Interest Rate Derivatives - Danske Bank · 2018. 11. 8. · Danske Research - Fixed Income &...

November 2007

Interest Rate Derivatives - A Primer

•

•

•

• This document is intended to serve as a handbook on interest rate derivatives

• We describe some of the basic theoretical and practical foundations of derivatives

Danske Research - Fixed Income & Derivatives Research

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1 D A N S K E B A N K

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Contents ..................................................................................................................................................................................................................................................................................................................... 3 Preface

.......................................................................................................................................................................................................................................................................................................... 4 Introduction...............................................................................................................................................................................................................................................4 Why use derivatives?

...........................................................................................................................................................................................................................5 How to think about derivatives...............................................................................................................................................................................5 What to consider before entering into derivatives

.................................................................................................................................................................................................................................................................6 A bit of theoryWhat is a derivative strategy?............................................................................................................................................................................................................................7

.....................................................................................................................................................................................................................................................7 Derivative markets..............................................................................................................................................................................................................................................................8 Documentation

....................................................................................................................................................................................................................................................................................................... 9 Interest rates..........................................................................................................................................................................................................9 The basic definitions and conventions

............................................................................................................................................................................................................................................................. 11 Pricing swaps........................................................................................................................................................................ 12 Yield curves: Can identical cash flows be different?

................................................................................................................................................................................................................... 13 Yield curves and their dynamics.........................................................................................................................................................................................................................................................................................15 Risk management

........................................................................................................................................................................................................................................ 15 Measuring uncertainty................................................................................................................................................................................................ 15 Yield curve risk � the classical approach

................................................................................................................................................................................................... 16 Yield curve risk � the modern approach................................................................................................................................................................................. 17 Risk managing options or just speaking Greek?

................................................................................................................................................................................................................................................................................. 17 Delta........................................................................................................................................................................................................................................................................... 17 Gamma

.................................................................................................................................................................................................................................................................................. 18 Vega................................................................................................................................................................................................................................................................................ 18 Theta

................................................................................................................................................................................... 18 Choosing your risk profile � interacting greeks............................................................................................................................................................................................................. 21 Pricing options � the put-call parity

......................................................................................................................................................................................................................................................................................22 Outright strategies...................................................................................................................................................................................................................................... 22 Where are rates going?

............................................................................................................................................................................................................................................................................................23 Curve strategies..................................................................................................................................................................................................................................... 23 Introducing relative bets

............................................................................................................................................................................................................................................. 23 Swaps versus bonds....................................................................................................................................................................................................... 25 Spot versus forward � hedge stability

....................................................................................................................................................................... 27 Spot versus forward � convenience of no payments.......................................................................................................................................................................................................................................... 27 Introducing swaptions

What determines the price of a swaption?............................................................................................................................................................................................ 28 ........................................................................................ 29 Conditional versus unconditional � using swaptions in a one-sided steepener strategy

.................................................................................. 29 Conditional versus unconditional � using swaptions to take positions on the entire curve...................................................................... 31 Conditional versus unconditional � using caps and floors to take positions on the entire curve

................................................................................................................................................................................... 33 Protection: Extendable and retractable swaps.......................................................................................................................................................................................................................... 34 Constant maturity strategies

.....................................................................................................................................................................................................................................................................................................37 Asset Swaps........................................................................................................................................................................................................................... 37 Levelling out the playing field

...................................................................................................................................................................................................................................... 37 What is an asset swap?.................................................................................................................................................................................................................. 37 What are asset swaps used for?

............................................................................................................................................................................................................................................ 37 Types of asset swaps....................................................................................................................................................................................................................................................................................39 Volatility Strategies

.................................................................................................................................................................................................................................................... 39 What is volatility?........................................................................................................................................................................................................................................... 39 Why look at volatility?

........................................................................................................................................................................................................................................................ 39 Implied volatility................................................................................................................................................................................................................... 40 Implied versus realised volatility

................................................................................................................................................................................ 40 Skews, smiles and term structure of volatilities............................................................................................................................................................................................................. 42 The dynamics of implied volatilities

............................................................................................................................................................................................................................ 43 An example of the dynamics.............................................................................................................................................................................................................................. 44 Tracking an options market

................................................................................................................................................................................................. 45 What determines the price of volatility?........................................................................................................................................ 45 Strangles and straddles - simple volatility bets using caps/floors

.................................................................................................................................................................................................... 46 Selling strangles or buying butterflies?........................................................................................................................................................................................ 47 A 1X2 Zero cost structure using swaptions

.................................................................................................................................................................................................................................................... 48 Calendar spreads.................................................................................................................................................................................................................................................................................50 Combined Strategies

..................................................................................................................................................................................................................................... 50 Taking relative positions............................................................................................................................................................................................................................................. 50 Swap spread trading

................................................................................................................................................................................................................................... 51 Relative curve strategies..................................................................................................................................................................................................... 51 Picking up the DKK-EUR risk premium

................................................................................................................................................................................................................. 53 Shorting options: Range accruals..................................................................................................................................................................................................... 54 Pricing range accruals: Digital options..................................................................................................................................................................................................... 54 Pure curve plays: CMS spread options

.......................................................................................................................................................................................................................................................................................56 Derivative formats..................................................................................................................................................................................................................... 56 Which wrapping do you prefer?

.................................................................................................................................................................................................................................................... 56 Structured swaps...................................................................................................................................................................................................................................................... 57 Structured notes

Structured deposits............................................................................................................................................................................................................................................... 57 ..........................................................................................................................................................................................................................................................................................................58 Conclusion

D A N S K E B A N K 2

Preface This Primer on Interest Rate Derivatives is written with the intention

of providing the reader with a detailed insight into the use of interest rate derivatives. We focus on strategic use, risk management and implementation. It is our aim and hope that this primer will be accessible to a broad range of practitioners.

The primer is organised as follows. Section 1 discusses the use of derivatives for both hedge and speculative purposes. Section 2 defines the basic interest rate concepts used. Section 3 presents the key risk management metrics known as �the greeks�. Section 4 and 5 introduces the use of derivatives for both outright and curve strategies. Section 6 provides a brief introduction to asset swaps and their use. Section 7 presents the use of derivatives for volatility strategies. Section 8 describes some combined strategies that offer a yield pick-up in different interest rate / volatility scenarios. Finally, section 9 discusses various ways to package derivative exposure while section 10 concludes the primer.

Copenhagen, November 2007 Danske Markets Derivatives Research Martin D. Linderstrøm Analyst Tel: +45 45 12 85 20 E-mail: [email protected]



Introduction Why use derivatives?

Basic concepts and arguments for using derivatives. Difference between cash and derivative instruments

A derivative security is a financial contract whose cash flows depend on the value of some other asset. What makes a derivative security different from holding this underlying asset? After all, one could argue that it must be the same fundamental economic forces of supply and demand that determine their value.

The first fundamental difference is the way the value changes over time. If you buy one unit of the underlying asset, a �1 change in its value will change the value of your holdings with a 1:1 ratio. Typically, the value of a derivative contract will change value relative to the underlying asset, at a ratio different from one. This is known as the leverage effect. Although to some people leverage may have a speculative ring to it, it is important to stress its benefits. While leverage is the reason a speculator can increase his exposure relative to his capital, it is also the reason that he can insure his portfolio at a fraction of its total market value.

Derivatives offer high flexibility and thus the possibility of tailoring exposure

Given that many fixed income derivatives are traded as over-the-counter (OTC) contracts there is a high degree of flexibility for the individual client to tailor his exposure. Relative to cash-only markets, this means that an investor can effectively have his derivative counterparty create securities that are not currently trading in the market place.

As the volumes traded in fixed income derivative markets have surged, the liquidity and thus the cost-effectiveness of participating in these markets has increased as well. For many investors this means that the derivative markets offer flexibility at very competitive terms.

Finally, derivatives used specifically for hedging purposes can be offered favourable accounting treatment. If certain conditions listed in IAS39 are met, changes in the value of a hedge position will be off-set against the changes in value of the hedged asset or liability � even if this is only projected. Specifically, this removes the timing mismatch that otherwise can arise in earnings. Hedge accounting thus helps to remove volatility in income statements.

Note finally that many investors already have a strong (short) presence in the fixed income derivatives markets through the embedded options in their mortgage bond portfolios. As such, taking on pure derivative positions does not for many investors present any fundamental change.


How to think about derivatives

Derivatives are a form of insurance and are therefore no more speculative than other types of insurance

Basically, derivatives can be used to gain or reduce exposure against specific risk factors. That is, derivative contracts can be used to transfer risks between counterparties. Derivatives can thus be used either for investment or hedging purposes. Economically, the two are equivalent � an investor buys or sells some derivative security that provides pay-off in certain scenarios. The simple idea when hedging an existing asset or liability is then to match this derivative pay-off to offset the changes in value on the underlying asset.

The economic equivalence is a good way of thinking about derivatives. Whether you use derivatives for investment or hedging purposes, you are effectively making a bet � the only difference lies in the context. Are we talking about a risk or a chance of some event taking place?

A homeowner who buys fire insurance makes a bet on whether his house will burn � presumably hoping that it never will. The person who buys a lottery ticket makes a bet that his number will be drawn, hoping that it will. While most people are perfectly happy to make the first bet, many will argue that the second bet is �speculative� and bring some negative connotation to it. The economic equivalence tells us that this is wrong � purchasing insurance or lottery tickets are both speculative bets.

What to consider before entering into derivatives

Important features to consider about risk exposure

As with all securities it is important for a client to fully understand the risk characteristics when entering into a derivatives contract. Given that derivatives can have more complex risk profiles than cash instruments, this point cannot be stressed enough. When using derivatives to hedge existing risks it is important to realise that while some exposures (eg, pure interest rate risk) can be reduced or even removed, other risks such as volatility or correlation risk can be introduced from the hedge position.

For the investor using derivatives to implement a given view on, say, expected yield curve changes, the same can be true � gaining one type of risk can, depending on the choice of instrument, be accompanied by other types of risk. The important lesson is that many derivative strategies are related. Making a bet (or buying insurance if you prefer) on interest rate moves may also be a more or less implicit bet on rate volatility and / or correlation.


Finally, it is worth noting that derivative investments are not necessarily any more speculative or any safer than investments in traditional cash assets. While there have been numerous highly publicised �scandals� over the years involving derivatives, it is important to understand that all of these had some common traits. Most of these scandals arose because of poor risk management either because the risks embedded in the derivative in question were not completely understood or the risks were simply excessive relative to the particular investor�s capital base. The use of derivatives or any other security therefore starts and ends with prudent risk management.

A bit of theory

No arbitrage = No free lunches, upside comes at a cost

Modern asset pricing theory is built around the concept of no-arbitrage ie, the economic intuition that any true arbitrage opportunities would be exploited instantaneously and thus be removed. A rather general approach to this is to employ some replication strategy. Suppose we can perfectly replicate the cash flows of a given instrument (say, an interest rate swap) by taking positions in other traded instruments (say, forward rate agreements), then the no-arbitrage principle tells us that the cost of setting up this replicating portfolio must equal the cost of the instrument we are trying to replicate � otherwise we would have arbitrage. While this can seem very abstract, there are two important points to take away:

Loosely speaking, no arbitrage implies that you get what you pay for. The price you pay in the market to have a specific risk removed (insured) must reflect the market�s view on this risk. Equally, the price the market will pay you to take on some risk when investing must reflect the consensus of this risk. Either way, having the opinion that something is cheap or expensive is equivalent to disagreeing with the market on relative risk.

While the no-arbitrage argument can be used to price some derivative instruments without a model, others will be heavily dependent on the chosen model. In general, we use models for two purposes:

Firstly, given that we can observe only a limited number of prices of specific instruments in the marketplace, we are often interested in knowing what the price of related instruments should be. Say we know the price of two options with differing strikes and we want to find a reasonable price for a third option with a strike in between the two other options. To do this we need a model with which we can interpolate as well as extrapolate between the prices that we observe.


Secondly, we need models to perform risk management of derivatives. While we might be able to observe the price of a given instrument in the market, we will in general need models to help us hedge the risks embedded in that instrument. Understanding the benefits and especially the shortcomings of the models is therefore not just of academic interest � it can be critical.

What is a derivative strategy?

Below we will go through a number of different strategies that use derivative instruments. Before we go into detail, we will highlight some distinctions and common characteristics. Basically, we distinguish between two types of derivative instruments: Linear instruments such as forwards, futures and plain-vanilla swaps and instruments with optionality such as swaptions, caps, floors and various exotics.

We will look extensively at different combinations of derivative instruments that can be designed to provide pay-off if the yield curve changes in a pre-specified way � referred to here as curve strategies. As many different combinations of instruments will typically be available to provide this pay-off, the final choice will often depend on the additional risk characteristics mentioned above. Many strategies will therefore have an element of both yield curve and volatility strategies. Besides these combined strategies, we will also look at more pure volatility strategies. Finally, we will look at more specialised strategies that look at utilising various spread and risk premia moves.

Derivative markets

Linear versus option instruments, listed versus over-the-counter markets

For both linear instruments and options we often have several markets through which we can trade similar or related risks. Depending somewhat on the historical background of a specific instrument, trading activity can be concentrated either in listed products or in OTC contracts. Listed derivatives are standardised contracts that are registered on an exchange. This has the potential benefit of providing high liquidity and transparent pricing. On the other hand, the standardised nature of the contracts means that only a limited number of contracts trade, which in turn gives less flexibility compared to an OTC market. Examples of listed products are government bond futures (and their related options) and money market futures. Examples of OTC products are swaps, swaptions, caps, floors and various exotics. In this primer we focus primarily on OTC products.


Documentation

Documentation is important when trading derivatives. Standard versus customised documentation framework

As derivatives are contingent claims it is obviously important to carefully document which cash flows are to be paid to whom and when for every contingency. Also, given that a cash flow is indeed to take place, one counterparty is effectively granting credit to the other party. As with any other loan agreement, this extension of credit will need documentation. Documentation is thus immensely important when dealing with derivatives � especially with individually negotiated OTC contracts.

Basically, there are two ways to set up the necessary documentation. Either counterparty can negotiate tailored terms of trading on a bilateral basis or the parties can agree to use standard documentation. In order to facilitate smooth transactions and mitigate legal risks, a number of standard legal documentation packages are commonly used. The International Swaps and Derivatives Association (ISDA) provides a documentation framework that is widely used for OTC contracts.

Building blocks of the ISDA framework: Master Agreement, Definitions and Credit Support Annex

This basis of the framework is the so-called ISDA Master Agreement that gives the general conditions under which two parties can engage in derivative transactions. Importantly, the Master Agreement underlies the netting principle, under which payments in the same currency between two counter parties are netted against each other.

In addition to the Master Agreement, there are a number of specialised ISDA Definitions that standardise some of the economic terms for different categories (eg, Government bond options). Finally, ISDA has published a framework of credit terms called the Credit Support Annex (CSA), which describes how to conduct marking-to-market, collateral posting and margin calls. The specific financial terms of this document will (as with any credit facility) be subject to individual negotiation.

A fundamental prerequisite for taking derivative positions is thus to have the necessary documentation in place. Once this is done, the additional documentation relating to each individual transaction can be reduced to a minimum that only relates to the financial characteristics. This is very important, as exposure is often removed by entering off-setting positions rather than cancelling existing trades. The volume of trades can thus become quite significant over time.


Interest rates The basic definitions and conventions

The definition of zero-rates and how to compute PV of certain cash flows

This primer deals with interest rate derivatives ie, various forms of financial contracts whose value depends on interest rates. The term �interest rates� is used with slightly different meanings in different settings but here it obviously deserves a broader definition.

As �1 today is more valuable than �1 10 years from now, most people require some compensation for giving up the �1 today in exchange for having it back 10 years later. Interest rates represent this compensation. To keep track of the value of payments at different points in time we use the concept of zero-coupon bond (ZCB) prices. A zero-coupon bond is a bond that makes a single payment of �1 at maturity. In mathematical terms we denote by P(t,T) the price at time t of a zero-coupon bond paying �1 at time T. Note that ZCB prices allow us to compare cash flows taking place at different points in time by computing the present value (PV). Note also that with this apparatus we can price any stream of certain cash flows � we just need to multiply the cash flows with the appropriate ZCB price and add these discounted cash flows together. Using discrete compounding we can represent this ZCB price as:

tTTtrTtP −+

=)),(1(

1),(

That is, instead of talking about ZCB prices we can express the same meaning by ZCB yield or rate. Supposing that the 10Y ZCB yield today (at time t=0) is 4% we can now calculate the ZCB price:

6756.0

%)41(1

)10,0( 010 ≈+

= −P

Why there is no single rate of interest. Defining the yield curve

By plotting the ZCB rates across maturities we obtain the ZCB yield curve. Note that it might not make sense to talk about �the� rate of interest � there can be one rate for every horizon. As an example the ZCB yield curve for the EUR swap market can be seen in Figure 1.


Figure 1 The EUR Swap market Zero Coupon Bond yield curve

Term Structure 11/04-2007

3.00%

3.25%

3.50%

3.75%

4.00%

4.25%

4.50%

4.75%

0 5 10 15 20 25 30

Maturity (Years)

The definition of forward rates and the relation to zero rates

The ZCB yield is, however, only one of several ways to describe the term structure of interest rates. In an arbitrage free world, one could just as well describe the term structure by forward or par rates. By F(t,T ,T1 2), we denote the rate of interest at which you can negotiate at time t to lend or place money between two future points in time T ,T1 2 � this is the forward LIBOR rate. This is related to the ZCB prices by:

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−= 1

),(

),(1),,(

2

1

1221 TtP

TtP

TTTTtF

Typically, when talking about forward LIBOR rates we refer to either three- or six-month rates. Suppose now that we assume that the ZCB yields are flat across maturities at 4%, what is then the 6M forward LIBOR rate between 10Y and 10.5Y seen from today (t=0)?

%9608.3

16624.06756.0

5.01

1)5.10,0(

)10,0(105.10

1)5.10,10,0(

≈

⎟⎠⎞

⎜⎝⎛ −≈

⎟⎠

⎞⎜⎝

⎛−

−=

PP

F

How to describe the term structure using forward rates. A short discussion on par rates

Note that a flat ZCB term structure at 4% does not give a forward LIBOR rate of 4%. Just as we have a complete term structure of ZCB rates, we can generate a term structure of forward rates � a forward curve � consisting of a series of, say, 6M rates.

For coupon-bearing instruments (a coupon bond or the fixed leg in an interest rate swap) it is common to talk about par rates. The par rate of a coupon-bearing instrument is the discount rate at which the instrument would trade at a price equal to its notional value. Suppose a bullet bond pays a coupon of C on an annual basis between years t and T. The par rates y(t,T) is then defined by


∑= +

++

=T

tsTs TtyTty

C

)),(1(

100

)),(1(100

The most common conventions for calculating interest rate payments

Note that the swap rate used to quote interest rate swaps can be thought of as a par rate. Finally, note that when quoting an interest rate we need to specify two things: The compounding frequency (ie, how often interest is paid) and the day count basis (ie, the convention used to determine the period length between two dates). The most common conventions are listed in the table below.

Convention Compounding Day Count Money Market 3M or 6M Act360

Semi Annual or Annual

Bond Basis 30-360

Note that a plain-vanilla interest rate swap uses the money market convention for the floating LIBOR leg and the bond basis convention for the fixed leg. Sometimes we are interested in converting a rate quoted with one convention into a rate with another convention. The following approximations can therefore be useful:

( )11

360*2

365

11*2*365360

2

36030360

30360360

−⎟⎠⎞

⎜⎝⎛ +≈

−+≈

SemiAnnualAct

Annual

AnnualSemiAnnualAct

CC

CC

Pricing swaps

How one can price swaps directly off the zero-coupon yield curve � even though future LIBOR rates are uncertain

As we will use interest rate swaps extensively below it is worthwhile understanding exactly how these are priced. In a plain-vanilla swap, a series of floating payments linked to a LIBOR reference rate are exchanged for a stream of fixed coupons. In the euro market it is standard to have semi-annual payments linked to 6M EURIBOR using money market conventions on the floating leg fixed in advance, paid in arrears and with annual payments on the fixed leg using bond basis conventions. In order to be general we look at a forward-starting swap and outline its cash flows below. For notional simplicity, we look at a swap with the same payment frequency and day count basis on both legs. Formally, we are looking from time t at a payer swap starting at time T maturing at time TS E with payments being exchanged at dates spaced by a period of δ. Note that the replicating strategy for the floating leg of a swap is to enter into a forward rate agreement for each payment date � we can therefore use the definition of forward LIBOR rates to find the PV of the floating leg.


Payments PV

Time Fixed Floating Fixed Floating

C*δ )*δ TS+1 F(t,TS,T P(t, T )*C*δ )- P(t, T ) P(t, TS+1 S+1 S S+1

C*δ )*δ TS+2 F(t,TS+1,T P(t, T )*C*δ )- P(t, T ) P(t, TS+2 S+2 S+1 S+2

� � � � �

C*δ )*δ )*C*δ F(t,T ,T P(t, TT )- P(t, T ) P(t, TS+E-2 S+E-1 S+E-1S+E-1 S+E-2 S+E-1

C*δ )*δ TS+E F(t,TS+E-1,T P(t, T )*C*δ )- P(t, T ) P(t, TS+E S+E S+E-1 S+E

∑+

+=

ES

SuuTtPC

1

),(*δ P(t, T )- P(t, T ) Sum S S+E

Defining the par swap rate Now, as the standard in the swap market is to select the fixed coupon C such that the net value of the swap is zero, we solve the following equation

∑

∑

+

+=

+

+=

−=

⇔−=

ES

Suu

ES

ES

ES

Suu

TtP

TtPTtPC

TtPTtPTtPC

1

1

),(

),(),(

),(),(),(*

δ

δ

With reference to the definition of par yields, we call this fixed coupon the par swap rate. The denominator in the last line is known as the accrual- or annuity factor of the swap. This tells us what the value is of an extra basis point paid on the fixed coupon. Sometimes, it is therefore also referred to as the PV01 of the fixed leg. Note finally, that although the future cash flows on the floating leg are uncertain we can price the swap using only today�s ZCB yield curve.

Yield curves: Can identical cash flows be different?

How do we treat payments in different currencies and of varying credit quality? We use different yield curves

We have now presented the basic definitions of interest rates and have showed how to compute present values of certain cash flow streams. But what do we do if the cash flow stream is not completely certain? Suppose a financial counterparty has promised us a stream of payments but we are not sure that the counterparty will be able to pay, ie, how do we treat credit risk? Although a general treatment of credit risk is outside the scope of this text, we can still give a fairly straightforward answer: We simply use another yield curve to discount these uncertain cash flows on.

In general, we thus have different yield curves for payments from counterparties with different credit risk. In particular, we use one yield curve to price government bonds and another yield curve to price interest rate swaps.


Furthermore, we have different yield curves for different currencies. This reflects the fact that the alternative investment opportunities are different for, say, assets denominated in USD and assets denominated in EUR. These are in turn influenced by the monetary policies in each currency.

The general point here is that although assets have identical face values, they are not economically identical if they are not denominated in the same currency and are of the same underlying credit quality.

Yield curves and their dynamics

How do yield curves change over time? They move up and down, steepen and flatten and bend � they have complicated dynamics

Given that curves describe interest rates at a single point in time, it is natural to think of interest rates over time as a surface. In Figure 2, we see how the term structure of EUR swap interest rates has changed over the last five years. We can see how the curve changes; it steepens, flattens and bends over time. Similarly, we see the term structure evolution of US Government rates in Figure 3. Most often we will describe a change in the curve as the relative movement between two reference points on the curve. When talking about, say, a 5-10Y steepener we thus mean that 10Y yields will increase relative to 5Y yields.

It is important to note that the curve rarely makes the parallel-only shifts that are the basic premise of most (textbook) risk management practices. Since the yield curve moves in more complex ways there is good reason to consider alternative scenarios when managing interest rate risk.

Figure 2 The term structure of EUR swap ZCB yields 2003-07

2W 9M 3Y 12Y

Sep-2007

Sep-2006

Sep-2005

Sep-2004

Sep-2003

0%

1%

2%

3%

4%

5%

6%

Yie

ld

Maturity

EUR Swap Curve (ZCB yields)


How we can use derivatives to bet on specific changes in the yield curve

Now where do derivatives fit into this discussion? One big advantage of OTC interest rate derivatives is that you can tailor a derivative to deliver payoff in any specific scenario. Compared to cash-only instruments, OTC derivatives can thus help you bet on very specific yield curve movements. Given that yield curve movements can be very complex this is indeed an important feature of derivative strategies.

Figure 3 The term structure of US Government rates 2003-2007

2W9M

3Y 12Y

Sep-2007

Sep-2006

Sep-2005

Sep-2004

0%

1%

2%

3%

4%

5%

6%

Yie

ld

Maturity

US Govt. Curve (ZCB yields)


Risk management Measuring uncertainty

Why do we need risk management? We need some way of quantifying risks to financial contracts

Before we present the various risk management metrics in detail it is perhaps useful to take a step back and think about why we are concerned with risk management. Basically risk means that we are somehow exposed to uncertainty. This is typically uncertainty about the size, timing or discounting of future cash flows. In turn such uncertainty will affect the present value of financial contracts. When performing risk management we are basically trying to quantify uncertainty and use various metrics to identify which positions or strategies can help us reduce or expand risk.

When dealing with derivatives, risk exposure is often expressed in terms of the so-called greeks. The basic idea is to measure how the value of a position changes as the inputs to our pricing model are changed one at a time. These inputs can either be parameters in a mathematical model or the term structure as observed today. Below we will first describe the classical metrics relating to term structure changes and subsequently present each of the greeks and their interaction.

Yield curve risk � the classical approach

Classical measures of interest rate risk � exposure to parallel shifts in the entire curve

The main driver of changes in value of both simple fixed income instruments such as bonds, swaps and forward agreements and more advanced derivatives such as swaption, caps/floors and exotics is changes to the yield curve. The most simple change we can look at is a parallel change in the ZCB yield curve. In bond terms we often express this risk as the Basis Point Value (BPV):

100*r

PVBPV

∂∂

−=

That is, the change in present value of an instrument by making a parallel shift of the entire ZCB yield curve scaled to a change of 100bp. This can be thought of as first order ie, a linear approximation of the change in value. Sometimes one encounters a very similar metric called the Present Value of a basis point (PV01) this is equal to

10001

BPVPV −=


Convexity � how to measure second-order risks

Again, this is, however, only a first order approximation. This is a relevant point of concern since the PV of simple fixed income instruments such a coupon bonds and swaps are not linear in ZCB yield changes. To address this issue of non-linearity we often measure the convexity of a position. This is defined as

rBPV

rPV

Convexity∂

∂−=

∂∂

= 2

2

This risk metric thus tells us by how much the BPV changes when the ZCB yields increase. Convexity represents second order changes to the PV of an instrument when the ZCB yields increase. Mathematics tells us that we can approximate total changes in PV as

K+Δ⋅+Δ⋅−≈Δ 2

2

1rConvexityrBPVPV

The additional terms become progressively smaller and smaller, so usually using the first two orders gives a high degree of precision.

Yield curve risk � the modern approach

Why BPV is not a sufficient risk metric. The definition of delta vectors

As hinted in the above section on yield curves and their dynamics, measuring interest rate risk by parallel-only shifts to the ZCB yield curve might not give a sufficiently detailed description of reality. Were we to rely exclusively on the measures introduced above we might still be exposed to risks of non-parallel yield curve changes � without being able to quantify these risks. Put simply, the fact that a position has a BPV of 0 does not mean that it is immune to interest rate risk. It therefore makes good sense to evaluate PV changes of a position relative to piece-wise changes to the ZCB yields or forward LIBOR rates.

By changing ZCB yields one at a time in a series of triangular shifts we obtain a so-called Delta vector. Each element in this vector represents the change in value of a position induced by bumping a certain part of the ZCB yield curve. By adding these vector elements together we get the total interest rate risk to a parallel increase as measured by PV01. By mapping out the total risk, we can measure how the PV of a position would be affected by, say, a steepening of the 2Y-10Y segment of the curve.


Risk managing options or just speaking Greek?

Once we leave the world of instruments that can be priced using only today�s yield curve, ie, instruments with optionality, we have to expand our risk management universe. This is done by measuring changes one at a time in PV induced by changing the parameters used in the mathematical models needed to price these instruments. These changes are traditionally named using letters from the Greek alphabet � the risk measures are therefore known as �greeks�. While doing so is mathematically well defined and consistent, the relevance of these risk metrics depends on how accurate the models are as a description of reality. Suppose for example that the input parameters never change one at a time but they always vary together � is the risk management exercise then valid? There is no general answer to this question but it is important to keep in mind. Below we introduce each of the traditional Greek metrics for the PV of an option written on some interest rate R

Introducing the greeks. How we can use mathematical models to express our risk exposure. Why interest rate options are more tricky than equity options

T expiring at time T. Note that interest rate options compared to, say, equity options are a little more complicated as the underlying (ie, interest rates) are also used to discount cash flows. For the moment we will ignore this effect on discounting and look exclusively at the underlying rate.

Delta

Delta � first order risk on the underlying rate

The delta of a position measures changes to its PV with respect to small changes in the underlying asset. Mathematically speaking we are thus calculating:

TRPV

Delta∂∂

=

This serves as a linear approximation to changes in the PV. Note that the term delta is sometimes used both to refer to changes in PV with respect to the entire yield curve and to refer to changes in a single underlying rate. The two uses are not identical as a change in yield curve will affect both the underlying rate as well as the discount factors.

Gamma

Gamma � second order risk on the underlying rate

The gamma of a position measures the curvature in the change in value of the derivative for changes in the underlying. The more curvature, the worse will a linear approximation of the derivative perform. Gamma is defined as:

TT R

Delta

R

PVGamma

∂∂

=∂∂

=2

2


Note that this is similar to the convexity measure introduced above. Any contract that is not linear in the underlying will have gamma exposure, more specifically any convex (read: options) pay-off will have positive gamma. For these contracts, gamma thus measures the sensitivity to larger movements in the underlying.

Vega

Vega � implied volatility risk It turns out that the value of any contract that is a non-linear pay-off of the underlying will depend on the volatility of the underlying. We measure the sensitivity of a contract against changing volatility by vega:

σ∂∂

=PV

Vega

This metric tells how much the value of the contract will change if the implied volatility increases by one percentage point.

Theta

Theta � carry exposure As many financial contracts accrue or lose value as time passes, although certain in nature, time itself can be considered a risk factor. We measure this time decay in the value of a derivative contract by theta:

T

PVTheta

∂∂

=

Choosing your risk profile � interacting greeks

How the greeks are related to each other. The importance of dynamic hedging

It turns out that the greeks are related. When choosing instruments a client should therefore consider which risk profile is most suitable. Let us look at an example. Table 1 shows the greeks for ATM EUR payer swaptions (for now just think of these as call options on interest rates) with varying time-to-expiration. There are four important lessons to learn from this example.

• The longer the time-to-expiration, the more sensitive will a position to changes in volatility be. This is because over long periods of time, higher volatility will have a large effect on the distribution of interest rates.


• When gamma is strongly positive (the investor is long convexity) a position will have a strong negative theta. A strong positive gamma means that an investor benefits relatively more from large movements than small movements in the underlying. As time-to-expiration shortens, large movements become less likely. Coming one day closer to expiration will therefore have a relatively larger impact on a high gamma position relative to a low gamma position. With this intuition, we can thus explain why theta and gamma as seen in

No arbitrage revisited � why one cannot be long convexity and long theta

Figure 4 Figure 6 and are inversely shaped.

• Also, as theta is measured for a fixed time horizon (one business day) theta is of lesser importance to contracts with long time-to-expiration, as a shortening of one day will mean relatively less.

• Note that the gamma and vega exposure has the same sign � thus if you are long vega you are also long gamma. Why is this? As mentioned, gamma measures how convex a contract is in the underlying. Since volatility exposure and thus vega arises from this convexity the two greeks come hand in hand for options.

Comparing the greeks of various options

Table 1 Greeks for ATM EUR bought payer swaptions on 5Y swaps with a notional of �100M

Expiration PV PV01 Delta Gamma Vega Theta1M 436,923 23,181 22,330 69,541 22,979 -6,9843M 680,988 23,212 22,437 43,913 39,549 -3,5286M 908,428 23,127 22,456 32,169 54,636 -2,3311Y 1,106,557 22,737 22,202 25,240 75,621 -1,3662Y 1,411,471 21,871 21,599 18,048 103,417 -7913Y 1,606,823 20,891 20,905 14,445 122,076 -5355Y 1,847,070 18,828 19,396 10,388 147,936 -27810Y 1,913,749 13,883 15,564 6,115 173,760 -2120Y 1,423,026 7,107 9,874 3,058 141,340 89

In the market jargon a short-dated (usually less than 1Y) volatility position is called a gamma vol position while a long-dated volatility position is called a vega vol position. For an investor with a gamma vol position the realised volatility of interest rates is important while implied volatility is important for the vega vol position. Why is this?

Beware of your risk metrics � gamma spikes at maturity

Close to expiration, the probability distribution of the underlying on the expiration date will be relatively narrow regardless of the level of implied volatility. The determinant of the option�s value therefore lies in gamma and the volatility that is realised up to expiration. This is the reason we see the so-called gamma-spike in Figure 4 prior to expiry if the underlying is trading close to the ATM level.


Figure 4 Gamma for a long position in a �100M 3M5Y receiver swaption

515

2535

4555

6575

85-103

-78-52

-26ATM

2652

78104

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� 150,000

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� 300,000

Days to expirationMoneyness (bp off-set)

What are you looking for: gamma or vega exposure?

For long-dated options the probability distribution of the underlying at the expiration date will be much more dependent on implied volatility as this widens the distribution. As time passes the importance of implied volatility declines, which explains the decreasing vega shown in Figure 5.

Figure 5 Vega for a long position in a �100M 3M5Y receiver swaption

5

30

55

80

-103-78-52-26ATM265278104� 0

� 5,000

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Days to expiration

Moneyness (bp off-set)


Figure 6 Theta for a long position in a �100M 3M5Y receiver swaption

5

30

55

80

-103-78-52-26ATM265278104-� 12,000

-� 10,000

-� 8,000

-� 6,000

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� 2,000

Days to expiration

Moneyness (bp off-set)

Pricing options � the put-call parity

The put-call parity � a basic relationship of option pricing. Holds true regardless of the choice of model

Note that one general relationship between option prices holds true regardless of models. Specifically, as a call option is the right (but not the obligation) to buy some asset R at time T at fixed price K and a put option is the ditto right to sell, we can in mathematical terms write pay-offs as:

( )( )+

+

−=

−=

T

T

RKPut

KRCall

Suppose now that you buy the call and sell the put, you then get

( ) (( )KR

RKKR

PutCall

T

TT

−=

−−−= )−

++

But this is exactly the pay-off of a forward contract expiring on time T. We thus have that

PutForwardCall

ForwardCallPut

ForwardPutCall

+=−==−

This is known as the put-call parity. It tells us that we can create a put by a short position in a forward and a long position in a call or we can create a call with a long position in both a forward and a put.


Outright strategies Where are rates going?

Using swaps to take a simple bet on the general levels of interest rates

The simplest bet we can make with interest rate derivatives is an outright on the direction of rates. Should a client wish to take the view that rates are going to fall, he will, in risk management terms, seek to obtain some negative delta or PV01 exposure. This can be done in many ways: by buying bonds (spot or in a future), receiving fixed in an interest rate swap or buying a receiver swaption. Suppose the client chooses to receive fixed in a 20Y swap starting spot. By doing so the client obtains the exposure to parallel changes in the ZCB yield curve shown in Figure 7.

Figure 7 PV of 20Y swap arrangements for shifts in ZCB yields.

�100M 20Y Plain Vanilla Swap

-�100

-�80

-�60

-�40

-�20

�0

�20

�40

�60

�80

�100

-400 -200 0 200 400

Shift in ZCB Yield Curve (bp)

Sw

ap

PV

�M

Payer Receiver

Why swaps have convexity We see that should ZCB yields fall, the client receiving fixed stands to gain. Furthermore, we note that the rate at which the client gains or loses changes with the magnitude of the shift in yields � ie, the swap is a convex instrument in ZCB yields. Finally, we see that paying fixed in the swap would give the exact opposite exposure and thus position the client to rising rates. We sum up the position�s risk in Table 2.

Table 2 Exposure on �100M 20Y receiver swap

Key figure BPV �12.585M Convexity �2.376M PV01 -�125,850


Curve strategies Introducing relative bets

Using derivatives to take relative bets on the yield curve

In this section we look at how a client can use derivatives to implement views of flattening or steepening yield curves. We reiterate the point that all of the below strategies require that the investor has an expectation that is different from the market�s.

Swaps versus bonds

Say an investor thinks that the 2Y to 10Y segment of the yield curve will steepen. That is, the investor thinks that 10Y rates will rise relative to 2Y rates. The problem is, however, that these rates are not traded instruments as such; the investor thus needs to find traded securities that are exposed to these rates. Two basic candidates are bonds and swaps (for the moment we assume that they are priced on the same curve).

The investor could obviously buy 2Y bonds and sell 10Y bonds. If 10Y rates rise relative to 2Y rates, the 10Y bonds will depreciate relative to 2Y bonds. However, for each yield curve there are only a limited number of bonds trading, and among these there might not be bonds with 2Y or 10Y maturities that are sufficiently liquid. Furthermore, when choosing specific bonds there might be idiosyncrasies that can lead to changes in relative value. Specifically, one could imagine liquidity effects if the one bond is deliverable into a future contract or if one bond drops out of a major index while the other remains.

Swaps are cleaner instruments for curve plays � they do not observe �specialness�

Alternatively, the investor could enter into two swap agreements receiving fixed on a 2Y swap and paying fixed on 10Y swap. By using swaps to implement the steepener strategy the investor eliminates the two problems outlined above. As swaps are OTC contracts, any maturity can in principle be agreed upon with the swap counterparty. Moreover, their generic nature means that only changes in the yield curve will change the relative value. One could thus say that the swap alternative offers a cleaner strategy compared to cash bonds. Finally, swap markets are highly liquid and screen prices are readily available through data providers such as Bloomberg and Reuters.

Should the strategy be built on top of existing assets or liabilities, the swap agreements can furthermore be structured to include a suitable amortisation schedule to fit these.


Think about BPV � how to make a curve strategy duration neutral

No matter whether bonds or swaps are ultimately used, the investor will introduce outright interest rate risk if he uses the same notional on 2Y and 10Y leg of the trade. Why is this? A 10Y swap or a 10Y bond has a higher BPV than a 2Y ditto. The investor will thus often choose to make his strategy BPV neutral by scaling the notionals on each leg in order to remove the outright risk. Note that BPV neutrality is often used interchangeably with duration neutrality. With a BPV neutral position, the overall level of interest rates can go up or down without affecting its value � only relative changes in rates matter. As shown in Table 3 the strategy is made duration neutral by scaling the notional on each swap.

Table 3. Making a swap strategy BPV neutral

BPV per �100M Notional

Notional BPV

2Y Receiver Swap � 19,636 � 100.00M � 19,63610Y Payer Swap -� 84,513 � 23.23M -� 19,636

� 0Total Position

Expressing a strategy�s risk profile in terms of delta vectors

As mentioned in the risk management section, the basis point value (BPV) of a position can be decomposed into delta vectors that measure the change in value for a change in a series of interest rates. Figure 8 shows the delta vectors for the combined position. As the investor is positioned to a rise in 10Y rates relative to 2Y rates, we see that the position has a strong positive delta in the 10Y point and a strong negative delta in the 2Y point. The same mechanism will however work against the investor should the 2Y-10Y segment flatten or even invert � this is a potential risk that the investor should be comfortable with. In addition to the delta risk, the position will be slightly gamma positive in the 10Y point and have some albeit less (in absolute terms) negative gamma exposure in the 2Y point.

Figure 8 Delta vectors of the 2Y-10Y steepener swap strategy

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0 1 2 3 4 5 6 7 8 9 10

Year


Spot versus forward � hedge stability

Hedge stability over time � how to play a steepening of the curve using forward starting swaps

Regardless of the choice of instrument, the investor is faced with a timing issue when implementing a given view on the yield curve: Should the strategy be implemented spot or forward? That is, as an alternative, the client could consider using bond futures or forward starting swaps. What should be considered when making this choice?

Suppose the investor believes that his view will materialise over the next year. As time passes the hedge ratio that we found above will generally change. That is, the position is unlikely to stay duration neutral. One year from now, the position will effectively be a 1Y receiver swap and 9Y payer swap. As can be seen in Table 4 the position has acquired duration sensitivity through the passage of time. The problem is that the short maturity swap loses duration faster than the long maturity swap.

Table 4 Hedge ratio after 1Y


Notional BPV


-� 8,007Total Position

Suppose the investor instead had chosen to implement his strategy on a 1Y forward basis. That is, by entering into a 1Y forward starting 2Y receiver swap and a 1Y forward starting 10Y payer swap the investor will at inception be exposed to the 3Y-11Y segment. One year later the investor will again be exposed to the 2Y-10Y segments. The question is now; at what point in time should the investor aim to make the position duration neutral? As can be seen in Table 5, it turns out that it does not make a big difference whether you make it duration neutral now or one year from now. The table shows how to set up a duration neutral position at the start of the swaps. Furthermore we can see that this choice almost makes the position duration neutral at inception.


Table 5 Duration neutrality of a forward starting strategy


Notional BPV


� 0


Notional BPV

1Y2Y Receiver Swap � 18,900 � 100.0M � 18,9001Y10Y Payer Swap -� 81,222 � 23.23M -� 18,872

� 28

Total Position

Total Position

The forward starting strategy obviously gives rise to delta vectors different from the ones we saw in the previous example � as can be seen in Figure 9. Characteristically, a forward starting swap has a delta exposure at the forward start point and a strong delta exposure of the opposite sign at the maturity point.

Figure 9 Delta vectors of the 2Y-10Y steepener strategy with a 1Y2Y receiver swap and a 1Y10Y payer swap

-� 1,000

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0 1 2 3 4 5 6 7 8 9 10 11

Year

10Y Payer Swap 2Y Receiver Swap Position Total


Spot versus forward � convenience of no payments

Another argument for using forward starting swaps � the convenience of not using liquidity

Although the strategy using spot swaps outlined above has zero NPV at inception, there will be net cash in- or outflows to the investor at every payment date. For an investor running an unfunded strategy, making net payments (should there be any) will use scarce liquidity. Furthermore, having to make payments tied to eg, 6M EURIBOR introduces fixing risk on the floating leg (once the reference rate has been fixed the investor obtains interest rate risk on the associated payment). For this reason and for pure administrative convenience, the investor could choose to implement the strategy with forward starting swaps thus eliminating payments during the forward period. For example, the investor could enter into a 6M forward starting 2Y receiver swap along with a 6M forward starting 10Y payer swap. At inception, this will effectively be a position based on the 2½Y- 10½Y segment of the yield curve but six months from now, the strategy will be 2Y-10Y position. If the investor�s beliefs have materialised at this time, the position can be closed down and profits realised without ever having made a single payment.

Introducing swaptions

Options on swaps = Swaptions. Simple OTC interest rate options on forward swap rates

As we saw above, the key to implementing the 2Y-10Y steepener strategy was to get negative delta exposure in the 2Y point and positive delta exposure in the 10Y point. While we saw how this worked with (forward starting) swaps, we could also obtain a similar exposure profile with swaptions. Swaptions are OTC contracts that give their holder the right but not the obligation to enter into a predetermined swap agreement. The investor can at inception choose between physical or cash settlement. Depending on the underlying swap, we refer to swaptions as being either receiver or payer swaptions (does the client have the right to pay or receive fixed?).

As with all options the swaption contract can be exercisable at maturity (European) or at specific intermediate dates (Bermudan). European swaptions are the market standard but Bermudan swaptions do trade in relation to the mortgage bond market. Following the usual option terminology we talk about swaptions being in-, out- or at-the-money.

When choosing a suitable swaption structure, the investor should consider what maturity, strike level and option type to employ in his strategy. Due to their OTC nature, swaptions can be traded at any strike, however the market is concentrated around ATM swaptions. In terms of liquidity and transparency (screen prices are available on Reuters and Bloomberg) these contracts can thus offer slightly better terms than their OTM counterparts.


Obviously, by choosing a swaption that is further out-of-the-money the investor will need a larger move in par-swap rates to make profit but will on the other hand pay a lower premium at inception. Similarly, swaptions will in general be more expensive in terms of premia the longer their maturity. On the other hand the theta or time decay will generally be higher for short maturity options, which makes them more expensive to hold. The investor should thus choose a maturity within which he feels comfortable that his views will materialise and over which he is comfortable with the carrying cost. Also, the investor could choose between the different types of options. The added flexibility of Bermudan options over their European counterparts come at a (weakly) higher cost � the investor thus needs to consider whether he needs this added optionality. For speculative purposes it will most often be sufficient to use European options as the investor can always sell his position in the market to take profit. The final choice of structure is thus a matter of how large and how fast the investor expects changes in rates to be.

What determines the price of a swaption?

The linkage between swaptions and the forward curve

In the introduction we mentioned that ZCB yields and forward rates are two different ways of characterising the term structure of interest rates. At every future point in time, we can (from the ZCB yield curve) calculate eg, 3M money market interest rates. These interest rates constitute the forward curve, which can be seen in Figure 10. At every future fixing date, the forward curve tells us what we can expect to pay on the floating leg of a swap. In the example on how to price interest rate swaps we saw that it is exactly these forwards that determine the price (ie, the fixed coupon) on the swap.

Figure 10 The forward rate and ZCB yield characterisation of the US swap curve

4.00%

4.25%

4.50%

4.75%

5.00%

5.25%

5.50%

5.75%

0 5 10 15 20 25 30

ZCB Yield 3M Forward Curve


As discussed previously, (the convex) option pay-off means that not only the expectation but also the volatility of forward rates matters when valuing swaptions. A 5Y option on a 10Y swap (which we will refer to as 5Y10Y swaption) will thus depend on the segment of the forward curve between the two red lines in Figure 10. Now, as pointed out earlier, the par swap rate is the coupon that equates the value of the floating side to the value of the fixed side in a swap. This means that we can think of swap rates as being a weighted average or a portfolio of forward rates. In turn, this insight tells us that the value of a swaption must depend on the correlation between forward rates as the value of this portfolio is correlation dependent.

Conditional versus unconditional � using swaptions in a one-sided steepener strategy

Swaptions can be used instead of forward starting swaps to make curve strategies conditional on the level of interest rates

When implementing the steepener strategy with swaptions, directionality will begin to play a role. Suppose that the investor buys an ATM 1Y2Y receiver swaption and sells an ATM 1Y10Y receiver swaption. This strategy will (one year from now) have the desired negative delta in the 2Y point and positive delta in the 10Y point.

Suppose furthermore that one year from now there has been a general rise in forward rates as well as a steepening of the 2Y-10Y segment. In this case the two swaptions will expire out-of-the-money. If forward rates, however, have experienced a general fall with the 0-2Y segment falling more than the 0-10Y segment, the investor will profit from the position. The investor has thus made a conditional bet on a steepening of the 2Y-10Y segment. As with the previous examples, the total position can be made duration neutral at inception or at expiration of the swaptions by scaling their relative notional. Again, the risk to the investor lies in the 2Y-10Y segment flattening although this will also be dependent on directionality. Should the general level of forward rates increase enough, even a significant inversion will not give the investor a loss beyond any initial premium paid since both swaptions will expire OTM.

Conditional versus unconditional � using swaptions to take positions on the entire curve

Since swap positions are constructed to have zero NPV at inception, the no-arbitrage principle tells us that if there is a potential for profit there must also be a potential for losses. Suppose the investor feels uncomfortable with the downside exposure, an alternative would be to use conditional options-based strategies.


Using swaptions instead of swaps allows the downside risk to be reduced by paying an upfront premium

Assume that an investor expects a general decline in interest rates on the entire curve over the course of the next two years. A position that would benefit from this movement would be, say, a 2Y20Y forward starting receiver swap. As rates fall, the investor�s swap will become very valuable. Suppose however that rates do not fall but rise instead. The investor will then be faced with a big loss. By using swaptions the investor can reduce this downside risk potential by paying an upfront premium.

The above investor could thus enter into an option on a 20Y receiver swap. If there is a general decline in interest rates, the par swap rate will also drop and the receiver swaption will gain value. Should interest rates move against the investor the downside is limited to the initial premium. What expiration should the investor choose for the swaption? Obviously, he could choose 2Y to match the horizon over which he is expecting changes, but it is worth stressing that there are many other choices as the investor can take profit on yet-to-expire options by selling them back to the counterparty. Choosing longer dated swaptions will generally be more expensive upfront but they will on the other hand generally be less theta sensitive ie, their negative carry will be lower. As outlined in the section on the greeks the investor needs to consider if he wants a gamma- or vega vol position.

Swaptions can generate much more convexity than swaps. This gives more leverage to the swaption position

It is worth noting that choosing the conditional swaption based strategy over the unconditional forward starting swap-based strategy also means a change in the investor�s risk profile. First of all, the investor gains vega exposure since the swaption is a volatility dependent product but also the BPV and convexity exposure changes. In Figure 11 we see why and how the BPV and convexity exposure changes by looking at how the value of the two contracts change as the zero coupon yield curve is shifted in parallel. Note that the BPV for each contract can be measured as the slope of the two curves. Similarly, convexity can be measured as the curvature of the two curves. We see that the swaption has a lower BPV than the swap but that delta increases as rates drop. By taking the conditional swaption position, the investor thus obtains a less BPV-sensitive position but at the same time gains much more convexity. Besides insuring against downside risk the conditional position is well suited for plays on large moves in interest rates around the ATM level as the strong positive gamma kicks in, the larger the move in rates (this is the gamma-spike we saw in Figure 4).


Figure 11 Price-interest rate relationship with parallel shifts for a 2Y20Y fwd starting rec. swap and a 2Y20Y rec. swaption on �100M notional

-� 300,000

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-300 -200 -100 0 100 200 300

Bp Offset

Swaption Swap

Conversely, expectations of higher rates can be transformed into a payer swaption strategy. While the delta of a payer swaption will be positive (as it is a call option on the par swap rate) it will, like the receiver swaption, also have a positive gamma. As mentioned in the section on greeks, this is because vega and gamma come hand in hand with non-linear pay-offs.

How the passage of time influences a swaption position

Finally, the conditional position will have a different theta or sensitivity to the passage of time. In a scenario with a normal (upward sloping) yield curve, the forward swap rate will be decreasing as the forward period is shortened. As time passes this means that the roll on the yield curve will be beneficial to the receiver swaption but will go against the payer swaption. For both options, however, shorting the time-to-expiration will, all things being equal, reduce the value of the contract.

Conditional versus unconditional � using caps and floors to take positions on the entire curve

The definition of a cap and a floor. Why these interest rate options are different from swaptions

As an alternative to the swaption-based strategies outlined above, an investor could also express the view of generally falling or rising interest rates through the market for caps and floors. Should the investor believe that interest rates are about to rise, he could buy a cap. If rates do indeed rise, the cap appreciates in value and thus generates profit. How is this different from using swaptions?


The volatility wedge � the difference between cap/floors and swaptions

A cap is a portfolio of caplets where each caplet is a call option on a reference rate (eg, 6M EURIBOR). This is contrary to a payer swaption that can be thought of as an option on a portfolio of reference rates. While the former can be valued as independent parts, the latter depends on the correlation between the rates. If forward rates are less than perfectly correlated a payer swaption will cost less than a similar forward starting cap since there is a diversification effect between rates. This difference is known as the volatility wedge and is described in detail in �What is the volatility wedge?�.

Note that while we can naturally think of swaptions as being the instruments that can hedge the fixed leg in a swap we can think of caps and floors as being the instruments that can hedge the floating leg.

A basket option analogy � due to correlation, an option on a portfolio is not the same as a portfolio of options

The correlation argument above is the same reason why a basket option is cheaper than a basket of similar options. If the investor uses swaptions rather than caps or floors he will thus have the mentioned implicit long position in the correlation between rates. When does this matter?

Suppose we have a scenario with a flat 6M EURIBOR forward curve at 4%. An investor expects rates to increase and therefore contemplates buying either a 1Y forward starting 10Y cap or a payer swaption both struck at 4% (disregarding the difference in conventions). Suppose now that the investor�s expectation is only partially fulfilled; one year from now the curve has steepened. Rates in the 0-5Y segment have fallen to 3%, while rates in the 5Y-10Y segment did in fact rise to 5%. This example is illustrated in Figure 12.

Figure 12 Example of 6M EURIBOR forward rates

0%

1%

2%

3%

4%

5%

6%

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

Forward Euribor @ year 0

Forward Euribor @ year 1


http://danskeresearch.danskenet.net/link/Whatisthevolwedge/$file/what_is_the_vol_wedge.pdf

Why there is more optionality (and thus vega) in a cap compared to a swaption

The steepened forward rate structure that is realised at year 1 implies a par swap rate on the 10Y EUR swap of 3.84%. The receiver swaption thus expires OTM. Even though the investor�s expectations were partially fulfilled, the strategy did not pay-off. Using swaptions to make the bet on increased interest rates is thus a one-shot game.

What would have happened had the investor instead chosen the cap? The first five caplets are now OTM while the last five caplets are ITM. If these forward rates are realised, the investor will profit. It is however important to understand that the cap�s cash flows (and thus interim market value) is still uncertain at year 1 � contrary to the swaption, the cap maintains optionality until year 10. The partial realisation of expectations will be reflected in the pay-off profile � on each caplet for which the investor�s view on interest rates proves to be correct he will profit. With a cap the investor has more shots (and more time) to get his view right. The same arguments can of course be used to compare receiver swaptions to forward starting floors.

Protection: Extendable and retractable swaps

By combining swaps with swaptions, one can tailor protective strategies to use alongside existing swaps

In many cases it can be interesting to use combinations of swaps and swaptions. Let us go back to the 2Y-10Y steepener strategy that used 1Y forward starting swaps. Suppose that market conditions go against the investor, that is, the 2Y-10Y segment flattens instead of steepens and thus leaves the investor with a massive loss. As long as these changes happen gradually and as long as liquidity is abundant the investor can always enter into off-setting positions and thus take stop-loss. If market moves do however exhibit jumps or deteriorating liquidity (ie, widening of bid-offer spreads to prohibitively expensive levels) it could be interesting for the investor to hedge his bet with OTM swaptions. Having the swaption hedge in place, means that the investor does not have to constantly monitor the swap market in order to fulfil a stop-loss obligation. The investor can in this sense use swaptions to protect his positions.

In the given example, the investor could thus buy a 1Y2Y OTM payer swaption and 1Y10Y OTM receiver swaption. The more out-of-the money the swaptions are, the less protection they offer but the cheaper they will be. When an off-setting swaption is taken along with a swap, we say that the swap is retractable. Similarly, a swaption position that allows an investor to extend the conditions of an existing swap arrangement is called an extendable swap.


Constant maturity strategies

The background for constant maturity swaps

The cash flows that take place in a regular interest rate swap depend on two things; some short floating rate and a fixed coupon that is determined by the forward curve at inception. As time passes the duration of the swap will obviously decrease, which means that the investor has to continuously monitor and hedge whatever duration risk he might have on a portfolio of swaps.

To avoid this problem, the investor can take a position based on a Constant Maturity Swap (CMS). A CMS is a swap agreement in which a floating reference rate eg, the 5Y par swap rate, is paid or received in exchange for a floating money market rate +/- a spread. The payments in a CMS are thus variable on both legs. Market convention denotes the swap relative to the CMS leg ie, in a receiver CMS an investor receives the reference rate and pays the floating rate. Contrary to the plain-vanilla swap, the cash flows on the CMS will depend on the forward curve at every fixing. This means that duration will be fairly constant over the life of the swap.

Swapping high volatility LIBOR rates into low volatility long rates

Also, a CMS swap can be used to swap exposure against short LIBOR rates that can be highly volatile into exposure against long-term reference rates that are usually less volatile.

We can think of a CMS as an agreement to exchange the spread between some floating rate, say 6M EURIBOR +/- a spread, and the reference swap rate. Intuitively, as this spread is immune to parallel shifts in the yield curve, a CMS has very limited duration exposure. This furthermore implies that the CMS swap has very limited convexity risk.

From the perspective of the investor, the two features outlined above are very attractive. When entering into a CMS he will obtain a duration neutral position that will remain so over time. This is especially attractive if the investor is planning to keep the strategy on his books over a prolonged period of time. Suppose we are again interested in implementing a strategy that benefits from a steepening of the 2Y-10Y segment. How can this be done with a CMS?

Using CMS swaps to take a curve position

Suppose the investor enters into a 2Y CMS receiving the 10Y swap rate while at the same time entering into 2Y CMS paying the 2Y swap rate (in effect a dual CMS leg swap). In Figure 13 we see the delta risk for the investor. We see that the investor will benefit from rates falling the short end (<5Y) and from increasing rates in the long end (>5Y). Compared to the 2Y-10Y strategy we saw earlier on, the CMS strategy will spread out the investor�s risk over a broader segment of the curve.


Figure 13 Delta vectors for a 2Y CMS, receiving 10Y swap rates while paying 2Y swap rates on a�100m notional

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This broader delta exposure means that a CMS position can be used to take a combined view on the slope and curvature of the yield curve. As seen in Figure 14, implementing a flattener strategy using a 5Y EUR CMS paying the 5Y swap rate, positions the investor for rising rates in the 0-5Y segment and falling rates in the 6Y-10Y segment ie, a flattening of the curve. The strategy however also gives a strong exposure towards the 5Y rate, implying that the investor will benefit from more curvature in the yield curve.

Why CMS swaps have vega risk while forward swaps do not

Besides the delta exposure shown in the above figures, CMS positions also have vega exposure. Why is this? We saw earlier that a (forward) swap is a convex instrument in ZCB yields. Moreover, a swap with a fixed coupon is also convex in par swap rates. Every payment made on the CMS leg in the CMS swap is however linear in par swap rates. Intuitively, one can therefore not hedge the CMS leg payments using only forward starting swaps. In particular, swaptions across a broad segment of strikes are needed to completely replicate a single CMS cash flow. From this replicating strategy in swaptions arises the vega exposure of CMS swaps.


Figure 14 Delta vectors for a 5Y swap EUR CMS paying the 5Y swap rate on �100m

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Finally, note that constant maturity swaps can be used to take a view between short rates, say 3M or 6M CIBOR, and an average interest rate over some suitable interval (as discussed we can think of the swap rate as an average interest rate). Suppose an investor thinks that an average of EUR rates up to and including the 2Y point will increase relative to 3M EURIBOR over the next year. The investor could then enter into 1Y EUR CMS receiving the 2Y swap rate. This also means that CMS swaps can be used together with plain vanilla interest rate swaps to convert outright interest rate risk into curve risk.


Asset Swaps Levelling out the playing field

Asset swaps � a clever way of comparing different bonds

Until now we have discussed how to implement various interest rate views using plain-vanilla interest rate swaps and swaptions. One argument for using derivatives in the first place is the fact that they provide much more flexibility compared to cash instruments. Often it will, however, make sense to use derivatives alongside cash instruments. In particular one can use so-called asset swaps alongside cash bonds.

What is an asset swap?

An asset swap is a swap arrangement in which one leg of the swap is structured to match an existing asset. In its simplest form this asset could be a regular fixed rate bond whose cash flows are swapped in return for LIBOR +/- a spread. Intuitively, such a transaction converts a fixed rate asset into a floating rate instrument.

What are asset swaps used for?

Using asset swaps to lock in carry of a position

Generally speaking, asset swaps allow investors to compare bonds across maturities, coupons and credit quality by quoting them as a fixed spread to a reference LIBOR rate. From this comparison investors can trade underlying cash bonds by either paying or receiving in the asset swap. Relative value investors can thus use asset swaps to lock in carry differences between different bonds. Similarly, asset swaps allow investors to remove their interest rate risk on regular fixed-rate bonds by converting their cash flows to floating-rate cash flows. Effectively, this means that investors can obtain bond exposure without any interest rate risk. Finally, asset swaps provide an easy way to trade swap spreads (sometimes referred to as spreadover risks).

Types of asset swaps

Basically, an asset swap can match the underlying bond both in terms of payment details (frequency and interest rate conventions) and coupons. Should the coupon on the bond differ from the par rate in the swap, using the bond coupon in the swap will introduce either a premium or a discount in the asset swap.

In a par-par asset swap the party paying or receiving the off-market coupon will have to pay or receive an upfront premium cancelling out the PV arising from the off-market coupon. This upfront payment is in turn amortised over time via the off-market fixed rate.


Alternatively, the investor can use a yield-to-maturity asset swap in which any premium or discount to par on the bond is offset by another off-market coupon in the swap. This off-market rate is chosen to match the premium in the bond when swapped against LIBOR flat. The payment details between the bond and the fixed coupon in the bond still matches � only the coupon differs.


Volatility Strategies What is volatility?

The background to looking at volatility: Some definitions and why this is related to derivatives

Before we start describing volatility strategies let us take a step back to understand why volatility is important and where it comes from. Intuitively, volatility represents the size of changes. When the daily changes in, say, 10Y yields are big we say that volatility is high and vice versa when the changes are small. Mathematically speaking, volatility is the standard deviation of the change in value of a given asset over a period of time (usually quoted for a period of one year). This metric thus tells us how much some uncertain asset price or interest rate varies over time. Looking into the future we might have some expectation of what, say, 5Y interest rates will be a year from now. Given that this is indeed uncertain, we can use volatility to describe the magnitude of this uncertainty � the bigger the uncertainty the higher the volatility. Uncertainty is thus the source of volatility.

Why look at volatility?

Why is volatility important when dealing with derivatives? As mentioned, the value of a financial contract that is a non-linear payoff written on some underlying asset will depend on the asset�s volatility. Investors that have positions in contracts with these non-linear pay-offs will thus have volatility exposure. For this simple reason, understanding volatility risk is necessary to understand daily movements in mark-to-market value. Furthermore, volatility is a tradable commodity � just as one can make speculative investments in yield curve changes, one can also make speculative investments in volatility.

Implied volatility

The definition of implied volatility � a way of comparing option premia across different options

The value of all options will be volatility dependent since they are non-linear pay-offs. Moreover, the value of these options will be increasing in the volatility of the underlying. That means that there is a 1:1 relationship between a given options price and its implied volatility. This is why options are often quoted in terms of their implied volatility. The implied volatility is the volatility that equates the theoretical price from an option pricing model (some variation of the Black-Scholes-Merton formula) to the price observed in the market.


Implied versus realised volatility

Why is implied volatility most often above the volatility realised in the market? The role of risk premia

Now it seems reasonable to assume that there must be some link between how much the underlying is expected to change over time and the implied volatility. However, it turns out that implied volatility is a poor predictor of realised volatility. Why is this? When an options trader quotes an option he will implicitly indicate how much variation (ie, volatility) he expects to be realised over the term of the option as this will determine his cost of hedging. However, he is likely to attach some risk premium to this expected volatility. This presence of risk premia on volatility will drive a wedge between realised and implied volatility.

Skews, smiles and term structure of volatilities

Strike dependent implied volatility: Skews, smiles and smirks

Given that we now understand the concept of implied volatility we can quickly move on to look at how implied volatility depends on strike, the level of the underlying and time-to-expiration. It is well documented that most fixed income and equity options markets exhibit so called skews, smiles or smirks.

How these are related to the market�s expectation of future interest rates

Figure 15 shows an example of the volatility smile of a 6M1Y EUR swaption. Deep OTM options are quoted with a higher implied volatility than ATM options. These features can be attributed to extreme events. If these events occur more often than the (log-normal) models predict, then an option trader will charge a higher premium on deep OTM options, which is why these will have a higher implied volatility. If the true distribution of the underlying does not fit the log-normal distribution, we will see different implied volatilities for different strike levels. For a more thorough presentation of the link between market implied probabilities and options see �Introduction to option implied probabilities�.

Furthermore, option markets tend to be most liquid around the ATM level which tends to make ATM options cheaper than OTM options. Often, one will see the strike listed relative to the current spot level of the underlying. We will then talk about an option�s moneyness, an ATM option thus has a moneyness of 100%.


http://danskeresearch.danskenet.net/link/IntroToQDistributions/$file/IntroToQDistributions.pdf

http://danskeresearch.danskenet.net/link/IntroToQDistributions/$file/IntroToQDistributions.pdf

Figure 15 Implied volatility smile for a 6M option on a 1Y EUR swap.

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The volatility term structure Implied volatilities are also dependent on the time-to-expiration. For fixed income options, a picture like the one shown in Figure 16, ie, a hump-shaped volatility term structure is very common. From time to time the volatility term structure however inverts.

When options are kept on the books over longer periods of time the term structure will be important since an investor will � all else being equal � roll along the curve as time passes.

Figure 16 Term structure of implied ATM swaption volatilities on a 1Y EUR swap, 31/10-2006

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If we look at a single underlying asset, say, a 1Y swap, we can think of the two graphs above as spanning a surface. In Figure 17 we can see an example of such a volatility surface, we see that the smile is more pronounced for short option expiries.


Figure 17 Implied volatility surface of a 1Y EUR swaption

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The dynamics of implied volatilities

How to look at the dynamics related to implied volatility

When an investor buys an option on the 1Y swap, he effectively buys a point on the surface above. When taking speculative positions in volatility, the investor thus needs to consider where he thinks there is value on the surface. Over time the investor�s volatility exposure will be exposed to three factors:

• As time passes, the expiration date of the option will come closer, and the investor will roll along the time dimension as described above.

• As the spot of the underlying changes, the moneyness of the option will change (as the strike is constant). This will cause the option to move along the skew dimension of the volatility surface.

The relationship between the at-the-money-forward implied volatility and the underlying rate

• Finally, the volatility surface itself changes over time. It turns out that most of the movement can be explained by parallel shifts up and down. However the steepness of the skew and the term structure also changes. Generally, we see a negative relationship between ATM volatilities and the underlying interest rates. An example of this can be seen in

Figure 18, the line drawn into the graph is called the backbone.


Figure 18 Daily observations of 10Y20Y ATM implied swaption volatility and the 10Y20Y forward swap rate, September 2001 through November 2006.

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An example of the dynamics

Tracking implied volatility over time and across different levels of interest rates

Let us suppose that an investor buys a 5Y10Y ATM payer swaption today. One year later interest rates have fallen: how has this impacted on the investor�s position? Since a payer swaption is a call option on the forward swap rate, falling interest rates will drive the swaption out-of-money, This will decrease the value of the swaption. This movement in rates will be the most important value driver � we sometimes call this a first order effect. However other factors � second order effects � will also affect the position.

Given that 1Y has passed, the time value of the swaption has been decreased (ie, the swaption has negative theta). Against these two sources of lost value stand two gains in terms of implied volatility. As rates drop, ATM volatility is likely to have increased. In addition to this, the reduced moneyness of the swaption is likely to have led to an increase in the implied volatility � something that is beneficial to the holder of the swaption. When taking volatility positions an investor has to feel comfortable understanding dynamics like those described above.


Tracking an options market

Multiple ways of saying the same thing: How to quote option prices

Before we go into detail about various trading strategies using options, it is useful to briefly present various ways in which options can be quoted in the fixed income markets. Below in Table 6 is an example of equivalent ways to quote the price of a swaption. First of all, we can naturally talk about the price ie, the premium of an option. This will most often be quoted as a fraction of the notional and expressed in basis points. As noted above, we could just as well quote the price in terms of implied volatility. The market standard for caps, floors and swaptions is then to use the Black�76 formula to imply the volatility. This number can be thought of as the annualised volatility to be used in a log-normal model to find the distribution of the underlying swap rate at the expiration of the contract. Note that this dual way of quoting prices is similar to the bond market where we are used to seeing quotes both in terms of prices and yields.

A way of comparing implied volatility across different levels of interest rates � basis point volatility

Finally, taking into account the negative relationship between forward swap rates and implied volatility, some market participants express implied volatility relative to the prevailing level of swap rates. This measure is known as the basis point volatility and is simply the forward swap rate multiplied by the implied Black�76 volatility.

Table 6. 5Y5Y ATM EUR Receiver swaption

Keyfigure Premium 183 bp Implied Volatility (Black�76) 12.4% Forward Swap Rate 4.7693% Basis Point Volatility 59.1 bp

Note that for a cap we are implicitly looking at several options (ie, caplets) on the same time. We can either express the value of these through forward LIBOR volatilities or through a flat volatility that can be thought of as a kind of �average� volatility.


What determines the price of volatility?

Drivers of the volatility market: The underlying market and structural conditions

Having seen how to track prices of options, it is useful to sum up the drivers of the volatility market. First of all, realised volatility feeds into implied volatility. This especially holds true for short expiry options (ie, gamma vol positions) as there is a high degree of persistency in realised volatility. Secondly, risk premia feed into implied volatility � by how much does realised volatility change over time? The more time-varying realised volatility is, the more expensive will implied volatility tend to be. Finally, there are structural supply and demand factors in the volatility markets. Some segments are highly influenced by hedging flows coming from exotic desks, while others to a large degree are determined by Asset and Liability Management (ALM) of pension funds.

Strangles and straddles - simple volatility bets using caps/floors

Simple ways of combining options with different strikes in a play on high or low realised volatility

Suppose that an investor expects that a large change in 6M EURIBOR rates will take place over the next 2Y. The investor is, however, uncertain about the direction of the change. In other words the investor expects high volatility in 6M rates over the next two years. When looking at the cap market, the market has not priced the investor�s view � implied volatility thus looks cheap to the investor. This expectation can be translated into a volatility strategy that will be profitable if the expectation is realised.

Assume that over the next 2Y the market prices the 6M EURIBOR flat at 4% in forward terms. Suppose further that the investor buys a 2Y cap on the 6M EURIBOR with a strike of say, 5%, and at the same time buys a similar floor struck at 3%. This combination of a long position in both a call (the cap) and a put (the floor) is called a (200 bp wide) strangle. For each cap- and floorlet the investor is looking at the following pay-off of the combined portfolio:

Figure 19 Bought 200 bp wide 2Y EUR cap/floor strangle

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As can be seen, the investor stands to profit if the 6M EURIBOR moves sufficiently far away from the current level of 4%. Note that the downside of the strangle position is limited to the premia paid for the two options. Should the investor choose the same strike level for both the cap and the floor, the combined position is called a straddle. Compared to the strangle above, a straddle is more expensive to enter since the underlying EURIBOR rate has to move by less in order to generate a pay-off at expiration. By entering into a straddle, the investor thus increases his downside risk.

Selling strangles or buying butterflies?

Is it attractive to reverse strangle or straddle strategies? This can be quite risky � so how can one short volatility in other ways?

As with all derivative strategies, one could naturally consider the reverse of the above strategies, that is, selling a strangle or a straddle. Obviously, an investor taking a short strangle position will benefit from low volatility. However, the investor has assumed two naked option positions, which due to their unlimited loss potential can be quite risky. The premium received by the investor in return for selling the strangle will provide the investor with the means to cover the costs of delta hedging his position plus a risk premium for taking on the risk that the delta hedge might not be sufficient.

An alternative that entails much lower risk but still benefits from low volatility is the butterfly spread. This strategy consists of a portfolio of four options: two long and two (identical) short and can be done with both puts and calls. Suppose an investor believes that realised volatility will be low in the 10Y swap over the next 6M. The forward swap rate is currently 4.65%.

By entering into a butterfly spread using 6M call options on the 10Y swap rate (ie, a 6M10Y payer swaption), the investor buys two calls struck at 4.15% and 5.15%, respectively and at the same time sells two calls struck ATM ie, at 4.65%. We would call such a position a 100bp wide butterfly around ATM. Six months from now, the investor will have the pay-off profile shown in Figure 20.


Figure 20 Payoff to a sold 100bp wide ATM butterfly in EUR 6M10Y swaptions

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As can be seen, the investor stands to profit from limited movements in the 10Y swap rate yield with a downside risk limited to the initial cash outlay.

Now, how should the peak in the diagram be chosen? Remember that the yield of the underlying bond will roll on the curve as the maturity shortens. The investor should take this into consideration such that the peak is set at the level that the current yield curve will give rise to six months from now. Given that the strategy is a play on small changes, the assumption that we will roll on today�s yield curve is not unreasonable.

A 1X2 Zero cost structure using swaptions

Making zero cost bets using options. Playing on low volatility while limiting risk against either up- or downward movements in rates

The no-arbitrage principle tells us that the butterfly structure described above will require an upfront payment at inception (as we cannot have a position that gives profit without any liabilities with positive probability). Suppose therefore that we allow for future liabilities, how can we then create a position that requires no initial payments? A simple answer would be a 1X2 zero cost structure. An example of this can be seen in Figure 21, where we have bought an ATM 6M5Y EUR payer swaption and sold two OTM payer swaptions. The moneyness of the OTM swaptions are chosen such that the revenue from selling these two options exactly matches the premium of the ATM option. From the figure we see that this strategy is a bet that swap rates will rise but not rise too much. The upside potential on the strategy is limited to approximately �1M while the downside potential is theoretically unlimited. The investor stands to make profit if rates go up by between 0 and 50 bp. Since the structure is zero cost, this bet can be implemented without any up-front payment.


Figure 21 Pay-off diagram for a 1X2 zero cost structure using 6M5Y EUR swaptions relative to �1 on the ATM leg.

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By changing the relative notional between the long and short position, the investor can alter the interval in which profit is made. By increasing notional on the short position, the investor can widen the profit interval by taking on the increased tail risk. That is, should swap rates exceed the interval, the increased leverage will increase the rate at which the investor loses money.

Calendar spreads

Using strategies that combine options with varying maturities � playing the calendar

Until now we have looked at strategies that combine options with different strikes (eg, a straddle) or options with different underlying (conditional steepeners). But what about combining options that have different expirations? These so-called calendar spreads offer interesting strategies for investors.

Before we consider calendar trades, it is important to note that for short-dated options with, say, 1-3M to expiration, the value of the options can be dependent on which events (eg, central bank policy meetings, data releases and holidays) are included. It is therefore natural that an option that expires on the day of an FOMC meeting is more valuable than an option that expires over Christmas.

How can we take advantage of an inverted volatility term structure with a calendar spread?

One possible use of calendar spreads is to play the volatility term structure. Suppose for example that short-dated options have become relatively expensive (ie, they trade at a higher implied volatility) because of, say, nervousness in the market. This was exactly the case in the late summer of 2007. Comparing Figure 22 below with Figure 16 we see that during this period, the volatility term structure inverted.


Figure 22 Term structure of implied ATM swaption volatilities on a 1Y EUR swap, 29/08-2007.

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Structuring a calendar spread trade

If an investor expects markets to calm down, he could sell the expensive short dated swaption straddles eg, 3M2Y and partially hedge these by buying the relatively cheaper 6M2Y straddles. Effectively, the investor is now exposed to the 3M2Y and 6M2Y forward swap rates and thus a curve exposure. By choosing different underlying swaps (say 3M2Y against 6M21M), the investor can remove this exposure.

How should such a trade be structured? One way would be to scale the notionals by delta such that the investor obtains a higher notional on the short-dated option (an option�s delta increases as time-to-expiration shortens). Doing this will however leave the investor gamma short but theta long � ie, the position is a carry position. The short straddle position leaves the investor exposed to large movements in the underlying. To hedge this, the investor buys the longer maturity straddle and hopes that any large movement in rates will be offset in the long maturity contract. As the notional is bigger on the short leg of the trade, the hedge is only partial � the investor still bears the risk of a dramatic movement in rates before the first expiration and a subsequent normalisation before the last.


Combined Strategies Taking relative positions

Taking a step deeper � using derivatives to make more advanced relative bets

In this section we will look at various combined strategies. In all of the above, we have looked at one yield curve or volatility surface at a time. Often it can also be interesting to look at how two yield curves or volatility surfaces move relatively to each other. More specifically, investors can take positions in the spread between reference rates and/or volatilities.

Given that asset pricing in effect is a question of relative prices, quotes for various instruments are often given as spreads to some reference security. The quotes for mortgage bonds can for example be given as spreads to the swap curve which in turn can be quoted as a spread to some government curve. This relative quoting also offers a way to analyse and manage risk exposures.

Swap spread trading

Why is the swap spread a relevant risk factor and how to buy and sell it

As discussed earlier we have separate yield curves for swap-related assets and for government bonds. As these curves are usually not identical there will be a spread between the different interest rates (ZCB-, forward- or par rates) on each curve. In particular we refer to the spread between the same, eg, the 10Y par rate on each curve, as the swap spread. This spread changes over time and is interesting for investors either in relation to their mortgage- or covered bond portfolios or as a speculative object in its own right. For detailed information on the swap spread and its associated risks please see �Introduction to swap spread risks�.

Suppose that an investor expects the spread between the swap and government curve to narrow. Depending on the investor�s preferences there are several ways to trade the swap spread and benefit from the expected tightening. It can be done either using spot asset swap packages, via futures combined with forward starting swaps or finally as a single OTC derivative contract. For a thorough introduction to trading please see �Trading swap spread risks�.


http://danskeresearch.danskenet.net/link/IntroToSwapSpreadRisk/$file/IntroToSwapSpreadRisk.pdf

http://danskeresearch.danskenet.net/link/TradingSwapSpreads/$file/TradingSwapSpreads.pdf

Relative curve strategies

Playing the DKK-EUR yield curve spread using swaps

Suppose an investor has a view on interest rate differentials between currencies. This could for example be the view that the spread between 5Y DKK and EUR interest rates should increase relatively to what is priced in the forward curves (see Figure 23). By entering into a DKK 5Y payer swap and EUR 5Y receiver swap the investor gets a strong positive delta vector at the 5Y point in DKK and a strong negative delta vector in the 5Y point in EUR. Should the rate differential indeed increase, the investor�s position will benefit. As seen earlier, such a position can be made duration neutral by scaling the respective notional amounts (here the meaning of duration would however be slightly different as we are measuring shifts in two different yield curves). If a similar strategy were undertaken between two floating currencies, the investor would of course also be subject to currency risk.

Figure 23 Spread between 5Y yields on the EUR and DKK swap curves, historical and forward

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Picking up the DKK-EUR risk premium

How to use swaptions to pick up the risk premia between DKK and EUR implied volatility

Despite Denmark�s well-established fixed exchange rate regime, market participants continually charge a risk premium on DKK interest rates and volatilities relative to the Eurozone. These risk premia primarily reflect the � arguably highly remote � possibility that Denmark one day breaks away from the Euro peg and the lower liquidity of the DKK market.

For investors looking for enhanced yield pick-up these premia could be interesting. In particular, an investor who feels confident that Denmark will eventually join the EMU can express this view by taking positions in the swaption market.


Figure 24 The DKK-EUR risk premia on 10Y10Y ATM swaptions

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Structuring a volatility spread trade

The basic strategy is very simple: taking a short position in the more expensive DKK volatility and a long position in the cheaper EUR volatility will give the investor a positive cash flow at inception. The investor has thus obviously been compensated for taking on some risk. The question is now, what are these risks and is the investor comfortable with them?

Suppose that the strategy is implemented using two cash settling payer swaptions struck ATM in their respective markets. At the expiration of the swaptions the investor is left with a potential liability in DKK and a potential asset in EUR. The investor is therefore interested in seeing the EUR swap rates increase relatively to DKK rates � ie, a narrowing of the rate spread. Suppose the spread does indeed narrow. Should the DKK swaption expire in-the-money, the EUR position will more than offset the liability and even pay out cash (in addition to the premium received at inception). The downside risk of the strategy lies in a rate spread widening (as eg, a change in monetary policy could cause), this movement would increase the investor�s short DKK position without an offsetting increase in the long EUR position.

Daily mark-to-market risks against pay-off profile at expiration

Given that we now understand the risks at expiry, let us consider the daily mark-to-market risks that the investor will be exposed to. First of all, movements in the spread between DKK and EUR swap rates will have the same effect as at expiry � ie, the strategy will benefit from narrowing spreads and lose on widening spreads. Furthermore, the same holds true for the volatility spreads. That is, a decrease in DKK volatility relative to EUR will give rise to a mark-to-market profit and vice versa.

Note that should Denmark one day choose to join the EMU the spreads will collapse instantaneously. Taking on the strategy outlined above could thus be said to be an implicit bet that this will happen before the swaption expires.


Note also that the investor does not have to keep the positions until expiry. Since the strategy can always be closed down by entering into offsetting positions, the investor does not need to hold the positions until expiry. The option to close down means that the strategy can also be used to take tactical positions in the movements of long-term rate and volatility spreads between currencies. It also means that an effective stop-loss trigger level can be put in to limit the loss potential on the short DKK position.

Leveraging the trade � using different strike levels

Finally, note that the strategy can be set up with other strike levels than ATM. Striking both swaptions at eg., the EUR ATM level means that a higher premium will be paid up front (since the sold DKK swaption will be in-the-money at inception), this will however increase the risk to investor. Striking both swaptions at the EUR ATM level is a bet that the rate spread will be completely eliminated before the expiration of the swaption. This could be an interesting way of implementing a view on a future Danish entry into the EMU.

Shorting options: Range accruals

The definition of range accruals and some intuition as to why options are implicitly traded in a range accrual

The above strategy involved selling a swaption in one currency and hedging this by buying a swaption in another currency. Writing a call option (as a payer swaption is) has potentially unlimited downside if left unhedged. Even though effective stop-loss and hedging schemes can be set up, some clients will find that (short) positions in options are beyond their (regulatory) mandate. Can they, however, implement strategies like the one above in other ways that do not involve unlimited loss potentials?

Indeed they can. A range accrual is a way of calculating floating-rate coupons in which interest is only accrued on days where some condition depending on one or more underlying assets is met. These coupons can in turn be used in notes, swaps or other formats. Let us consider a simple example of a range accrual note.

Party A buys a 2Y accrual note from party B at par. B pays quarterly coupons to A. The (annualised) coupon paid each quarter is calculated as:

N

mcoupon *%5=

where m is the number of days that some condition (eg, that 10Y DKK swap rates are less than 5bp above 10 EUR swap rates) is met and N is the number of business days in the coupon period. After 2Y the bond is redeemed at par. A structure like this can thus offer returns between 5% and 0% � the maximum loss incurred by the investor is thus the carry.


These notes can be structured to offer high coupons by taking on the risk that the accrual condition specifies. Importantly, by designing the accrual condition the investor can get exposure that would otherwise require (short) positions in options. When buying into an accrual note, the investor implicitly takes long and short positions in digital options without ever risking more than the carry of the invested amount.

Returning to the DKK-EUR strategy described in the previous section, the same views could be implemented by buying the accrual note in the above example. If the spread between DKK and EUR does indeed narrow, the investor will find the accrual condition met going forward and will thus receive an above-market return.

Pricing range accruals: Digital options

Why are digital options related to range accruals and how are these different from plain-vanilla options?

As mentioned above a range accrual embeds positions in digital - or binary � options, but what are these? A digital option is an option that pays off one or nothing depending on whether the underlying is above (call) or below (put) the strike. It thus does not matter how �far� in- or out-of-the-money the option is on expiry � the option payoff is a binary (ie, 0 or 1 event).

The fact that the pay-off profile to digital options is discontinuous means that their greeks can be somewhat misbehaved up to expiration if the underlying is close to the strike.

In order to price a range accrual, one typically looks at the replicating strategy. In the above example this would be buying a 2Y 5% annuity and selling a series of digital options that pays 5% if 10Y DKK swap rates are more than 5bp above 10 EUR swap rates. To hedge the range accrual cash flow exactly, one would need to sell options expiring on every accrual day with a notional relative to the day count fraction.

Pure curve plays: CMS spread options

Using more advanced instruments to make cleaner curve strategies

We have shown how to position for steeper interest rate curves using both swaps and swaptions. Suppose however that we are interested in taking on a position that benefits from a steepening curve without the risk of losing money if the curve flattens. How can we do this?

If we use swaps to take on a curve play, we are obviously running the risk of the curve moving against us. If we use swaptions we can of course ensure that we do not experience losses beyond the premium paid should our bet go wrong. With swaptions we, however, also introduce directionality ie, we have to decide whether we think a steepening will come from lower short-end rates or higher long-end rates. What do we then do, if we want to play the slope of the curve on an unidirectional basis?


The cleanest way to this is to use CMS spread options. These are fairly actively traded OTC derivatives that are structured as plain-vanilla caps- and floors but whose pay-off depends on the spread between two CMS rates rather than a single LIBOR rate. Most of the options are written on either the 2-10Y or 2-30Y spread.

The risk factors involved in CMS spread options: Curve, volatility and correlation

What are the risks embedded in such an instrument? Obviously, the contract depends on the forward spreads between the two reference rates ie, the delta of the option. The option�s value, however, also depends on the volatility of the rate spread. This spread can in turn be decomposed into the volatility of each underlying index combined with the correlation between the two. CMS spread options thus introduce correlation risk between CMS rates. An investor who has bought a cap on, say, the 2-10Y spread will benefit from a lower correlation as this increases the chance that the spread will expire in the money. The same goes for floors � a long position in a CMS spread option thus gives a short position in correlation.


Derivative formats Which wrapping do you prefer?

Using derivatives as building blocks we can create any risk profile in any wrapping

Basically, we have so far looked at various combinations of derivative instruments to create different risk profiles. Suppose now that an investor likes a certain risk exposure but is not interested in or able to take positions in the particular instruments that generate the risk profile. What can such a client do?

An important part of a modern investment bank is to structure and create risk for its clients. We are therefore able to help clients obtain risk in the particular format that is most beneficial to them. Basically, we offer fixed income derivative risks in three formats: structured swaps, notes and deposits.

Structured swaps

Using structured swaps for unfunded investments or on top of existing plain-vanilla structures

Many clients have funding based on short money market rates. This can for example be a client who has swapped a fixed rate loan into floating. These clients can use structured swaps to take an active market view and use this to reduce their funding cost should their view be correct. Alternatively, a client can be interested in making an unfunded investment in which the client pays LIBOR against receiving a coupon that will be above-market if his view proves to be correct. Let us look at an example.

Suppose a client has some rolling funding tied to 3M EURIBOR but that the same client wants to express a view on the steepness of the swap curve. Such a client could enter into a so-called booster swap. This is a structured swap arrangement in which the client receives 3M EURIBOR while paying a coupon that becomes smaller and smaller the steeper the swap curve becomes. Effectively, this is equal to buying CMS spread options combined with a regular fixed-for-floating interest rate swap. For the client, this offers the possibility of taking the market view of a steepening curve and tying this to his funding.

3M EURIBOR

Client Danske Bank

5%-G*Max(CMS10Y-2Y-xbp,0)


Note that such a structure means that the client can never pay more than 5%. The booster swap thus offers a clearly defined downside to the client (and even offers the possibility of paying negative coupons while receiving EURIBOR).

Structured notes

A flexible way of giving retail investors access to derivative risk exposures with or without capital guarantees

Structured notes are bond-like instruments that offer a floating rate coupon depending on one or more underlying assets. Some sort of capital protection (ie, a promise that the full face value of the note will be repaid at redemption) is often attached.

The basic idea of a structured note is to offer an alternative investment opportunity in a bond-like format that provides an above-market coupon should some specific market view materialise. Also, high �teaser� (ie, initial) coupons are often combined with callability features to give investors more leverage in the underlying asset.

The embedded market view could be tied to everything from equity indices over commodity prices to interest rates. Combined with a capital protection guarantee this means that structured notes have become very popular with retail clients as these products provide access to a wide range of assets that otherwise would not be available to the retail segment.

Note that structured notes provide a way of trading exposure linked to derivatives without having set up specific documentation. Also, as these notes are often exchange listed they can come within reach of clients who are otherwise restricted from taking positions in risk factors that are only traded via OTC derivatives.

Structured deposits

How to use derivatives to create structured interest rate payments on deposits

As the flip side to the structured swaps, clients can also have their placement of liquidity tied to a specific market view through structured deposits. Instead of having interest accrue against some money-market rate, the funds on a structured deposit accrue above market interest according to the performance of some underlying asset. How can we build an active view on money market rates into a structured deposit?

Suppose that the short end of the curve is currently inverted. Contrary to the market, an investor believes that the inversion will continue for the next 12 months and that short-term rates will stay above 4.5%. The investor could implement his view by placing funds in a multi-look range accrual deposit that accrues a high coupon when

• 3M EURIBOR rates are above 12M EURIBOR

• 3M EURIBOR is above 4.5%.


Conclusion In the above sections we have discussed a broad range of investment

strategies based on interest rate derivatives. Each of these strategies provides a specific risk profile that can help the investor to generate excess returns if the investor�s expectations about future interest rates and / or volatilities are realised. A thorough understanding of these risk profiles is the key to successful investing and risk management using derivatives. It is our hope that this primer has been of assistance in providing this.


Index

Accounting..................................................................................................4 Moneyness..............................................................................................40 Annuity factor........................................................................................ 12 No-arbitrage .............................................................................6, 29, 47 Asset Swap............................................................................................. 37 OTC Backbone ................................................................................................. 42 Over the counter ..................................................4, 7, 8, 14, 27 Basis Point Value................................................................................ 15 Outright strategies.............................................................................22 Basis point volatility ......................................................................... 44 Par rates...................................................................................................10 Black�76 ................................................................................................... 44 Par-par asset swap............................................................................37 Black-Scholes-Merton ..................................................................... 39 Put-call parity........................................................................................21 Bonds ......................................................................................................... 23 PV01...........................................................................................................15 Butterfly spread................................................................................... 46 Range accrual .......................................................................................53 Calendar spread ................................................................................. 48 Replication strategy.............................................................................. 6 Cap............................................................................................................... 31 Retractable.............................................................................................33 Caplet ......................................................................................................... 32 Risk factors................................................................................................ 5 CMS Risk premium....................................................................... 40, 46, 51

Constant Maturity Swap................................34, 35, 54, 56 Skew............................................................................................................40 Conditional bet...................................................................................... 29 Smile ...........................................................................................................40 Convexity .................................................................................................. 16 Smirk...........................................................................................................40 Correlation ....................................................................... 5, 29, 32, 55 Speculative...................................................4, 5, 6, 28, 39, 42, 50 CSA Spread options .....................................................................................55

Credit Support Annex....................................................................8 Straddle ........................................................................................... 45, 49 Delta................................................................16, 17, 24, 26, 35, 36 Strangle ....................................................................................................45 Delta vector ........................................................................................... 16 Structured deposits .........................................................................57 Digital options ...................................................................................... 54 Structured notes ................................................................................57 Documentation.........................................................................................8 Structured swaps ..............................................................................56 Extendable .............................................................................................. 33 Swap spread...........................................................................................50 floor.............................................................................................................. 31 Swaps ............................................................................................... 11, 23 Forward curve....................................................10, 28, 29, 32, 34 Swaption ................................................................22, 27, 28, 33, 44 Forward LIBOR rate........................................................................... 10 Theta ...........................................................................................................18 Gamma.....................................................................................17, 20, 30 Time decay ..................................................................................... 18, 28 Gamma-spike................................................................................ 19, 30 Vega.................................................................18, 19, 20, 30, 31, 35 Greeks........................................................................................................ 15 Volatility ........................................................18, 39, 44, 45, 48, 55 Hedge stability ..................................................................................... 25 Volatility wedge.....................................................................................32 Implied volatility .................................................................................. 39 Yield-to-maturity asset swap.......................................................38

ZCB yield curve....................................................................................... 9 ISDA Zero-coupon bond.................................................................................. 9 International Swaps and Derivatives Association.......8

Leverage effect.........................................................................................4 Model................................................................................... 6, 15, 39, 44


This publication has been prepared by Danske Markets for information purposes only. It is not an offer or solicitation of any offer to purchase or sell any financial instrument. Whilst reasonable care has been taken to ensure that its contents are not untrue or misleading, no representation is made as to its accuracy or completeness and no liability is accepted for any loss arising from reliance on it. Danske Bank, its affiliates or staff, may perform services for, solicit business from, hold long or short positions in, or otherwise be interested in the investments (including derivatives), of any issuer mentioned herein. Danske Markets´ research analysts are not permitted to invest in securities under coverage in their research sector. This publication is not intended for private customers in the UK or any person in the US. Danske Markets is a division of Danske Bank A/S, which is regulated by FSA for the conduct of designated investment business in the UK and is a mem-ber of the London Stock Exchange.


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Interest Rate Derivatives - Danske Bank · 2018. 11. 8. · Danske Research - Fixed Income &...

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