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    Modelling Single-name and Multi-name Credit Derivatives
    Modelling Single-name and Multi-name Credit Derivatives
    Modelling Single-name and Multi-name Credit Derivatives
    Ebook1,036 pages10 hours

    Modelling Single-name and Multi-name Credit Derivatives

    By Dominic O'Kane

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    About this ebook

    Modelling Single-name and Multi-name Credit Derivatives presents an up-to-date, comprehensive, accessible and practical guide to the pricing and risk-management of credit derivatives. It is both a detailed introduction to credit derivative modelling and a reference for those who are already practitioners.

    This book is up-to-date as it covers many of the important developments which have occurred in the credit derivatives market in the past 4-5 years. These include the arrival of the CDS portfolio indices and all of the products based on these indices. In terms of models, this book covers the challenge of modelling single-tranche CDOs in the presence of the correlation skew, as well as the pricing and risk of more recent products such as constant maturity CDS, portfolio swaptions, CDO squareds, credit CPPI and credit CPDOs.

    LanguageEnglish
    PublisherWiley
    Release dateMar 8, 2011
    ISBN9781119995449
    Modelling Single-name and Multi-name Credit Derivatives

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      Modelling Single-name and Multi-name Credit Derivatives - Dominic O'Kane

      1

      The Credit Derivatives Market

      1.1 INTRODUCTION

      Without a doubt, credit derivatives have revolutionised the trading and management of credit risk. They have made it easier for banks, who have historically been the warehouses of credit risk, to hedge and diversify their credit risk. Credit derivatives have also enabled the creation of products which can be customised to the risk-return profile of specific investors. As a result, credit derivatives have provided something new to both hedgers and investors and this has been a major factor in the growth of the credit derivatives market.

      From its beginning in the mid-1990s, the size of the credit derivatives market has grown at an astonishing rate and it now exceeds the size of the credit bond market. According to a recent ISDA survey,¹ the notional amount outstanding of credit derivatives as of mid-year 2007 was estimated to be $45.46 trillion. This significantly exceeds the size of the US corporate bond market which is currently $5.7 trillion and the US Treasury market which is currently $4.3 trillion.² It also exceeds the size of the equity derivatives market which ISDA also estimated in mid-2007 to have a total notional amount outstanding of $10.01 trillion.

      In addition to its size, what is also astonishing about the credit derivatives market is the breadth and liquidity it has attained. This has been due largely to the efforts of the dealer community which has sought to structure products in a way that maximises tradabililty and standardisation and hence liquidity. The CDS indices, introduced in 2002 and discussed extensively in this book, are a prime example of this. They cover over 600 of the most important corporate and sovereign credits. They typically trade with a bid-offer spread of less than 1 basis point and frequently as low as a quarter of a basis point.³

      To understand the success of the credit derivatives market, we need to understand what it can do. In its early days, the credit derivatives market was dominated by banks who found credit derivatives to be a very useful way to hedge the credit risk of a bond or loan that was held on their balance sheet. Credit derivatives could also be used by banks to manage their regulatory capital more efficiently. More recently, the credit derivatives market has become much more of an investor driven market, with a focus on developing products which present an attractive risk-return profile. However, to really understand the appeal of the credit derivatives market, it is worth listing the many uses which credit derivatives present:

      • Credit derivatives make it easier to go short credit risk either as a way to hedge an existing credit exposure or as a way to express a negative view on the credit market.

      • Most credit derivatives are unfunded. This means that unlike a bond, a credit derivative contract requires no initial payment. As a consequence, the investor in a credit derivative does not have to fund any initial payment. This means that credit derivatives may present a cheaper alternative to buying cash bonds for investors who fund above Libor. It also makes it easier to leverage a credit exposure.

      • Credit derivatives increase liquidity by taking illiquid assets and repackaging them into a form which better matches the risk-reward profiles of investors.

      • Credit derivatives enable better diversification of credit risk as the breadth and liquidity of the credit derivatives market is greater than that of the corporate bond market.

      • Credit derivatives add transparency to the pricing of credit risk by broadening the range of traded credits and their liquidity. We estimate that there are over 600 corporate and sovereign names which have good liquidity across the credit derivatives market.⁴ The scope of the credits is global as it includes European, North American and Asian corporate credits plus Emerging Market sovereigns.

      • Credit derivatives shift the credit risk which has historically resided on bank loan books into the capital markets and in doing so it has reduced the concentrations of credit risk in the banking sector. However, this does raise the concern of whether this credit risk is better managed in less regulated entities which sit outside the banking sector.

      • Credit derivatives allow for the creation of new asset classes which are exposed to new risks such as credit volatility and credit correlation. These can be used to diversify investment portfolios.

      The relatively short history of the credit derivatives market has not been uneventful. Even before the current credit crisis of 2007–2008, the credit derivative market has weathered the 1997 Asian Crisis, the 1998 Russian default, the events of 11 September 2001, the defaults of Conseco, Railtrack, Enron, WorldCom and others, and the downgrades of Ford and General Motors. What has been striking about all of these events is the ability of the credit derivatives market to work through these events and to emerge stronger. This has been largely due to the willingness of the market participants to resolve any problems which these events may have exposed in either the mechanics of the products or their legal documentation. Each of these events has also strengthened the market by demonstrating that it is often the only practical way to go short and therefore hedge these credit risks.

      In this chapter, we discuss the growth in the credit derivatives market size. We present an overview of the different credit derivatives and discuss a market survey which shows how the importance of these products has evolved over time. We then discuss the structure of the credit derivatives market in terms of its participants.

      1.2 MARKET GROWTH

      The growth of the credit derivatives market has been phenomenal. Although there are different ways to measure this growth, each with its own particular approach, when plotted as a function of time, they all show the same exponential growth shown in Figure 1.1. Let us consider the three sources of market size data:

      1. The British Bankers’ Association (BBA) surveys the credit derivatives market via a questionnaire every two or so years. Their questionnaire is sent to about 30 of the largest investment banks who act as dealers in the credit derivatives market. Their latest report was published in 2006 and estimated the total market notional at the end of 2006 to be $20.207 trillion.

      2. The International Swaps and Derivatives Association (ISDA) conducts a twice-yearly survey of the market. In the most recent, they surveyed 88 of their member firms including the main credit derivatives dealers about the size of their credit derivatives positions. The collected numbers were adjusted to correct for double-counting.⁵ The mid-2007 survey estimated the size of the credit derivatives market to be $45.25 trillion, an increase of 32% in the first six months of 2007.

      3. The US Office for the Comptroller of the Currency conducts a quarterly survey of the credit derivatives market size. The survey covers just the US commercial bank sector. The June 2007 survey found that the total notional amount of credit derivatives held by US commercial banks was $10.2 trillion, an increase of 86% on the first quarter of 2006. This number is lower than the others partly because it excludes trades done by many non-US commercial banks and investment banks.

      Although these numbers all differ because of the differing methodologies and timings, what is beyond doubt is the rapid growth that has been experienced by the credit derivatives market.

      Figure 1.1 Evolution of the credit derivatives market size using estimates calculated in the BBA Credit Derivatives Report 2006. Source: British Banker’s Association

      004

      Although the size of the credit derivatives market is significant, it is important to realise that credit derivatives do not increase the overall amount of credit risk in the financial system. This is because every credit derivative contract has a buyer and a seller of the credit risk and so the net increase of credit risk is zero. However, credit derivatives can in certain cases be used to increase the amount of credit risk in the capital markets. For example, suppose a bank has made a $10 million loan to a large corporate and this loan is sitting on their loan book portfolio. As we will see later, to hedge this risk, the bank can use a credit default swap (CDS) contract. The credit risk of the corporate is therefore transferred from the bank to the CDS counterparty who may then transfer this risk on to another counterparty using another CDS contract. At the end of this chain of transactions will be someone, typically an investor, who is happy to hold on to the risk of the corporate and views that the premium from the CDS is more than sufficient to compensate them for assuming this risk. The loan that was sitting on the bank’s loan book is still there. However, the risk has been transferred via the CDS contracts to this investor in the capital markets. Because the credit risk has been transferred without any actual sale of the loan, the credit risk produced by a CDS is ‘synthetic’. If a default of the corporate does occur, the loss compensation is paid from the investor to the counterparty and on down the chain of contracts to the bank which was the initial buyer of protection. The bank has successfully hedged its credit exposure to this loan.

      The only way in which credit risk has increased is through the counterparty credit risk associated with each contract. This is the risk that the protection seller does not make good a payment of the default loss compensation to the protection buyer. In practice, this risk is usually mitigated through the use of collateral posting as explained in Section 8.8.

      1.3 PRODUCTS

      The simplest and most important credit derivative is the credit default swap (CDS). This is a bilateral contract which transfers the credit risk of a specific company or sovereign from one party to another for a specified period of time. It is designed to protect an investor against the loss from par on a bond or loan following the default of the issuing company or sovereign. In return for this protection the buyer of the CDS pays a premium.

      We note that someone who is assuming the credit risk in a credit derivatives contract like a CDS is called a protection seller. The person taking the other side of this trade is insuring themselves against this credit risk and is called a protection buyer.

      An important extension of the CDS is the CDS index. This is a product which allows the investor to enter into a portfolio consisting of 100 or more different CDS in one transaction. For example, one of the most liquid indices is the CDX NA IG index which consists of 125 investment grade corporate credits which are domiciled in North America. We call this a multi-name product because it exposes the issuer to the default risk of more than one credit or ‘name’. The considerable liquidity and diversified nature of the CDS index have meant that it has also become a building block for a range of other credit derivatives products.

      There are also a number of option-based credit derivatives. These include single-name default swaptions in which the option buyer has the option to enter into a CDS contract on a future date. More recently we have seen growth in the market for portfolio swaptions. As the name suggests, these grant the option holder the option to enter into a CDS index.

      Then there are the multi-name contracts such as default baskets and synthetic CDOs which are built on top of a portfolio of CDS. These contracts work by ‘tranching’ up the credit risk of the underlying portfolio. Tranching is a mechanism by which different securities or ‘tranches’ are structured so that any default losses in the portfolio are incurred in a specific order. The first default losses are incurred by the riskiest equity tranche. If the size of these losses exceeds the face value of the equity tranche then the remaining losses are incurred by the mezzanine tranche. If there are still remaining losses after this, then these are incurred by the senior tranches. The risk of this credit derivatives contract is sensitive to the tendency of the credits in the portfolio to default together. This is known as default correlation and, for this reason, these derivatives are known as correlation products.

      Finally, we have the credit CPPI structure and the more recent CPDO structure. These structures exploit a rule-based dynamic trading strategy typically involving a CDS index. This trading strategy is designed to produce an attractive risk-return profile for the investor. In the case of CPPI, it is designed to provided a leveraged credit exposure while protecting the investor’s principal. In the case of CPDO, the strategy is designed to produce a high coupon with low default risk.

      Table 1.1 Market share of different credit derivatives products measured by market outstanding notional. We compare the results of the BBA survey for 2002, 2004 and 2006. Source: BBA Credit Derivatives Report 2006

      005

      Table 1.1 shows a breakdown of the various credit derivatives by their market outstanding notional. The data is sourced from the BBA survey of 2006 mentioned earlier. Note that this survey does not consider the CPPI and CPDO products since these have only become important in the time since this survey was carried out. This table already enables us to make the following observations about the current state and also the trends of the credit derivatives market:

      • Many of the products which appeared in the 2004 and 2006 surveys did not exist in 2002. The most notable examples of this are the full index trades, which refer to trades on the CDX and iTraxx indices which were launched after 2002. We also see the establishment of a number of synthetic CDO categories and a tranched index trade category.

      • We see a relative decline in the importance of more traditional credit derivatives products such as credit-linked notes and spread options. However, this decline in market percentage share is actually an increase in absolute size given the fourfold growth of the credit derivatives market over the 2004–2006 period.

      • The market share of CDS fell from 51% to 32.9% over the 2004-2006 period. Over the same period, the portfolio indices rose from 9.1% to 30.1% of a much larger market. This suggests a trend away from single-name credit towards portfolio products. In absolute terms, the CDS market size actually grew significantly over this period.

      There is a clear trend towards portfolio index products, i.e. multi-name products.

      1.4 MARKET PARTICIPANTS

      There are several different types of participants in the credit derivatives market. Each has its own specific rationale for using credit derivatives. Table 1.2 presents a breakdown of the market share of the credit derivatives market by participant. This data is taken from the 2006 BBA Credit Derivatives Report. From this, we make the following observations:

      • Banks are the largest participant in the credit derivatives market, both as buyers and sellers of protection. Table 1.2 splits the category of banks into trading activities and loan portfolio which we now consider separately.

      • Many banks, in particular the securities houses, have significant trading activities as they act as dealers in the credit derivatives market. Dealers provide liquidity to the credit derivatives market by being willing to take risk onto their trading books which they then attempt to hedge. As a result, they buy roughly as much protection as they sell. They also act as issuers of structured products such as synthetic CDOs which they also hedge dynamically.

      • Commercial banks possess loan portfolios. They use credit derivatives to buy protection in order to synthetically remove credit risk concentrations from their loan portfolio. They sell protection on other credits in order to earn income which can be used to fund these hedges, and to diversify their credit risk. One of the main drivers of bank behaviour is their regulatory framework. Until recently, this was based on the 1988 Basel Accord in which the capital a bank had to reserve against a loan or credit exposure was linked to whether the issuer of the loan was an OECD⁶ member government, bank or other. However, the regulatory regime has recently changed to the Basel II capital accord in which the regulatory capital is linked to the credit rating of the asset.

      Table 1.2 Market share of different market participants. Source: BBA Credit Derivatives Report 2006

      006

      • Insurance companies mainly use credit derivatives as a form of investment which sits on the asset side of their business. They are principally sellers of credit protection and tend to prefer highly rated credits such as the senior tranches of CDOs.

      • Hedge funds have grown their credit derivatives activity and have become significant players in the credit derivatives market. They are attracted by the unfunded⁷ nature of most credit derivatives products which makes leverage possible. The fact that the credit derivatives market makes it easy to go short credit is another big attraction. Furthermore, credit derivatives also facilitate a number of additional trading strategies including cash versus CDS basis trading, correlation trading and credit volatility trading which hedge funds are free to exploit.

      • Mutual and pension funds are not particularly large participants in the credit derivatives market. As investors, they would be primarily sellers of protection. They buy protection to hedge existing exposures. Often, they have restrictions on what sort of assets they can hold which preclude credit derivatives. However, the exact permissions depend on both their investment mandate and the investment regulations governing the jurisdiction in which they operate. Typically, one of their main restrictions is that the credits owned should be investment grade quality.

      • Although credit derivatives could be used to try to hedge the credit risk of receivables, corporates have not become significant players in the credit derivatives market. There are a number of reasons why. First, standard credit default swaps do not trigger on the nonpayment of receivables since receivables are not classified as borrowed money–the term used to encompass the range of obligations which are covered. Typically a contract will only trigger if there is a default of bonds and loans. Second, the payout from a standard credit default swap may not be consistent with the actual loss since it is linked to the delivery of a senior unsecured bond or loan. Finally, the range of traded credits may not overlap with the companies to which the corporate has a credit exposure.

      Understanding the motivations of these different participants assists us in understanding why different products are favoured by different participants.

      1.5 SUMMARY

      In this chapter we have explained the features of credit derivatives which have led to their success and then discussed the market in terms of its size, the types of credit derivatives which are traded and who uses them.

      2

      Building the Libor Discount Curve

      2.1 INTRODUCTION

      This chapter does not focus on credit modelling or credit derivatives. It simply discusses one of the prerequisites for any credit derivative model–a Libor discount curve. Building the Libor discount curve is an important and often overlooked part of credit derivatives pricing and for this reason, we include a description of how it is typically done. In the process, we also introduce some of the instruments which will later be used to hedge the interest rate sensitivities of credit derivatives.

      We need a discount curve because the valuation of all credit derivatives securities with future cash flows requires us to take into account the time value of money. This is captured by the term structure of interest rates. The interest rate used to discount these future cash flows can then have a significant effect on the derivative price. The next question is–which interest rate should I use? Since the sellers of credit derivatives, typically commercial and investment banks, need to hedge their risks, the interest rate used to discount cash flows is the one at which they have to fund the purchase of the hedging instruments. This interest rate is known as Libor. It is a measure of the rate at which commercial banks can borrow in the inter-bank market. As a result, derivative pricing requires a discount curve which is linked to the level of the current and expected future level of the Libor index.

      2.2 THE LIBOR INDEX

      Within the derivatives market Libor is the benchmark interest rate reference index for a number of major currencies including the US dollar and the British pound. Libor stands for the London Inter Bank Offered Rate and is the interest rate at which large commercial banks with a credit rating of AA and above offer to lend in the inter-bank market. It is set daily in London by the British Bankers’ Association (BBA) using deposit rates offered by a panel of typically 16 banks. This is done for a range of terms out to one year.⁸ The process for determining Libor is as follows:

      1. For a given currency and term, the 16 rates are sorted into increasing order.

      2. The top and bottom four quotes are removed.

      3. The arithmetic average of the remaining eight quotes is calculated.

      4. The result becomes the Libor rate for that currency and term and is published on Reuters at 11.00am London time each day.

      For the euro currency the BBA produced rate is called euro Libor. However, the more widely used rate for the euro is known as the Euribor. This is calculated daily by the European Banking Federation, also as a filtered average of inter-bank deposit rates offered by a panel of 50 or more designated contributor banks.

      As most large derivatives dealers fund at or close to the Libor rate, it has become the reference interest rate for the entire derivatives market. However, Libor is also very important because of the broad range of interest rate products which are linked to it. These include:

      • Money market deposits in which short-term borrowing occurs in the inter-bank market.

      • Forward rate agreements which can be used to lock in a future level of Libor.

      • Exchange-traded interest rate futures contracts in which the settlement price is linked to the three-month Libor rate.

      • Interest rate swaps which exchange fixed rate payments for a stream of Libor payments.

      In this chapter we will describe in detail these Libor instruments. In particular, we will focus on the interest rate swap because it is a key component of many credit structures such as the asset swap. The interest rate swap is also the main hedging instrument for the interest rate exposure embedded in credit derivatives.

      Since the price of these instruments embeds market expectations about the future levels of Libor, we can use them to generate an implied Libor discount curve. As a result, we will show how these instruments can be used to construct the risk free discount curve which will be used in the pricing of credit derivatives in the rest of this book.

      2.3 MONEY MARKET DEPOSITS

      Money market deposits are contracts in which banks borrow in the inter-bank market for a fixed term of at most one year. Because this is the inter-bank market, in a standard money-market deposit contract, the amount is borrowed at a rate of return equal to the corresponding Libor index.

      The deposit rate is agreed at contract initiation time t. We denote the maturity date of the deposit with T. However, it is on the settlement date ts of the deposit that the deposit is made and the interest begins to accrue. The period between trade date and settlement date is two days for EUR, JPY and USD currency deposits, but can be different for other currencies. The standard terms of money-market deposits go out to one year, with the quoted rates including one week (1W), two week (2W), one month (1M), three months (3M), six months (6M), nine months (9M), and one year (12M) terms.

      2.3.1 Mechanics

      The mechanics of a money-market deposit in which $1 is borrowed are shown in Figure 2.1.

      The trade is done at time 0 and settles at time ts with the borrower receiving $1. At the end of the deposit term, the lender is repaid the borrowed cash amount plus interest. The amount paid at time T is given by

      V(T) = 1 + Δ(ts, T) L(ts, T)

      where Δ represents the year fraction, also known as the accrual factor, between the settlement date and the deposit maturity date defined according to some day count convention. It should be emphasised that every time we encounter a day count fraction, we must take care to apply the correct day count basis convention. In US dollars, Japanese yen and euros, the standard money-market deposit convention is Actual 360. For UK sterling, Actual 365 is used.

      Figure 2.1 Mechanics of a money-market deposit on a face value of $1

      007

      Actual 360 means that the value of Δ is given by calculating the number of calendar days between the settlement date and maturity date, and dividing the result by 360. As a result we have

      008

      where DayDiff(t1, t2) is a function that simply calculates the number of days between two times t1 and t2.

      We can calculate the current value of the deposit by discounting the expected cash flows using Libor discount factors Z(t, T). We have

      V(0) = − Z(0, ts) + (1 + Δ(ts, T)((ts, T)) Z(0, T).

      For a trade done at time 0, the expected present value of the deposit V(0) must be zero. The value of the discount factor from today time 0 until the maturity of the deposit is therefore given by

      009

      where Z(0, ts) is the discount factor from today until the settlement date of the deposit.

      However, there is a difficulty in determining the value of Z(0, ts) . Deposits in most currencies and for terms from 1W to 12M settle two days after trade date. This is known as a ‘T + 2’ settlement convention where T represents the trade date and is not to be confused with the deposit maturity date. It means that the associated rates do not provide any information about the discount curve between times 0 and ts. In order to determine Z(0, ts), we need to have a deposit or deposits which settle immediately and which can bridge the gap between 0 and ts. This can usually be done using the overnight (O/N) rate which settles today for maturity tomorrow, and ‘tomorrow/next’ (T/N) which is the rate that settles tomorrow for payment in two days. By compounding these two rates, we can determine Z(0, ts) .

      2.4 FORWARD RATE AGREEMENTS

      Forward rate agreements (FRAs) are over-the-counter bilateral agreements to borrow money for a forward starting period at a rate defined today. This rate is known as the FRA rate which we will denote with F0. FRAs are generally traded over the counter between banks in a broker market and the corresponding FRA rates are shown on electronic broker screens.

      2.4.1 Mechanics

      To describe the mechanics of an FRA contract we begin by defining the three dates shown in Figure 2.2:

      1. The trade date of the FRA contract, time 0.

      2. The forward start date T1 on which the interest period begins.

      3. The end date of the FRA, T2 on which the effective payment is equal to

      Δ(T1, T2) (L(T1, T2)–F0)

      where L(T1, T2) is the Libor fixing observed at time T1 for the period T1 to T2, and Δ(T1, T2) is the accrual factor from time T1 to T2 quoted according to the FRA basis. F0 denotes the strike of the FRA and can be thought of as the forward rate of borrowing/lending.

      Figure 2.2 Mechanics of a forward rate agreement on a face value of $1

      010

      While the final payment is defined as though it occurs at time T2, in practice the contract is cash settled at time T1. After all, since we know the value of the Libor rate L(T1, T2) at time T1, then we can use it to calculate the value of the payoff at time T2 and to discount this payoff to time T1. As a result, the actual payment on the FRA is made at time T1 and is equal to

      011

      The breakeven value of the strike of the FRA is the value of F0 for which the present value of the contract is zero. However, L(T1, T2) is the future Libor rate which has not yet been set. Essentially what we need to solve for is the expected value of this future Libor. The good news is that it is also possible to determine the fair-value of F0 using a replication strategy.

      2.4.2 FRA pricing by replication

      We begin by noting that the effective FRA payoff can be replicated using a combination of T1 and T2 maturity zero coupon bonds and a money-market deposit. The steps of the strategy are as follows:

      1. Pay Z(ts, T1) to purchase $1 face value of a T1 maturity zero coupon bond which settles at time ts.

      2. Sell short face value (1 + Δ(T1, T2) F0) of a T2 maturity zero coupon bond which settles at time ts.

      3. When the T1 maturity bonds mature at time T1, invest the $1 received in a T2 maturity money-market deposit. This pays (1 + Δ(T1, T2)L(T1, T2)) at a later time T2.

      The results of this strategy mean that:

      • At time ts we pay Z(ts, T1) − (1 + Δ(T1, T2)F0)Z(ts, T2).

      • At time T2 we receive Δ(T1, T2)(L(T1, T2) − F0).

      Since the payoff at time T2 is the same as an FRA, no-arbitrage requirements mean that the initial payment at time ts must be equal to zero. Hence we have

      (1 + Δ(T1, T2)F0)Z(ts, T2)–Z(ts, T1) = 0.

      This implies a fair value for the FRA rate given by

      012

      Hence, given a term structure of Libor discount factors, we can determine the FRA rate. However, once an FRA is transacted the FRA rate becomes fixed. However the value of the forward Libor rate

      013

      will continue to change as the Libor curve changes. Although it is beyond the scope of the book, it is possible to demonstrate the same result using a change of numeraire technique. We are then able to interpret F0 as the expected value of the forward Libor rate under the forward measure. For details see Brigo and Mercurio (2001).

      2.5 INTEREST RATE FUTURES

      Interest rate futures contracts are exchange-traded contracts where the price at maturity is equal to 100 minus the Libor rate observed on the maturity date. They can therefore be used to express a view on future Libor rates, or to hedge a forward starting interest rate exposure.

      The most liquid interest rate futures contracts reference the three-month Libor rate. At any moment in time, several contracts may trade with maturity dates occurring on the IMM date–defined as the third Wednesday of the delivery month. The most liquid delivery months are March, June, September and December. Examples of interest rate futures contract are the Eurodollar contracts which trade on the Chicago Mercantile exchange, plus the euro, UK sterling, Swiss franc and yen contracts traded at the Liffe exchange in London.

      2.5.1 Mechanics

      Consider a futures contract which matures at time T1 at a price equal to 100 minus the Libor rate to time T2. The mechanics of the contract are as follows:

      1. At initiation of the contract at time 0, the contract is worth nothing. However, a small percentage of the face value of the contract known as the initial margin will be deposited in the investor’s margin account at the exchange. The quoted price of the contract at any time t is given by

      P(t, T1, T2) = 100 − F(t, T1, T2)

      where F(t, T1, T2) is called the futures rate.

      2. On each date between time 0 and T1, variation margin is deposited in the investor’s margin account at the exchange equal to the daily change in the futures price. The size of the variation margin is calculated by multiplying the price change in ticks⁹ by the tick size of the contract and by the number of contracts. The tick size has been set by the exchange so that the change in the final payment from the contract is consistent with a 3M deposit contract with the same face value. The balance of the margin account is rolled over at Libor.

      3. At the contract maturity date T1, the price of the futures contract is given by

      P(T1, T1, T2) = 100–L(T1, T2)

      where L(T1, T2) is the Libor fixing for the period [T1, T2] observed and set at time T1.

      4. Time T2 is the end of the interest rate period referenced by the contract. No payments occur at time T2.

      It is important to realise that the price of a futures contract is not the same as the value of the contract. At the moment it is transacted, the value of the contract is zero to the investor. However, as the market changes its view about what value it expects for Libor at time T1, the futures price changes, and so does the value of the contract to the buyer. If the futures price falls, the investor will be required to post additional margin with the exchange. If the futures price rises, the exchange deposits the increase into the investor’s margin account.

      By requiring investors to post daily collateral equal to the change in value of the contract with the exchange, the exchange minimises any counterparty risk it may have to the investor. Equally, if the contract price moves in the investor’s favour, the exchange posts daily collateral to the investor’s margin account and so the investor minimises counterparty risk to the exchange. This margining process is important as it creates what is known as a convexity effect which is best explained using an example.

      Example An investor buys one futures contract which is trading at a price of 95.82, implying a futures rate of 4.18%. The contract face value is $1 million and the tick size is $25. Consider the following two scenarios:

      1. If the futures price falls to 95.55, the futures rate has increased to 4.45% and the investor is required to borrow (95.82 − 95.55) × 100 × $25 = $675 to post as collateral at a higher interest rate.

      2. If the futures price increases to 95.90, implying a futures rate of 4.10% then the investor’s margin account at the exchange shows a balance of (95.90 − 95.82) × 100 × $25 = $200 which is then invested by the exchange at the lower spot deposit rate.

      We have made the very reasonable assumption here that the futures rate F(t, T1, T2) and the spot rate at which the exchange rolls the margin account are very highly correlated.

      In both scenarios the margining always works against the futures contract holder. When they receive margin, this will coincide with a drop in the rate at which the balance in the margin account is rolled. When they have to deposit margin, this will coincide with an increase in the borrowing rate. This effect is known as negative convexity and the interest rate futures buyer will always lose relative to the investor in the comparable FRA who makes no payments until T1. If both the quoted futures rate and the FRA rate for the same period and currency are the same, then this will be exploited by arbitrageurs who will lock in a forward borrowing rate using the FRA and sell the futures contract against it in an attempt to profit from the negative convexity. This demand will drive down the futures price and so drive up the futures rate until this convexity bias is cancelled out.

      Since the value of the negative convexity depends on the volatility of the interest rate process, we need to have a model of the interest rate process in order to calculate its value. If we model the dynamics of the short interest rate process using the simple one-factor Ho and Lee (1986) model process, i.e. dr = θ(t)dt + σdW, where θ(t) is used to match the initial term structure, we assume that the short rate at which the margin account is rolled is perfectly correlated with the futures rate. We find that the relationship between the futures rate F and the FRA rate f is given by

      (2.1)

      014

      This equation is only exact if the forward and FRA rates are expressed as continuously compounded rates. However, even if they are not, the small size of the correction and the uncertainty in the value of the parameter σ means that this equation is a perfectly good first approximation which can be used to link futures contracts and FRAs.

      Example Assuming a basis point volatility for the short rate of 100 bp, the convexity correction for the futures contract with T1 = 1.0 and T2 = 1.25 is equal to

      015

      The effect is small but it does grow quadratically with the time to the forward date of the future, and also with the basis point volatility of the short rate process.

      The importance of having a convexity adjustment is that by adjusting the futures rate, it allows us to map the prices of highly liquid interest rate futures contracts onto FRAs which we are then able to use in the construction of a discount curve. This will be described later in this chapter.

      2.6 INTEREST RATE SWAPS

      The global interest rate swap market has been in existence since the early 1980s. Since then it has grown substantially and now has a total market outstanding estimated in mid-2007 to be around $347 trillion.¹⁰ The interest rate swap plays an extremely important role in the credit derivatives market. It is a building block of many popular credit structures including the asset swap. It is also the main interest rate hedging instrument for credit derivatives.

      An interest rate swap (IRS) is a bilateral over-the-counter contract. The standard swap contracts are transacted within the legal framework of the International Swap Master Agreement (ISMA) produced by the swap market’s International Swaps and Derivatives Association (ISDA). We now set out the mechanics of the standard interest rate swap contract.

      2.6.1 Mechanics

      An interest rate swap traded at time 0 settles at time ts. This typically occurs two days later, i.e the settlement convention is known as ‘T + 2’. There is, however, no payment on the settlement date as the coupon on the fixed leg has been set so that the interest rate swap costs nothing to enter into.

      Following contract initiation, the two parties exchange fixed rate interest payments for floating rate payments linked to a floating rate index, typically Libor. We call these two streams of payments the fixed leg and the floating leg. Figure 2.3 shows the payments on a five-year interest rate swap in which the fixed flows are assumed to be annual and at a fixed rate which we denote with H. The floating flows are semi-annual and are set at Libor. Each payment leg of the swap can have its own conventions in terms of coupon frequency and accrual basis. Both sets of flows terminate at the swap maturity.

      Market terminology is to distinguish between the two parties to an interest rate swap based on whether they pay or receive the fixed rate. The party who pays the fixed rate is known as the payer and the party who receives the fixed rate is known as the receiver. We now consider the valuation of the two legs of an interest rate swap separately.

      Figure 2.3 Mechanics of a five year interest rate swap which is paying semi-annually on the floating leg and annually on the fixed leg. We define the floating payments cm = Δ(tm − 1, tm) L(tm − 1, tm) where tm = t1,..., tM are the floating leg payment times. On the fixed leg we pay Δ(tn − 1, tn)H where H is the swap rate and tn = t1,..., tN are the fixed leg payment times. Note that tM = tN = T, the swap maturity

      016

      2.6.2 Valuing the Fixed Leg

      We begin by defining the N payment dates of the fixed leg of the swap as t1, t2, . . ., tN. All payments on the fixed leg of a swap are based on the swap rate H which is set at the start of the trade. If we assume that the notional of the swap is $1, then the cash flow paid on the fixed leg at time tn is given by

      H · Δ(tn − 1, tn),

      where we remind the reader that the accrual factor is also a function of the basis convention on the fixed leg. The value of the fixed side of the swap at time t is therefore given by discounting the fixed coupons according to a Libor discount factor and then summing over all of the fixed coupons to give

      017

      We now consider the floating leg.

      2.6.3 Valuing the Floating Leg

      The floating leg of a swap consists of a stream of M cash flows linked to the Libor interest rate index. We define the payment times on the floating leg of the swap as t1, t2, . . ., tM. This leg may have a different frequency and basis convention from the fixed leg of the swap.

      However, the final cash flows of both legs usually coincide so that we have tN = tM = T, the swap maturity date.

      In the standard swap contract, each floating rate coupon is set in advance, and paid in arrears meaning that the value of the next coupon payment is determined by observing the appropriate term Libor on the fixing date which typically falls two days before the immediately preceding coupon payment date. This is shown in Figure 2.4. Accrual of the Libor payment then begins on the previous coupon payment date.

      Figure 2.4 Each Libor floating rate payment is set at 11am on the fixing date, which occurs typically two days before the accrual period starts. The accrual period starts on the previous coupon date and is paid in arrears

      018

      The present value of the floating side of the swap is then given by discounting the future Libor cash flows and summing over them. We can write this formally as

      019

      where r(s) is the continuously compounded Libor short rate. In Section 2.9.1 we show that

      020

      This allows us to write

      021

      The situation becomes more complicated if the floating leg has already begun and we are part way through a coupon payment period. In this case we have a next Libor coupon which is already known and began to accrue at time t0 to be paid at time t1. Therefore

      VFloat (0)= Δ(t0, t1) L(t0, t1) Z(0, t1) + (Z(0, t1) − Z (0, tM)).

      2.6.4 The Swap Mark-to-market

      The mark-to-market of an interest rate swap is the value of money we would receive if we were to close out the swap contract. Since interest rate swaps usually settle according to a T + 2 convention, i.e. two days after trade date, then the mark-to-market must be based on exchanging the unwind value in two days’ time. We call this date the settlement date and it occurs at time ts.

      When an interest rate swap is initially traded, it has zero value. However, as soon as the market swap rates move, the value of the contract moves away from zero.

      Example Consider a scenario in which we have entered into a 5Y receiver swap contract at a swap rate of 7%. A year later, rates have declined so that the 4Y swap contract has a swap rate of 5%. The value of this initial contract will move into positive territory since we are receiving more than the 5% we would receive from a new 4Y swap contract. Indeed, we could enter into the 4Y payer swap and end up with a contract in which we have a net income of 2% on the fixed swap legs while the floating legs cancel out. The question is then–what is this worth?

      We consider an interest rate swap contract which was traded in the past at a time 0 and has a fixed rate leg which pays a swap rate H(0). We consider the valuation from the perspective of a receiver, the party to the swap who is receiving the fixed coupon. The mark-to-market from the perspective of a payer is simply the negative of this.

      (2.2)

      022

      Suppose that we are receiving the fixed leg and paying the floating leg, and suppose that the settlement date is on a coupon payment date for both the fixed and floating legs so that we don’t have to consider any floating payment that has already been set. The value of the swap on the settlement date is given by

      (2.3)

      023

      2.6.5 The Breakeven Swap Rate

      The breakeven swap rate at time ts is the value of swap rate H(ts) which makes the net present value of a swap equal to zero. We therefore have

      024

      Defining the PV01 as the present value of an annualised coupon of 1 dollar paid on the schedule of the fixed leg of the swap, we have

      025

      The breakeven swap rate at time t is therefore given by

      026

      We can use this to rewrite Equation 2.3 as

      WReceiver (ts)= (H (0) − H (ts)) PV01(ts, T).

      The value of an interest rate swap is roughly linear in the market swap rate H(ts). It is not exactly linear since the PV01(ts, T) of the swap also contains an implicit dependence on the term structure of swap rates and therefore on H (ts).

      In the following, we will use the terms ‘on-market’ and ‘off-market’. The term ‘on-market’ refers to a swap which is quoted in the market for one of the standard fixed terms such as 1Y, 3Y, 5Y, 7Y, 10Y. The breakeven swap rate is the swap rate which makes the net present value of this swap equal to zero. As soon as the swap is traded it goes ‘off-market’ and its value can move away from zero. This terminology is also used for other sorts of derivatives contracts.

      2.6.6 Interest Rate Swap Risk

      The value of an interest rate swap is sensitive to changes in the market Libor curve. We can quantify these sensitivities by considering the sensitivity of the swap mark-to-market to changes in market swap rates. We begin by writing the value of a payer interest rate swap as

      027

      Differentiating with respect to the market swap rate to the swap maturity H (ts), we get

      028

      For an on-market swap, we know that H (ts) = H (0) and the value of the swap is zero. As a result we have

      029

      The swap rate sensitivity of the swap value therefore equals its PV01. This result holds for both payer and receiver swaps although the sensitivities have opposite signs. The advantage of this result is that the PV01 is a sensitivity which can be computed analytically without the need to do any bumping of rates and rebuilding of the Libor discount curve.

      When the swap is off-market, i.e. H(ts) ≠ H(0), then its interest rate sensitivity must be computed by bumping the swap rates, rebuilding the Libor curve, and then repricing the swap. This requires us to have a methodology for building the Libor discount curve and this is the subject of the next section.

      2.7 BOOTSTRAPPING THE LIBOR CURVE

      So far, we have introduced and analysed the pricing of money-market deposits, forward rate agreements, interest rate futures and interest rate swaps. We have shown that their pricing is directly linked to the shape of the forward Libor discount curve at today time 0 which is given by Z (0, T). However, we now ask the reverse question. How can we construct a discount curve that is consistent with the prices of deposits, futures, FRAs and swaps? To begin, we write out the set of pricing equations for all of the Libor instruments, except the interest rate futures:

      • For deposits d = 1,..., ND we have

      Z(ts, Td)(1 + Δ(ts, Td)L(ts, Td)) = 1.

      • For FRAs f = 1,..., NF we have

      Z(ts,t1f) = Z(ts, t2f) (1 + Δ (t1f, t2f)F(t1f, t2f)).

      • For swaps h = 1,..., NH we have

      030

      These are all linear equations in Z (t, T). This is why we have omitted interest rate futures. Due to their convexity adjustment, interest rate futures do not have a linear pricing equation in Z (t, T). However, we can incorporate futures as a linear product by mapping them onto equivalent FRAs with the same start and end dates. To do this, we must convert the futures rate to FRA rates using a convexity adjustment such as the one given in Equation 2.1.

      These instruments present a system of ND + NF + NH linear equations. However, the number of discount factors to be solved for will typically be greater than this. Consider the following example.

      Example Suppose we have the following instruments:

      • 3M and 6M deposit rates. These have a dependency on discount factors at T = 0.25 and T = 0.5 years.

      • A 5 × 8M FRA with dependency on dates at T = 0.417 and T = 0.75 years.

      • 1Y, 2Y, 3Y, 4Y, 5Y swap rates with semi-annual coupons on the fixed leg. The discount factors needed are at times T = 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0.

      We have a total of 2 + 1 + 5 = 9 linear pricing equations, one for each instrument. However, there are a total of 13 different discount factors to find. The system is underdetermined and can only be solved if we make further assumptions.

      It is also possible for the system of equations to be overdetermined. Consider another example:

      Example Suppose we have the following instruments:

      • A 6M deposit with payment dates at T = 0.5.

      • A 6 × 12M FRA with dates at T = 0.5 and T = 1.0.

      • A one year swap with semi-annual coupon payment dates at T = 0.5, 1.0.

      We have a total of three pricing equations, but with only two unknowns which are Z (0, 0.5) and Z (0, 1.0). The system is overdetermined.

      When the system is overdetermined, we need to reject some instruments, i.e. we should decide which instruments take precedence when building a discount curve. In the example, it may be decided that the 1Y swap is a more liquid and more appropriate measure of the 6M to 1Y discount curve than a 6 × 12M FRA. We would then remove the 6 × 12M FRA in order to imply out the 12M discount factor from the 1Y swap rate.

      2.7.1 Interpolation

      While the case of an overdetermined system is certainly possible, it is much more likely that the system is underdetermined, with more unknowns than equations. We then need additional assumptions. Typically, these assumptions take the form of an interpolation scheme. Using interpolation, we simply reduce the number of unknown discount factors to the number of instruments and then solve for these. Any remaining discount factors are not solved for but simply determined by interpolating the ‘skeleton’ of solved discount factors.

      There are many possible interpolation schemes. Spline methods such as quadratic, tension and b-splines may be used and have been discussed in recent papers by Andersen (2005) and Hagan and West (2004). However, such approaches may be deemed to be overly sophisticated in the context of credit derivatives pricing since credit derivatives have a low interest rate sensitivity and are also fairly insensitive to the shape of the interest rate curve. We therefore choose to use a simple interpolation scheme which assumes that the continuously compounded forward rate is piecewise constant.

      2.7.2 The Bootstrap

      Given an interpolation scheme, the most common approach to building curves is to use a bootstrap. The idea of a bootstrap is that we start with the shortest maturity product. We then solve for the discount factor at its maturity date, relying upon our interpolation scheme to determine any earlier discount factors which may be required. We then move to the instrument with the next longest maturity and once again solve for its maturity date discount factors. Any cash flows which fall between the previous maturity date and this maturity date are also determined by interpolation. Hence each product requires us to solve for one discount factor and the system is exactly determined. This process is shown in Figure 2.5.

      Figure 2.5 The bootstrap methodology works through the instruments from left to right, in order of their increasing final maturity. Discount factors for cash flow dates with the solid circles are solved for, while those discount factors at the cash flow dates with empty circles are interpolated using a chosen interpolation scheme. This means that we have one unknown

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