approaches to the ongoing valuation of an existing (seasoned)

swap are demonstrated

Chapter 5 There are a wide variety of potential swap structures, and this chapter covers

the pricing and hedging of some of the more complex and popular ones These include

asset packages, credit swaps, mismatch swaps of various types including yield curve and

overnight average It concludes with a discussion on two less common structures,

inflation-linked (which are growing rapidly) and volatility swaps

Chapter 6 The earliest swap structures were cross-currency swaps, although this market

has long been overtaken by interest rate swaps Nevertheless, they possess some unique

characteristics and structures This chapter starts with the fundamental CCS building

block, the cross-currency basis swap, and explores its characteristics, uses, pricing and

hedging This employs a novel approach: worst-case simulation The role of CCBSs in the

derivation of cross-currency discount factors is also explored The other main types of

swaps are then discussed: fixed-floating, floating-floating, diff and quantodiff Fixed-fixed

swaps occupy a special place because they are a general case of long-term FX forward

contracts, so the pricing and hedging of these is considered in some detail Finally, swap

valuation is revisited because, in the CCS market, such swaps are frequently valued

annually and the principals reset to the current exchange rate

Preface and Acknowledgements

Chapter 7 There is an active market in many currencies in medium to long-term options

on forward interest rates, usually known as the cap & floor market Such structures are

intimately linked to swaps for two reasons: first, because combinations of options cancreate swaps and second, swaps are generally used to hedge them In many banks, they areactually traded and risk-managed together This chapter reviews a range of differentoption structures, and touches albeit briefly on option pricing Volatility plays a crucialrole, and various techniques for estimation, including transformation from par to forward,are described in detail

These options are also frequently embedded in many swap structures, and thebreakdown and pricing of a range of structures is discussed There is also an active market

in options on forward swaps (aka swaptions or swoptions) which, not unnaturally, isclosely related to the swap market The pricing and embedding of swaptions is described.The chapter concludes with two sections on FX options These options are mainlytraded OTC, although there is some activity on a few exchanges such as Philadelphia Thefirst section concentrates on the pricing of these options, and how it may be varieddepending on the method of quoting the underlying currencies The second section showshow traders would dynamically create a delta-neutral hedge for such an option, togetherwith the hedging errors through time

Chapter 8 In the early days of the swap market, swap portfolios were risk-managed

using either asset-liability methods such as gapping or the more advanced institutionsused bond techniques such as duration By the late 1980s a number of well-publicizedlosses had forced banks to develop more appropriate techniques such as gridpointhedging These (in today's eyes) traditional approaches have stood the banks in good steadfor the next decade

This chapter describes the main techniques of both gridpoint and curve hedging, takinginto account both first and second-order sensitivities In passing, mapping cashflows togridpoints is also discussed The use of swap futures, as a relatively new hedginginstrument, is also considered

The chapter then extends the risk management to interest rate options Most textsdiscuss the "greeks" using short-dated options; unfortunately, the discussion often doesnot apply to long-term options, and so their different characteristics, especially as afunction of time, are examined The effectiveness of some optimization techniques toconstruct "robust" hedges is examined as an alternative to the more traditional delta-gamma methods

Finally, the chapter shows how the same techniques can be used to create an inflationhedge for a portfolio of inflation swaps

Chapter 9 Risk management, however, is not a static subject, but has evolved rapidly

during the latter half of the 1990s and beyond Traditional risk management operatesquite successfully, but there is a very sensible desire by senior management to be able toassess the riskiness of the entire trading operation and even wider The traditional riskmeasures are not combinable in any fashion, and cannot be used Value-at-risk wasdeveloped as a family of approaches designed very much to address this objective It isnow being developed further to encompass not only market risk but also credit and evenoperational risks into the same set of measures1

1See the proposed Basel Accord (for details, see BIS website: www.bis.org) for the regulatoryrequirements using VaR-style approaches

xii Preface and AcknowledgementsThis chapter describes the major approaches used to estimate VaR: delta, historic and

Monte-Carlo simulations, as well as second-order delta-gamma approaches The

advantages and disadvantages of each approach are discussed, along with various

extensions such as extreme value theory and sampling strategies The measurement of

spread VaR and equity VaR using either individual stocks or a stock index are also

considered Finally, stress testing, or how to make significant moves in the properties of

the underlying risk factors (especially correlation), is described

The book is supported by a full range of detailed spreadsheet models, which underpin

all the tables, graphs and figures in the main text Some of the models have not been

described in detail in the text, but hopefully the instructions on the sheets should be

adequate Many of the models are designed so that the reader may implement them in

practice without, hopefully, too much difficulty

Many of the ideas, techniques and models described here have been developed over the

years with colleagues at both ANZ and Lombard Risk Systems, and through various

consulting assignments with a wide range of banks across the world Particular thanks go

to Ronny Moller, Richard Szwagrzak and Sean Register for their careful review and

insightful comments when the book was in a pretty unreadable shape I also wish to thank

the various editors and personnel at Wiley, especially Sam Whittaker who has remained

cheerful and supportive despite the range of missed deadlines Finally, for my wife

Marilyn, who has shown much forbearance during all the hours that I have disappeared,

and who hopes I will now regain a life

The parallel loan market requires a friendly US company prepared to provide thedollars, and at the same time requiring sterling in the UK, perhaps for its own subsidiary.

Two loans with identical maturities are created in the two countries as shown Usually thetwo principals would be at the prevailing spot FX rate, and the interest levels at the marketrates Obviously credit is a major concern, which would be alleviated by a set-off clause.This clause allows each party to off-set unpaid receipts against payments due As the spotand interest rates move, one party would find their loan "cheap", i.e below the currentmarket levels, whilst the other party would find their loan "expensive" If the partiesmarked the loans to market, in other words, valued the loans relative to the current marketlevels, then the former would have a positive value and the latter a negative one A

"topping-up" clause, similar in today's market to a regular mark-to-market andsettlement, would often be used to call for adjustments in the principals if the ratesmoved by more than a trigger amount

Swaps and Other Derivatives

As exchange controls were abolished, the parallel loan became replaced with the

back-to-back loan market, whereby the two parent organizations would enter into the loans

directly with each other This simplified the transactions, and reduced the operational

risks Because these loans were deemed to be separate transactions, albeit with an

off-setting clause, they appeared on both sides of the balance sheet, with a potential adverse

effect on the debt/equity ratios

The economic driving force behind back-to-back loans is an extremely important

concept called "comparative advantage" Suppose the UK company is little known in the

US; it would be expensive to raise USD directly Therefore borrowing sterling and doing a

back-to-back loan with a US company (who may of course be in exactly the reverse

position) is likely to be cheaper In theory, comparative advantage cannot exist in efficient

markets; in reality markets are not efficient but are racked by varieties of distortions

Consider the simple corporate tax system: if a company is profitable, it has to pay tax; if a

company is unprofitable, it doesn't The system is asymmetric; unprofitable companies do

not receive "negative" tax (except possibly in the form of off-sets against future profits)

Any asymmetry is a distortion, and it is frequently feasible to derive mechanisms to exploit

it — such as the leasing industry

Cross-currency swaps were rapidly developed from back-to-back loans in the late 1970s

In appearance they are very similar, and for an outside observer only able to see the

cashflows, identical But they are subtly different in that all cashflows are described as

contingent sales or purchases, i.e each sale is contingent upon the counter-sale These

transactions, being forward conditional commitments, are off-balance sheet We have the

beginning of the OTC swap market!

The structure of a generic (or vanilla) cross-currency swap is therefore:

• initial exchange of principal amounts;

• periodic exchanges of interest payments1;

• re-exchange of the principal amounts at maturity

Notice that, if the first exchange is done at the current spot exchange rate, then it possesses

no economic value and can be omitted

Interest rate, or single currency swaps, followed soon afterwards Obviously exchange ofprincipals in the same currency makes no economic sense, and hence an interest swap onlyconsists of the single stage:

• periodic exchanges of interest payments

where interest is calculated on different reference rates The most common form is withone side using a variable (or floating) rate which is determined at regular intervals, and theother a fixed reference rate throughout the lifetime of the swap

1.2 APPLICATIONS OF SWAPS

As suggested by its origins, the earliest applications of the swap market were to assist inthe raising of cheap funds through the comparative advantage concept The EIB-TVAtransaction in 1996 was a classic example of this, and is described in Box 1.1 Both partiesbenefited to the total of about $3 million over a 10 year period, and therefore were bothwilling to enter into the swap

It was quickly realized that swaps, especially being off-balance sheet instruments, couldalso be effective in the management of both currency and interest rate medium-term risk.The commonest example is of a company that is currently paying floating interest, and isconcerned about interest rates rising in the future By entering into an interest rate swap topay a fixed rate and receive a floating rate, uncertainty has been removed:

of the techniques used to structure such swaps

'Remember: legally these cashflows are not "interest" but contingent sales, but for clarity of exposition they will

be called "interest" as they are calculated in exactly the same way.

Swaps and Other Derivatives

A well-known and very early example of the use of swaps is the one conducted between

the World Bank and IBM in August 1981—described in Box 1.2 This swap has the

reputation of kick-starting the swap market because it was performed by two extremely

prestigious organizations, and received a lot of publicity which attracted many other

end-users to come into the market It was the first long-term swap done by the World Bank,

which is now one of the biggest users of the swap market

Box 1.1 Comparative Advantage: European Investment Bank-Tennessee Valley

Authority Swap, September 1996

Both counterparties had the same objective: to raise cheap funds The EIB, being a

European lender, wanted Deutschmarks The TV A, all of whose revenues and costs

were in USD, wanted to borrow dollars Their funding costs (expressed as a spread

over the appropriate government bond market) are shown below:

EIBTVASpread

credit

If both organizations borrowed directly in their required currency, the total

funding cost would be (approximately — because strictly the spreads in different

currencies are not additive) 37 bp over the two bond curves However the

relative spread is much closer in DEM than it is in USD This was for two

reasons:

• the TVA had always borrowed USD, and hence was starting to pay the price of

excess supply;

• it had never borrowed DEM, hence there was a considerable demand from

European investors at a lower rate

The total cost if the TVA borrowed DEM and the EIB borrowed USD would be

only 34 bp, saving 3bp pa

The end result was:

• EIB issued 10-year $1 billion bond;

• TVA issued 10-year DM1.5 billion bond;

• they swapped the proceeds to raise cheaper funding, saving roughly S3 million

over the 10 years

This was a real exercise in comparative advantage; neither party wanted the currency

of their bond issue, but it was cheaper to issue and then swap

Box 1.2 World Bank-IBM Swap, August 1981

This is a simplified version of the famous swap The two counterparties had verydifferent objectives

IBM had embarked upon a world-wide funding programme some years earlier,

raising money inter alia in Deutschmarks and Swiss Francs The money wasremitted back to the US for general funding This had created an FX exposure,because IBM had to convert USDs into DEMs and CHFs regularly to make thecoupon payments Over the years the USD had significantly strengthened, creating again for IBM It now wished to lock in the gain and remove any future exposure

The World Bank had a policy of raising money in hard currency; namely DEM,

CHF and Yen It was a prolific borrower, and by 1981 was finding that its cost offunds in these currencies was rising simply through an excess supply of WB paper.Its objective, as always, was to raise cheap funds

Salomon Brothers suggested the following transactions.

(a) The WB could still raise USD at relatively cheap rates, therefore it shouldissue two Eurodollar bonds:

• one matched to the principal and maturity of IBM's DEM liabilitiesequivalent to $210 million;

• the other matched to IBM's Swiss Franc liabilities equivalent to $80million

Each bond had a short first period to enable the timing of all future cashflows

to match

(b) There was a two-week settlement period, so WB entered into an FX forwardcontract to:

• sell the total bond proceeds of $290 million;

• buy the equivalent in DEM and Sw Fr

(c) IBM and WB entered into a two-stage swap whereby:

USD coupons

IBM

DM/Sw Fr coupons

DM/Sw Fr coupons

At maturity USD principals

DM/Sw Fr principals

World Bank

DM/Sw Fr principals

USD coupons

USD principals

so that IBM converted its DEM and Sw Fr liabilities into USD, andthe WB effectively raised hard currencies at a cheap rate Both achieved theirobjectives!

Swaps and Other Derivatives

87 88 89 90 91 92 93 94 95 96 97 98 99 00

Source: ISDA Reproduced by permission of the International Swaps and Derivatives Association.

Figure 1.1 Size of the swap market (notional principal outstanding $tr)

1.3 AN OVERVIEW OF THE SWAP MARKET

From these earliest beginnings, the swap market has grown exponentially As Figure 1.1

shows, the volume of interest rate swap business now very much dominates cross-currency

swaps2, suggesting that risk management using swaps is commonplace

The graph is shown in terms of notional principal outstanding, i.e the principals of all

swaps transacted but not yet matured For the cross-currency swap described above, this

would be recorded as [$100m + £60mxS]/2 where S is the current spot rate The market

has shown a remarkable and consistent growth in activity, although there has been a

significant decline in the CCS market in 2000 due to the emergence of the single currency

Euro

It is arguable whether this is a very appropriate way of describing the current size of the

market, although it certainly attracts headlines Many professionals would use "gross

market value" or total replacement cost of all contracts as a more realistic measure This

measure has been in broad decline as banks improve their risk management, and are

unwilling to take on greater risks due to the imposition of capital charges:

Gross market value ($tr)

9899

Jun-00 Dec-00

Source: BIS.

the Bank for

IRS1.5091.1501.0721.260

Reproduced

CCS

0.200 0.250 0.239

0.313

by permission of International Settlement.

"The original source of these data was the International Swaps and Derivatives Association (ISDA) which for

many years conducted a semiannual survey of its members In 1995, the Bank for International Settlement (BIS)

started a triennial survey of OTC derivative activity via the central banks In 1997 ISDA stopped their original

survey, whilst BIS expanded theirs to cover currently 48 central banks and monetary authorities The jump in the

graph from 1997 to 1998 is in part due to the shift from ISDA to BIS The BIS also conducts a semiannual review

of global derivative activity across a smaller range of participants to provide more regular indications At the time

of writing, the last available statistics were for the triennial survey of end-June 2001.

A brief overview of the OTC derivative market is shown in Box 1.3 Probably the mostimportant statistic is that, despite all the publicity given to more exotic transactions, theoverwhelming workhorse of this market is the relatively short-term interest rate swap.The derivative markets continue to grow at an astounding rate — why? There are twomain sources of growth—breadth and depth

• Financial markets around the world have increasingly deregulated over the past 30years, witness activities in Greece and Portugal, the Far East and Eastern Europe Asthey do, cash and bond markets first develop followed rapidly by swap and optionmarkets

• The original swaps were done in relatively large principal amounts with high creditcounterparties Banks have however been increasingly pushing derivatives down intothe lower credit depths in the search for return It is feasible to get quite smalltransactions, and some institutions even specialize in aggregating retail demand into awholesale transaction

Box 1.3 A Brief Overview of the Current State of the Derivative Market

• The total OTC derivative market was estimated by the latest triennial survey to be

$100 trillion, measured in terms of outstanding principalshown below:

1995

FX contracts 13.1outfights and swaps 8.7CCS 2.0options 2.4

IR contracts 26.6FRAs 4.6swaps 18.3options 3.5Credit

Other* 7.8Other**

Total 47.5Exchange-traded 10.3

FX 0.1

IR 9.7equity 0.5equity and commodity related

^estimated non-regular reporting

• Currently growing at 11 4% pa

199822.1

14.72.35.048.16.632.98.50.12.011.480.313.90.112.81.0

I counterparties

, slowing down from 15%

amount, broken up as

200120.413.34.32.875.97.757.210.90.72.712.995.219.50.117.41.9

in 1998

• The total gross value was $3 trillion, or 3% of the notional amount

• If netting between reporting institutions is taken into

exposure drops to only about 1 3% of notional amount

account, total market

• Removing the irregular reportees, IR products constitute some 76% and growing

at 16.4% pa

(Continued)

Swaps and Other Derivatives

Box 1.3 (Continued)

• Whereas traditional FX forward contracts, which are predominantly short-term,

are only 13% and have declined by 3% pa (due particularly to the advent of the

Euro)

• Although CCS have nearly doubled over the period

• All other products are very small in comparison, although they may attract

considerably more publicity: credit derivatives have expanded very rapidly from

which is very much the easier part of the curve to hedge

The percentage share of each currency:

Currency Percentage of market share of IR derivatives

Source: Extracted from BIS survey to end June 2001, published December 2001.

Reproduced by permission of the Bank for International Settlement

1.4 THE EVOLUTION OF A SWAP MARKET

The discussion below refers to the evolution of the early swap market in the major

currencies during the 1980s It is however applicable to many other generic markets as they

have developed

There are typically three phases of development of a swap market

1 In the earliest days of a market, it is very much an arranged market whereby two swap

end-users negotiate directly with each other, and an "advisory" bank may well extract

an upfront fee for locating and assisting them This is obviously a slow market, with

documentation frequently tailored for each transaction The main banks involved are

investment or merchant banks, long on people but low on capital and technology as of

course they are taking no risk Typical counterparties would be highly rated, and

therefore happy to deal directly with each other:

In the second phase, originally early to mid 1980s, commercial banks started to take

an increasing role providing traditional credit guarantees:

T Bank T

The counterparties would now both negotiate directly with the bank, which wouldstructure back-to-back swaps but take the credit risk, usually for an ongoing spreadnot an upfront fee The normal lending departments of the bank would be responsiblefor negotiating the transaction and the credit spread The documentation is now morestandardized and provided by the bank This role is often described as acting as an

"intermediary", taking credit but not market risk

The role of intermediary may also be encouraged by external legislation In the UKfor example, if a swap is entered into by two non-bank counterparties, the cashflowsare subject to withholding tax This is not true if one counterparty is a bank.The concept of a market-making bank originally developed by the mid to late 1980s,whereby a bank would provide swap quotations upon request This means they would

be dealing with a range of counterparties simultaneously, and entering into a variety ofnon-matching swaps With increased market risk, such banks required considerablymore capital, pricing and risk management systems, and very standardizeddocumentation The swap market became dominated by the large commercial bankswho saw it as a volume, commoditized business

These banks would typically be off-setting the market risk by hedging in anothermarket, usually the equivalent government bond market as this is the most liquid.Therefore banks with an underlying activity in this market are likely to be at acompetitive advantage Local domestic banks usually have close links with the localgovernment bond market, and hence they are frequently dominant in the domesticswap market Probably the only market where this is not the case is the USD market,where the markets are so large that a number of foreign banks can also be highlyactive and competitive

It might be worth making the point here that banks frequently and misleadingly talkabout "trading" swaps, as if a swap were equivalent to a spot FX transaction which issettled and forgotten about within two days A swap is actually a transaction whichhas created a long-term credit exposure for the bank The exposure is likely to remain

on the bank's books long after the swap "trader" has been paid a bonus and left the

10 Swaps and Other Derivativesbank From this perspective, swaps fit much more comfortably within the traditional

lending departments with all the concomitant credit controlling processes and not

within a treasury which is typically far more lax about credit

This link with the bond market has meant that a bank may well adopt different roles in

different markets For example, a Scandinavian bank such as Nordea Bank would be a

market-maker in the Scandinavian and possibly some of the Northern European

currencies On the other hand, it would act as an intermediary in other currencies For

example, if a customer wanted to do a South African Rand swap, it would enter into it

taking on the credit risk, but immediately laying off the market risk with a Rand

market-making bank

In this context, the 1996 EIB-TVA swap was interesting The deal was brokered by

Lehmann Brothers, but they played no role in the swap At one point the swap had been

out for tender from a bank but (rumour has it) the bid was a 1 bp spread Why, asked the

two counterparties, do we need to deal with a bank at all, especially given that we are both

AAA which is better than virtually all banks? So they dealt directly! As the relative credit

standing of banks declines, the market may well see more transactions of this nature —

back full circle

One cannot really talk about a "global" swap market There are obviously some global

currencies, notably USD, Yen and the Euro, which are traded 24 hours a day, and then it

would be feasible to get swaps But most swap markets are tied into their domestic

markets, and hence available only during trading hours

Swap brokers still play an important role in this market Their traditional role has been

to identify the cheapest suitable counterparty for a client, usually on the initial basis of

anonymity This activity creates liquidity and a uniformity of pricing, to the overall benefit

of market participants However, as the markets in the most liquid currencies continue to

grow, the efficiency provided by a broker is less valued and their fees have been

increasingly reduced to a fraction of a basis point They are having to develop more

electronic skills to survive

1.5 CONCLUSION

The story of the swaps market has been one of remarkable growth from its beginnings

only some 30 years ago This growth has demonstrated that there is a real demand for the

benefits swaps can bring, namely access to cheap funds and risk management, globally

Furthermore, the growth shows little sign of abating as swap markets continue to expand

both geographically as countries deregulate and downwards into the economy

Hopefully this book will play some small role in the continued expansion, assisting the

orderly development of the market by ensuring that people are well-trained in their

understanding of the pricing, structuring and risk management of swaps and related

derivatives

Short-term Interest Rate Swaps

OBJECTIVE

The main objective of this chapter is to provide an introduction to the construction andpricing of short-term IRS using futures contracts However, because a simple swap may beregarded as an exchange of two streams of cashflows which occur at different points in thefuture, extensive use is made of the concept of discounting The chapter therefore beginswith a brief discussion on the time value of money, and demonstrates how implieddiscount factors may be derived from the cash market Because rates are only available atdiscrete maturities, interpolation is a necessary technique, and there are a number ofdifferent approaches which end up with different results The chapter then discusses how

to estimate forward rates, and how to price FRAs first off the cash market and then off thefutures market This leads naturally to the pricing and hedging of short-term IRS off afutures strip Examination of the hedging reveals a convexity effect which is discussed inmore detail in Chapter 5 Finally, an alternative approach to pricing swaps withoutdiscounting is briefly discussed

2.1 DISCOUNTING, THE TIME VALUE OF MONEY

AND OTHER MATTERS

Today's date is Tuesday 4 January 2000, and you have just been offered a choice oftransactions:

Deal 1: to lend $10 million and to receive 6.25% for 3 months

Deal 2: to lend $10 million and to receive 6.70% for 12 months

Which do you find more attractive?

The current London rates at which you could normally deposit money are 61/32% pa and

619/32% pa for 3 and 12 months respectively, and we will assume that the creditworthiness ofthe counterparty is beyond question Comparing the transactions with these market rates,the 3 month deal is 22 bp above the market, whilst the 12 month deal is only 10bp.Intuitively you favour the first transaction, but wish to do some more analysis to be certain.These market rates suggest that the following transactions are currently available1:

3mo Cash

- 10,000,00010,152,457payments, positive

12mo Cash-10,000,00010,674,028

or no sign receipts

Please note that the calculations for all the numbers are replicated on the accompanying CD.

12 Swaps and Other DerivativesNote the following:

(a) Whilst the rates are being quoted on 4 January, they are with effect from 6 January In

other words, there is a two-day settlement period between the agreement of the

transaction and its start This is the normal convention in the USD market, although it

is feasible to organize a "same day" transaction Conventions vary between markets;

for example, the GBP convention is normally "same day"

(b) Interest rates are invariably quoted on a "per annum" basis, even if they are going to

be applied over a different period It is therefore necessary to have a convention that

translates the calendar time from, say, 6 April 2000 back to 6 January 2000 into years

The USD money market, in common with most money markets, uses an "Actual/360"

daycount convention, i.e calculates the actual number of days:

6 April 2000 - 6 January 2000 = 91 daysand then divides by 360 to convert into 0.252778 years The other common convention

is "Actual/365", which is used in the sterling market and many of the old

Commonwealth countries The cashflow at the end of 3 months is given by:

$10,000,000 x (1+6.03125% x 0.252778) = $10,152,456.60(c) 6 January 2001 is a Saturday, and a non-business day in London Payments can only

be made on business days, and therefore a convention has to be applied to determine

the appropriate date The most popular is the "modified following day" convention,

i.e the operating date moves to the next business day unless this involves going across

a month-end, in which case the operating date moves to the last business day in the

month Using this convention, the 12 month transaction ends on the next business

day, i.e Monday 8 January 2001, and interest is calculated accordingly:

$10,000,000 x (1 + 6.59375% x 368/360) = $10,674,027.78The concept of discounting will be used extensively throughout this book The "time value

of money" suggests that the value of money depends upon its time of receipt; for example,

$1 million received today would usually be valued more highly than $1 million to be

received in 1 year's time because it could be invested today to generate interest or profits in

the future If C t represents a certain cashflow to be received at time r>0, then a discount

factor df, relates this cashflow to its value today (or present value) Co by:

C0 = C, x dft

Note that this does not presuppose any source or derivation of the discount factor

The present value of each of these two market-based transactions may easily be

calculated as:

-$10,000,000 + $10,152,456.60 x df3

-$10,000,000+ $10,674.027.78 x df12

where df3 and df12 are the 3 and 12 month discount factors respectively The market rates are

obviously freely negotiated, and we will assume that, at the moment of entering into the

transactions, they represent no clear profit to either party In other words, at inception the

transactions would be deemed to be "fair" to both parties, and hence have a zero net value

This is of course ignoring market realities such as bid-offer or bid-ask spreads (or

"doubles" as they are frequently called) In practice, most analysis uses mid-rates, i.e the

Short-term Interest Rate Swaps

arithmetic average between bid and offer, simply to enable the statement of "fairness" to

be made, and subsequently adjusted for various spreads These issues will be discussed inmore detail later; for the current discussion they will be ignored

If the present values are both zero, we can solve for the two discount factors, i.e

df3 = 0.984983 and df12 = 0.936853 respectively A general formula for discount factorsfrom the money markets is:

J j f l / / 1 i J \ / "^ 1 \

where dt is the length of time (in years) and r t is the rate (expressed as % pa)

Turning back to the two original transactions, these will generate the followingcashflows:

Dates4-Jan-006-Jan-OO6-Apr-008-Jan-01

Deal 1-10,000,00010,157,986

Deal 2

- 10,000,00010,684,889

These cashflows may be present valued using the discount factors derived from the marketrates, giving:

The current money market data readily available is:

14 Swaps and Other Derivatives

Table 2.1 Calculation of discount factor on 21-Sep-OO

Interpolation of rates Interpolation of DFs

Rates

DFs

Linear6.374%

0.956153

Cubic6.330%

0.956443

Linear0.955996

Cubic0.956184

Log-linear0.955860

You are now offered the opportunity to purchase a riskless $100 million on 21 September

2000 What value would you place on this transaction? To answer this question, the

discount factor on 21 September is required — but how to calculate it? The obvious

approach is "interpolation", but this raises two questions:

• what is interpolated: cash rates or discount factors?

• how is the interpolation calculated: linear, polynomial, exponential, etc.?

with associated questions "do the answers change the valuation?" and "are there any

'right' answers?" The simple answers to the latter questions are "yes" and "no, but some

are better than others"! The results from some popular methods are shown in Table 2.1

where:

• "linear" is simply straight-line interpolation;

• "cubic" implies fitting a cubic polynomial of the form a + bt + ct 2 + dt 3 through the four

neighbouring points and solving for {a,b,c,d};

• "log-linear" is the straight-line interpolation of the natural logarithm of the discount

factors (this last one is often suggested since a discount curve is similar to a negative

exponential curve)

The deal value fluctuates by some $50,000 or roughly 5 bp, which is perhaps not significant

but worthwhile It is more common practice to interpolate rates rather than discount

factors at the short end of the curve This is probably because it would be perfectly feasible

to get a quote for a rate out to 21 September for depositing, and of course the two

transactions should be arbitrage free

Cash rates are of course spot rates, i.e they all start out of "today" The cash curve may

be used to estimate forward rates, i.e rates starting at some point in the future For

example, if we knew that we would receive $100 million on 6 April 2000 for, say, 3 months,

we could lock in the investment rate today by calculating the 3/6 rate2 Forward rates are

usually estimated using an arbitrage argument as follows:

1 we could borrow $100 million for 3 months at 61/32% which would cost:

Short-term Interest Rate Swaps 15

Table 2.2 Calculation of discount factor on 6-Oct-00

0.9538653mo

6.3100%

6.4216%

6.9544%

Linear0.953362Forward rates

6.3100%

6.6317%

6.7485%

Interpolation ofCubic0.953556

6.3100%

6.5505%

6.8280%

DFsLog-linear0.953222

6.3100%6.6901%6.6913%

The break-even or implied 3/6 rate is therefore given by the equation:

$100,524,566 x (1 + r3/6 x d3/6) = $103,143,924 => r3/6 = 6.3100%

A general expression for a forward rate F t /T, from t to T, is:

F t/T = ([1 + r T x d T }/(1 + r t x d t }] - 1}/(T - t) (2.2) However to use this expression, zero coupon spot rates are required with maturities t and

T This is acceptable for when T is under 1 year, but they are unlikely to be available for

longer maturities A more widely used expression for longer-dated forward rates is:

The impact of the different methods on the forward rates is quite dramatic, showingdifferences of up to 30 bp See Figure 2.1 Contrast this with the difference in the discountfactors, which in the previous example only reached 5 bp

To understand why, rewriting Equation (2.3) as:

Linear interpolation of rates Cubic interpolation of rates Linear interpolation of DFs Cubic interpolation of DFs Log-linear interpolation of DFs

Figure 2.1 3-Monthly forward rates

16 Swaps and Other Derivatives

Table 2.3 15-Day forward curve

highlights the fact that a forward rate is related to the gradient of the discount curve and is

therefore much more sensitive to small differences in the estimates To demonstrate this more

clearly, Table 2.3 calculates a 15-day forward rate curve using all the five different methods of

interpolation The average difference between the highest and lowest curves is 7.8 bp

In practice, whilst there is no "right" method, most people interpolate the cash rates

using either linear if the cash curve is relatively flat, or polynomial if the curve is quite

steep

2.2 FORWARD RATE AGREEMENTS AND

INTEREST RATE FUTURES

An FRA is an agreement between two counterparties whereby:

• seller of FRA agrees to pay a floating interest rate and receive a fixed interest rate;

• buyer of FRA agrees to pay the fixed interest and receive the floating interest;

• on an agreed notional principal amount;

• over an agreed forward period

For example, a company is a payer of 3 month floating interest on $10 million of debt The

company is concerned about interest rates rising, and on 4 January 2000 it buys a S10

million 3/6 FRA at a fixed rate of 6.31% from a bank The following operations occur:

Short-term Interest Rate Swaps

4 April 2000: 3mo $ Libor is fixed out of 6 April 2000

6 July 2000: net cash settlement (L-6.31%) x $10m x (6 July-6 April)/360 is paid.

This is shown from the point of view of the company, and will be positive

{(L - 6.31 %) x $10m x (6 July - 6 April)/360}/[l + L x (6 July - 6 April)/360]

and paid then The usual reason given for this market convention is a reduction in thecredit exposure between the two parties:

(a) On 4 January, the current exposure is assumed to be zero, i.e the FRA would have azero valuation

(b) However, there is a "potential future exposure" over the period from 4 January to 4April which would fluctuate as the estimate of the Libor fixing on 4 April varies If theestimate rises, then the FRA has a negative value for the bank and hence the companyhas a credit exposure on the bank Conversely, if the estimate falls, then the FRA has apositive value for the bank, and it has a credit exposure on the company

(c) On 4 April, the official Libor fixing is known, which then fixes the net settlementamount and crystallizes the residual credit exposure

(d) The two parties could wait until 6 July with one of them having this known residualexposure By making the payment immediately on 6 April, this residual is removed.Discounting the net settlement amount appears to favour the bank, as it implies that, for agiven credit limit, in the case above of the 3/6 FRA the bank could effectively do twice thetotal business This impact of discounting on reducing the total credit exposure obviouslydeclines as the time to the fixing date lengthens The benefit to the company is less clear.Whilst the value of the net settlement remains constant whether discounted or not, mostcompanies neither mark-to-market nor are overly concerned about credit exposures Thecashflows from the FRA and from the underlying debt are not on the same dates,therefore creating a mismatch which may cause accounting and tax problems It is highlyunlikely that the company could reproduce the undiscounted net settlement, as it wouldnot be able to deposit or borrow at Libor flat for an odd cashflow, irrespective of itscreditworthiness It is perfectly feasible for banks to provide non-discounted FRAs3 at aprice, but this is seldom done

We saw in Section 2.1 how a forward rate may be created by spot money markettransactions However FRAs are off-balance sheet whereas cash trades are on-balancesheet, which is not a good mix If a liquid interest rate (or deposit) futures market exists,then this is much more likely to be used to price and hedge FRAs A brief reminder aboutfutures contracts:

• equivalent to standardized FRA contracts, traded through exchanges;

• standardized notional principal amounts, maturity dates and underlying interest rates;

• futures are deemed to be credit risk free as each contract is guaranteed by the

-'Which could of course be thought of as a single period swap!

18 Swaps and Other Derivativesexchange — to achieve this, when entering into a contract, each party must place an

initial margin with the exchange (sufficient to cover an extreme movement in the

market) plus variation margin because each contract is valued and settled daily

For example: the most liquid contract in the world is 3 month Eurodollar traded on

Chicago Mercantile Exchange:

• notional principal amount is $1 million;

• maturity dates: third Wednesday of a delivery month;

• delivery months: March, June, September and December;

• in theory, 40 contracts (i.e spanning the next 10 years) are open at any time; in practice,

there is good liquidity in the near 20 contracts;

• underlying interest rate: 3 month USD Libor quoted on a "price" basis; on 4 January

2000, the quote for the March contract was 93.80, implying that the market was

anticipating the 3 month Libor rate out of 15 March 2000 to be 6.20%;

• variation margin: $1,000,000 x 1 bp x (90/360) = $25 per basis point movement has to

be paid or received daily (notice this simplistically assumes 3 months is equal to 0.25 of

a year, and does not use Actual/360)

The current quotes for the Eurodollar futures contracts are:

MarchJune

Maturity date

15 March 2000

21 June 2000

Price93.8093.50

Implied rate6.20%

6.50%

Given these rates, we wish to price the FRA above by estimating the fair 3 month rate out

of 6 April: this is usually done by simple linear interpolation between the neighbouring

implied futures rates as shown in Figure 2.2

Since 6 April—15 March = 22 days and 21 June —6 April = 76 days, linear interpolation

gives:

(76/98) x 6.20% + (22/98) x 6.50% = 6.267%

The reason for writing the interpolation in this fashion is that it provides a clear indication

of the contribution of each futures contract, i.e March provides 78%, June 22%, to the

6.6 6.5 6.4

6.1

6.0

Figure 2.2

Short-term Interest Rate Swaps 19

6.6 6.5- 6.4-

i6-3'6.2- 6.1-

How good is this hedge? This will be discussed conceptually first, and then in detaillater Consider first of all a 10 bp parallel shift in the 3 month forward rate curve SeeFigure 2.3 The bank would have to pay $10,000,000 x 10bp x (91/360) = $2,258 extra

on the FRA and would receive $25x10 contracts x 10 bp = $2,500 from the futures

So the hedge is fairly effective, given the slight daycount mismatch In theory, the size

of the futures hedge could have been adjusted slightly, but this is obviouslyimpractical

Next a rotational shift, pivoting around 1 April 2000 See Figure 2.4 This results in thefollowing shifts:

value =value =value =

-$1,326+ $1,782-$505-$55

20 Swaps and Other Derivatives

The hedge appears to be quite effective against both parallel and rotational shifts

However, if the rates move to increase their curvature, for example both futures rates

decrease but the FRA rate remains constant, then the hedge will fail

As time passes, the hedge needs to be rebalanced as the proportions of the two contracts

change Eventually the March contract will expire, leaving the FRA hedged only with the

June contract This exposes the bank to rotational risk for the remainder of the contract

This may be reduced by selling a small amount of September contracts, but this is unlikely

to be very effective given the short time to the FRA fixing By this, we mean that the

correlation between the remainder of the FRA contract and the September contract is

likely to be quite small, and hence a large degree of curve risk has been introduced The

time of greatest risk therefore when hedging a FRA with futures is when one of the

bracketing contracts has matured The only way of removing this residual risk completely

is to sell an IMM FRA, i.e when the FRA fixing date falls on a futures maturity date, so it

may be hedged with a single contract

2.3 SHORT-TERM SWAPS

There are some other issues that we need to discuss, and these will be done in the context

of a more complex example A money market swap is a short-dated swap typically priced

and hedged using a futures strip The swap will be:

• notional principal amount of $10 million;

• 1 year maturity, starting on 4 January 2000;

• to receive fixed F annual Actual/360;

• to pay 3 month Libor quarterly

In this context, the "fair price" of a swap is the fixed rate F such that the net present value

of the swap is zero The structure of the swap is:

4-Jan-00 First Libor fixing ( = current 3mo cash rate 6.03125%)

6-Jan-00 Start of swap (start of interest accruing on both sides)

4-Apr-00 Second Libor fixing

6-Apr-00 First floating payment = $10,000,000 x 6.0313% x (6 Apr-6 Jan)/360

= $152,4574-Jul-00 Third Libor fixing

6-Jul-00 Second floating payment = $ 10,000,000 x L x (6 Jul - 6 Apr)/360

4-Oct-00 Fourth Libor fixing

6-Oct-OO Third floating payment

8-Jan-OO Final cashflow: fourth floating payment

& single fixed receipt = $10,000,000 x Fx (8 Jan 01 -6 Jan 00) 360

Note that, whilst described in detail above, the distinction between the fixing date of a

floating reference rate and the start of the accruing period will generally be ignored unless

it has some special significance Future examples will tacitly assume that fixing takes place

on the start date of each period

The current market information out of 4 January 2000 is:

Short-term Interest Rate Swaps 21

6-Jan-00 13-Jan-00 7-Feb-00 6-Apr-OO 6-Jul-00 8-Jan-01

Cash rates 5.5313%

93.80 93.50 93.27 93.05 93.03

We will look at various ways of determining F, and then will return to hedging As with all

swaps, there are two main issues:

1 What to do about the unknown forward floating rates?

2 As the cashflows occur at different points in time, they need to be made comparable insome fashion; usually by discounting all the cashflows back to the present

Whilst there are a variety of approaches that may be used to address the first issue, as weshall see later, using the futures to estimate the forward Libor rates as we did on the FRA,and subsequently to hedge them is a very natural choice The first Libor is of course fixedtoday to be the current 3mo cash rate The second Libor is none other than the 3/6 rate:

we estimated this using the March and June futures above to be 6.267% Similarly, we canestimate the other two Libor fixings, which are the 6/9 and 9/12 rates respectively, asfollows:

• The value of the fixed side for F= 1 is simply:

$10,000,000 x [(8 Jan 01-6 Jan 00)/360] x 0.936853 = $9,576,724

For the swap to be fairly priced, F= $628,655/$9,576,724 = 6.5644% This will work in

this situation as the present value of the fixed side is linear in terms of the fixed rate; this

is true in many relatively simple structures

• Alternatively, we could guess the fixed rate, construct the cashflow in column [6],calculate the present value of the fixed side, and if it is not the same as the floating PV,adjust the guess A good starting guess would be to use the average of the floating rates,i.e 6.40% and adjust from quarterly to annual using the formula (l+rq u/4)4

= (l + >rann), i-6 6.5566%; this gives a net PV of —$747 But the starting point seldommatters as the iterations are well behaved When pricing transactions in a spreadsheet,most people make extensive use of the goal seeking or solver functions to do this type ofcalculation There are probably two reasons why this is so popular:

Worksheet 2.1 Pricing a money market swap using cash

Today's date: 4-Jan-OO

March 93.80 June 93.50 September 93.27 December 93.05

1 Swap details:

10 million USD

1 year 6.5644% ann, act/360 3mo Libor qu, act/360 Futures Implied Swap dates forward dates

rates

6-Jan-OO 15-Mar-00 6.200% 6-Apr-OO 21-Jun-00 6.500% 6-Jul-OO 20-Sep-OO 6.730% 6-Oct-OO 20-Dec-OO 6.950% 8-Jan-Ol

Initial guess 6.5566%

Final value 6.5644%

Estimated Floating 3mo Libor cashflows polated fixings cash

Inter-rates [1] [2] [3]

Cash Fixed DFs cashflows

[4] [5]

0.984983 0.969519 0.953522 0.936853 671,028 628,655 628,655

cr

3.

» rt

Shift (bp) Shift (bp) Futures Implied Swap

in prices in prices dates forward dates

rates[7] [8]

Y 0

6-Jan-OO-25 0 15-Mar-OO 6.450% 6-Apr-OO-50 0 21-Jun-00 7.000% 6-Jul-00-100 0 20-Sep-OO 7.730% 6-Oct-OO-75 0 20-Dec-00 7.700% 8-Jan-01Ol

Total change in value =

Hedge ratio

DiscountedNew Libor Change change Paymentsfixings in in or receipts

cashflow cashflow from

on swap on swap futures[9] [10] [11] [12]

6.0313%

6.5735% 0 4,8477.1203% -7,738 -7,502 13,2467.7247% -14,884 -14,192 24,725

-24,963 -23,387 3,297-47,585 -45,081 46,115

1.03 0.98

24 Swaps and Other Derivatives(i) it directly generates the actual cashflows likely to happen under this swap, which is

extremely useful for checking the structure;

(ii) the method may easily be modified to enable the pricer to calculate a fixed rate that

will generate a desired profit (non-zero net PV) for the transaction

As before, the hedges for the three unknown Libor fixings may be calculated:

6/9: (76/91) x 10 = 8.35 June and (15/91) x 10 = 1.65 September

9/12: (75/91) x 10 = 8.24 September and (16/91) x 10 = 1.76 December

A total of 30 contracts are required, as shown in Box 2 of Worksheet 2.1

The effectiveness of this hedge is explored in Box 3 The futures prices are shifted, either

individually in column [7] or in parallel in [8] The new Libor estimates are calculated in

[9], and the resulting change in the swap cashflows in [10]; obviously the cashflow

corresponding to the first Libor fixing does not change The margin cashflows from the

futures hedge are calculated in column [12]; for example:

March: 7.76 contracts x —25 bp shift in price x $25 per bp = $4,847 received

We can see that the total changes in the swap cashflows shown in column [10], and the

total receipts under the futures hedge in [12], are very similar They should be equal if the

hedge is completely effective; the reason why they are not is because of the differences in

daycounts as discussed above [the resulting hedge ratio of 1.03 is roughly the ratio of the

length of 3 months under the swap convention of Actual/360 and under the futures

convention being equal to 1/4 of a year, which suggests that about 31 contracts are actually

required]

However, column [10] ignores the timing of the cashflows and simply adds them up The

hedge is said to be a "cash hedge" In practice, the futures would pay the receipts on

margin received today, whilst the additional payments under the swap would only occur

on the payment dates To make the results comparable, the changes in swap cashflows

need to be discounted as in column [11] In that case, the swap is overhedged, i.e the

changes in the value of the swap will always be smaller than the off-setting changes in the

value of the futures receipts, so that the net effect is that we are short futures contracts, as

shown in Figure 2.5

There is however a serious practical flaw in the model, and this refers back to the second

issue The model uses futures for estimating the future Libor fixings, and cash for deriving

the discount factors Both markets are providing information over the 12 month period;

some of the information must therefore be redundant, and it may also be contradictory

Change in futures prices (bp)

Figure 2.5 Hedge effectiveness on a discounted basis

Short-term Interest Rate Swaps 25

The hedge only protected against movements in the fixings, whereas shifts in theunderlying interest rates should also affect the discounting process These effects have beenignored If we were to attempt to introduce this effect, we would have to link shifts infutures to shifts in the discounting

The discounting process is going to have to be rebuilt, this time using the following redundant or parsimonious set of market information:

non-6-Jan-00 15-Mar-00 6-Apr-00

Cash rates 5.99%

6.03125%

March June September December March 01

Futures prices 93.80 93.50 93.27 93.05 93.03

Notice that there is a short cash rate from today to the maturity date of the first futurescontract: this is often called the "cash stub" or "cash to first futures" (CTFF) and in this

case would be roughly equivalent to a 2% month cash rate.

Initially, let us assume that the implied futures rates apply from the maturity of onefutures contract to the next one, e.g the implied rate of 6.20% applies from 15 March until

21 June, the rate of 6.50% from 21 June until 20 September, etc In this case we can build adiscount curve as follows:

• define D F ( t1 ,t2) to be the discount factor at time t2 that will discount back to t1;

• converting the DFs into zero coupon rates by z t = — ln(DF t )/t: see column [2];

• linearly interpolating the zero rates in [6];

• finally transforming back to discount factors using DFt = exp{ — z t t} in [7].

This method of interpolating the discount curve is widely used, often under the name of

"continuously compounded interpolation" Its implications will be explored later The fairrate for the swap can now be calculated as before to be 6.5635% —see column [8] — which

is very slightly different to the earlier rate

In practice futures are not exactly 3 months apart; sometimes they gap, sometimesoverlap If it is deemed necessary, then one approach is as follows (see Worksheet 2.3 fordetails):

26 Swaps and Other Derivatives

Worksheet 2.2 Pricing a money market swap off a futures strip

Today's date: 4-Jan-00

1 Swap details:

Principal amountMaturity

From cash

Futures Implieddates forward

rates

6-Jan-OO15-Mar-OO 6.200%

DFs Z-c rates

[1] [2]

10.988649 5.9559%

Short-term Interest Rate Swaps 27

[4] [5] [6] [7] [8]

-152,457 5.9812% 0.984995 -150,169-158,425 6.0908% 0.969677 -153,621-167,080 6.2238% 0.953734 -159,350-176,738 670,935 6.3495% 0.937156 463,139

Present value = 0.00

2 Hedge

PV01 Hedge

[9] [10]Cash stub 0.452March -193.555 7.72June -265.737 10.63Sep -252.915 10.12Dec -45.033 1.80

30.27