Abstract
The fixed income market is a sector of the global financial market in which various interest rate-sensitive instruments are traded, such as bonds, forward rate agreements, various forms of swaps, swaptions, caps and floors. It makes up a large portion of the global financial market. The recent financial crisis, of which the key features are counterparty and liquidity/funding risk, has heavily impacted the entire financial market and the fixed income market in particular. Inspection of quoted interest rates and derivative prices reveals that some classical relationships have broken down, which has induced the actors on the fixed income market to model as separate objects rates that correspond to different maturities (multi-curve models). In this chapter we review the basic notions and concepts in use before the crisis and describe how they have found an extension to the multi-curve setup.
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The terminology fixed-income market designates a sector of the global financial market on which various interest rate-sensitive instruments, such as bonds, forward rate agreements, swaps, swaptions, caps/floors are traded. Zero coupon bonds are the simplest fixed-income products, which deliver a constant payment (often set to one unit of cash for simplicity) at a pre-specified future time called maturity. However, their value at any time before maturity depends on the stochastic fluctuation of interest rates. The same is true for other fixed-income derivatives. Fixed-income instruments represent the largest portion of the global financial market, even larger than equities. Developing realistic and analytically tractable models for the dynamics of the term structure of interest rates is thus of utmost importance for the financial industry. From the theoretical point of view, interest rate modeling presents a mathematically challenging task, in particular due to the high-dimensionality (possibly even infinite) of the modeling objects. In this sense interest rate models substantially differ from equity price models.
The credit crisis in 2007–2008 and the Eurozone sovereign debt crisis in 2009–2012 have impacted all financial markets and have irreversibly changed the way they functioned in practice, as well as the way in which their theoretical models were developed. One may thus distinguish between a pre-crisis and a post-crisis setting. The key features that were put forward by the crises are counterparty risk, i.e. the risk of a counterparty failing to fulfill its obligations in a financial contract, and liquidity or funding risk, i.e. the risk of excessive costs of funding a position in a financial contract due to the lack of liquidity in the market. The fixed-income market has been particularly concerned by both of these issues. The reason for this is the following: the underlying interest rates for most fixed-income instruments are the market rates such as Libor or Euribor rates and the manner in which the market quotes for these rates are constructed (see Sect. 1.1 for details) reflects both the counterparty and the liquidity risk of the interbank market. Inspection of quoted prices for related instruments reveals that the relationships between Libor rates of different maturities that were previously considered standard, and held reasonably well before the crisis, have broken down and nowadays each of these rates has to be modeled as a separate object. Moreover, significant spreads are also observed between Libor/Euribor rates and the swap rates based on the so-called overnight indexed swaps (OIS), which were following each other very closely before the crisis. Simultaneous presence of these various interest rate curves is referred to in the current literature as the multiple curve issue and the post-crisis interest rate models are often referred to as the multiple curve models (multi-curve models). This book aims at providing a guide for development of interest rate models in line with these changes, accompanied by a detailed overview of some current research articles, in which, to the best of our knowledge, such a development has been studied. The recent monograph by Henrard (2014) and the article collection Bianchetti and Morini (2013) also concern post-crisis multiple curve modeling, reflecting in particular the practitioners’ perspective.
Pre-crisis interest rate models can be divided into various classes, in particular the following: the short-rate models, where the short interest rate is modeled, including pricing kernel models; the Heath–Jarrow–Morton (HJM) framework, where the zero coupon bond prices, or equivalently, the forward instantaneous continuously compounded rates are modeled; the Libor market models, where the market forward rates are modeled directly. The books by Björk (2009), Brigo and Mercurio (2006), Cairns (2004), Filipović (2009), Hunt and Kennedy (2004) and Musiela and Rutkowski (2005) provide an excellent introduction to interest rate theory, as well as an extensive overview of the existing modeling approaches in this field. The first short-rate models were introduced in the seminal papers by Vasiček (1977), Cox et al. (1985) and Hull and White (1990). The HJM framework was developed in Heath et al. (1992). Furthermore, rational pricing models were pioneered by Flesaker and Hughston (1996), models using potential approach were developed in Rogers (1997) and Markov functional models in Hunt et al. (2000). Finally, Libor market models were proposed by Brace et al. (1997) and Miltersen et al. (1997) and later developed further especially by practitioners. Various extensions and generalizations of all these model classes can be found in the literature, which is too vast to mention it all here.
In order to give a first flavor and illustrate the issues presented above, we shall give a closer look at a prototypic interest rate derivative, a forward rate agreement, which is a building block for all linear interest rate derivatives and is also related to the underlying rate of nonlinear derivatives, as its price represents the market expectation about the future value of the Libor rate. Precise definitions of all the notions used below, as well as a more detailed treatment of the pricing of FRAs will be presented in Sect. 1.4.1.
Let us firstly recall the classical, pre-crisis connections between zero coupon bonds, FRA rates and Libor rates. As mentioned above, a zero coupon bond is a financial contract which delivers one unit of cash at its maturity date \(T >0\). Its price at time \(t \le T\), denoted by p(t, T), represents therefore the expectation of the market concerning the future value of money. Obviously, \(p(T, T)=1\). Traditionally, interest rates are defined to be consistent with the zero coupon bond prices p(t, T). For discretely compounding forward rates this leads to
This formula can also be justified as representing the fair fixed rate at time t of a forward rate agreement (FRA), where the floating rate received at time S is
The rate (1.1) is therefore also called the FRA rate. Note that we have assumed, without loss of generality, the notional equal to 1 here, as we are interested only in the rates. The arbitrage-free price at time t of such an FRA is, using the forward martingale measure \(Q^S\) (see Sect. 1.3.2),
where R denotes the fixed rate of the FRA. This price is zero for
Therefore, one obtains the following key relation between the bond prices and the forward rates thanks to the connection between the floating rate and the zero coupon bond prices (1.2):
The discretely compounded forward rate at time \(t\ge 0\) for the future time interval [T, S], where \(t \le T \le S\), is the rate given by
Before the crisis, the Libor rate, which is an interest rate obtained as an arithmetic average of submitted daily quotes from a panel of banks participating in the London interbank market (see Sect. 1.1), was assumed to be equal to the floating rate defined using zero coupon bond prices, i.e.
where L(T; T, S) stands for the Libor rate at time T for the period [T, S]. This was rightfully done so, since the Libor panel, which is refreshed on a regular basis in such a way that the banks of deteriorating credit quality are replaced with the banks of a better credit quality, contained virtually no counterparty and liquidity risk, making thus plausible the assumption of risk-freeness, which is implicitly made when assuming (1.5). Consequently, the FRA rate F(t; T, S) was also called the forward Libor rate (since it represented the market expectation of the future value of the Libor rate) and often denoted by L(t; T, S). Hence, the forward Libor rate was given either as a conditional expectation of the spot Libor rate under the forward martingale measure, or expressed using the quotient of the bond prices, cf. (1.4):
Due to the crisis and in view of the manner in which the spot Libor rate quotes are produced, in the post-crisis market the assumption that the Libor rate is free of various interbank risks is no longer sustainable and hence the connection (1.5) to the zero coupon bonds, which are assumed risk-free, is lost:
The question what these zero coupon bonds are in the post-crisis setup is also far from a trivial one and is tackled in Sect. 1.4.4; here for simplicity we do not enter into it. Let us now consider again the same type of the forward rate agreement as above, to exchange a payment based on a fixed interest rate R against the one based on the spot Libor rate L(T; T, S), cf. Definition 1.3. The payoff of the FRA at maturity S being equal to
its value at time \(t \le T\) is calculated as the conditional expectation with respect to the forward measure \(Q^{S}\) associated with zero coupon bond \(p(\cdot , S)\) as numéraire and is given by
We use the same symbol for the value of the FRA here as in (1.3) since it is basically still the same contract, only the underlying rate L(T; T, S) does not satisfy (1.5) anymore. Hence, the key quantity is the conditional expectation of the spot Libor rate that we denote by L(t; T, S) and define
The forward Libor rate at time \(t \ge 0\) for the future time interval [T, S], where \(t \le T \le S\), is the rate given by
The crucial difference with respect to the classical pre-crisis forward Libor rate is the following one:
Typically, the former quantity will be greater than the latter and this provides the very first example of the post-crisis spreads (or equivalently multiple curves):
This spread depends, moreover, also on the difference \(\varDelta :=S-T\), i.e. the length of the period to which the Libor rate applies, also known as tenor. In practice, the tenor \(\varDelta \) ranges from one day to several months (up to twelve months) and the observed market spreads are typically increasing with respect to the tenor, i.e. the function \(\varDelta \mapsto S(t; T, T+\varDelta )\) is an increasing function for fixed t and T. This and other types of spreads will be defined in Sect. 1.4.4.
1.1 Types of Interest Rates and Market Conventions
In this section we describe the most important market rates and their main characteristics. Note that all these rates are quoted as annualized rates. This means for example that a quote of \(1\,\%\) for a 3-month interest rate corresponds to the following interest: 1 unit of cash invested at this rate yields \(1 + \frac{3}{12} 0.01 = 1.0025\) units of cash in 3 months.
1.1.1 Basic Interest Rates: Libor/Euribor, Eonia/FF and OIS Rates
The most widely known market rates are the Libor rates because they are reference rates for a variety of fixed income products, but even a person without any experience in the financial industry might have seen this rate as an underlying floating rate in bank loans for example. Therefore, we begin this section by giving a description and the main characteristics of the Libor rates. The LIBOR stands for London Interbank Offered Rate and the description below is taken from the ICE Benchmark Administration (IBA) webpage https://www.theice.com/iba/libor, which is an independent entity administering the Libor as of February 1st, 2014. The Financial Conduct Authority (FCA) serves as a regulator, which supervises the panel banks and has a power to take individuals to court for benchmark-related misconduct. We quote from the ICE Benchmark Administration:
“ICE Libor is designed to reflect the short term funding costs of major banks active in London, the world’s most important wholesale financial market. Like many other financial benchmarks, ICE Libor (formerly known as BBA Libor) is a polled rate. This means that a panel of representative banks submits rates which are then combined to give the ICE Libor rate. Panel banks are required to submit a rate in answer to the ICE Libor question: At what rate could you borrow funds, were you to do so by asking for and then accepting inter-bank offers in a reasonable market size just prior to 11 a.m.? Although banks now use transaction data to anchor their submissions, having a polled rate is crucial to ensure the continuous publication of such a systemic benchmark, even in times when liquidity is low and there are few transactions on which to base the rate. Currently only banks with a significant London presence are on the ICE Libor panels, yet transactions with other non-bank financial institutions can often inform panel banks’ submissions. Reasonable market size is intentionally unquantified. The definition of an appropriate market size depends on the currency and tenor in question, as well as supply and demand. The current wording therefore avoids the need for frequent and confusing adjustments. 11 a.m. was chosen because it falls in the most active part of the London business day. It is also sufficiently early in the day to allow the users of ICE Libor to use each day’s rates for valuation processes, which may take place in the afternoon. All ICE Libor rates are quoted as annualized interest rates. This is a market convention. For example, if an overnight Pound Sterling rate from a contributor bank is given as 0.5000 %, this does not indicate that a contributing bank would expect to pay 0.5 % interest on the value of an overnight loan. Instead, it means that it would expect to pay 0.5 % divided by 365.”
Note that the ICE Libor rates are produced each business day for five different currencies (US Dollar, Euro, British Pound Sterling, Japanese Yen, Swiss Franc) and seven maturities (1 day, 1 week, 1, 2, 3, 6 and 12 months). For each currency, a different panel of representative banks is selected, ranging from 11 to 18 banks. The ICE Libor daily quote for each currency and each maturity is the “trimmed arithmetic mean” of all of the panel banks’ submissions, i.e. the highest and lowest 25 % of the submissions are removed and the rest is averaged.
In the Eurozone, an interest rate with very similar features to those of the Libor is called the Euribor, see http://www.emmi-benchmarks.eu for details. The entity administering the Euribor is the European Money Markets Institute (EMMI) as of June 20th, 2014. Similarly to the Libor, the Euribor is also produced from quotes submitted by a panel of banks from EU countries, as well as large international banks from non EU-countries, participating in Eurozone financial operations. The choice of banks quoting for Euribor is based on market criteria and the panel consists currently of 26 contributing banks, which submit a rate at which they believe “Euro interbank term deposits are offered by one prime bank to another prime bank within the EMU zone”. The Euribor rates are quoted for eight different maturities (1 and 2 weeks, 1, 2, 3, 6, 9, 12 months) and are “calculated at 11:00 a.m. (CET) for spot value (T+2)” as a trimmed average of the quotes submitted by the panel banks. Figure 1.1 displays the historical series of the Euribor rates for maturities 1, 2 and 3 months. The starting month in the graph is January 2010 and the last month is September 2015.Footnote 1
Historical series of the Euribor rates for maturities 1, 2 and 3 months. The starting month in the graph is January 2010 and the last month is September 2015
The reference rate for the shortest maturity of one day in the Eurozone is the Eonia (Euro OverNight Index Average) rate. The Eonia rate is computed as a “weighted average of all overnight unsecured lending transactions in the interbank market, undertaken in the European Union and European Free Trade Association (EFTA) countries by the panel banks. It is reported on an act/360 day count convention and is displayed to three decimal places”. The panel banks for the Eonia rate are the same as the ones for the Euribor. Note that these banks contribute daily data on their total volume of transactions in unsecured overnight money and the average interest rate for this daily volume; the Eonia rate is then calculated from these contributions as a weighted average interest rate, where the weighting is done according to the transaction volumes of the contributors. In the left graph of Fig. 1.2 the yearly average values of the Euribor rates with maturities 1, 2 and 3 months are displayed and in the right graph the average Eonia rates.Footnote 2
Left Yearly average for Euribor rates with maturities 1, 2 and 3 months, 2009–2014. Right Yearly average for Eonia rates, 2009–2014
A corresponding overnight rate in the United States to the Eonia rate in the Eurozone is the Federal Funds (FF) effective rate, which is the weighted average across all overnight transactions between depository institutions trading balances held at the Federal Reserve, which are called federal funds. Similarly to the transactions contributing to the Eonia rate, these transactions are also unsecured.
Finally, the name OIS rate refers to a market swap rate of an overnight indexed swap (OIS), which is, as any interest rate swap, defined on a discrete tenor structure and in which, at every tenor date, payments based on a fixed rate are exchanged for payments based on a floating rate. This floating rate is a discretely compounded rate obtained by compounding the overnight rates over the corresponding intervals between two subsequent tenor dates (the reason why the swap is called overnight indexed swap). The reference overnight rate in the Eurozone is the Eonia and in the US the FF rate. Overnight indexed swaps and corresponding OIS rates are precisely defined and studied in detail in Sect. 1.4.4.
1.2 Implications of the Crisis
As already mentioned in the introduction to this chapter, a number of anomalies that were not previously observed in the fixed income markets appeared due to the financial crisis. The interest rates whose dynamics were very closely following each other have started to diverge substantially, thus prompting the introduction of various spreads measuring this divergence.
1.2.1 Spreads and Their Interpretation: Credit and Liquidity Risk
The first type of post-crisis spreads, mentioned already at the beginning of the chapter, concerns the spreads between the Libor rates and the OIS rates of the same maturity (see Sect. 1.4.4 and in particular, Eq. (1.33) for the precise definition), which have been far from negligible since the crisis. Moreover, also the spreads between the swap rates of the Libor-indexed interest rate swaps and the OIS rates (see Sect. 1.4.4, Eq. (1.37) for definition) have appeared. The former type of spreads is known as the Libor-OIS spread and the latter as the Libor-OIS swap spread. In Fig. 1.3 Footnote 3 (left) the historical Euribor-Eonia swap spreads in the period 2005–2010 are plotted for maturities ranging from 1 month to 12 months. As one can clearly see, before the crisis these spreads were practically negligible, whereas at the peak of the crisis they were greater than 200 basis points for some maturities. Furthermore, the Libor rates of different maturities have exhibited notably diverse behavior, reflected in basis swap spreads, which are appearing in connection to basis swaps (cf. Sect. 1.4.5 for definition of a basis swap and Eq. (1.40) for the related basis swap spread). Figure 1.4 Footnote 4 shows the evolution of the Euribor-OIS spread, as well as the 3-month versus 6-month basis swap spread, for a time period from January 2004 until April 2014. As the graphs indicate, even after 2012 these spreads did not revert back to their pre-crisis levels. This is why practitioners nowadays tend to produce different yield curves for different tenors; compare Remark 1.2 and see Fig. 1.3 (right), which displays discount curves related to the Eonia swap rates, the 3-month and the 6-month Euribor rates.
Left Historical Euribor-Eonia swap spreads 2005–2010. Right Discount curves bootstrapped on September 2, 2010
Spread Development from January 2004 to April 2014
The question that immediately comes to mind is if such observed large spreads present arbitrage opportunities in the market. However, as pointed out by Chang and Schlögl (2015), these spreads have persisted since the crisis, implying that such opportunities have not been exploited. The reason is that the spreads are due to the various risk levels, therefore Chang and Schlögl (2015) conclude that the expected gains from a possible arbitrage have to be offset by the expected losses at the different risk levels, which would imply that those arbitrage opportunities are only illusory and hence cannot be exploited.
In the chapters below we shall present models for the dynamic evolution of the spreads. Notice that the first approaches to spread modeling considered them as deterministic quantities (see Henrard 2007, 2010). The advantage of this deterministic modeling is that it allows the pre-crisis single curve pricing formulas for both linear and optional derivatives to be applied in the same form also to the multiple curve setup. In particular, for the case of optional derivatives, it suffices to modify the strike by simple deterministic shifting or scaling (see also more detailed comments in Sect. 4.3). For this reason, this way of modeling seems to be still often used in practice in spite of contrary evidence from empirical data (see Fig. 1.4), as well as of possible anomalies introduced by these simplistic assumptions. One such anomaly is mentioned e.g. in Mercurio and Xie (2012) (see also the last part of Sect. 4.1.1) who point out that a deterministic spread assumption would wrongly lead to a zero price for out-of-the-money Libor-OIS swaptions.
The various risks driving the spreads are the risks related to the interbank market, in particular to the banks participating in the Libor panel. Therefore, these risks are sometimes jointly referred to as interbank risk. One important component of this risk is default risk. The rolling construction of the Libor panel is intended to reduce the possibility of actual defaults within the panel. However, the deterioration of the credit quality of the Libor contributors during the length of a Libor-based loan is greater with longer tenors. Moreover, interbank risk arises also from liquidity risk. Strategic gaming can also play a role (Michaud and Upper 2008). Such considerations might from time to time incite a bank to declare as its Libor contribution a number different from its internal conviction regarding “The rate at which an individual Contributor Panel bank could borrow funds, were it to do so by asking for and then accepting interbank offers in reasonable market size, just prior to 11.00 London time” (the definition of Libor). All this results in a spread between the Libor rates of different tenors (OIS rates in the limiting case of an overnight tenor).
Filipović and Trolle (2013) analyze the decomposition of the interbank risk driving spreads into default and liquidity risk components. Using a data set covering the period August 2007–January 2011, the authors show that the default component is overall the main dominant driver of interbank risk, except for short-term contracts in the first half of the sample (see Figs. 3 and 4 in their paper). The remaining risk is attributed to liquidity risk, an observation made in Morini (2009) as well. The liquidity risk component driving the Libor-OIS spreads is studied and explicitly modeled in Crépey and Douady (2013). A recent work by Gallitschke et al. (2014) provides an endogenous explanation of spreads and constructs a structural model for Libor rates, deriving them from fundamental risk factors, namely: interest rate risk, credit risk and liquidity risk. The emphasis is on liquidity risk, which is shown to be mainly induced by the tenor basis.
1.2.2 From Unsecured to Secured Transactions
Broadly speaking, in finance the term collateral refers to assets or cash posted by a borrower to a lender in order to secure a loan. In other words, if for various reasons the borrower fails to make the promised loan payments, the lender can cover (partially or fully) the occurred losses using the collateral. If this has not been the case, the collateral is returned to the borrower after the loan has been fully repaid, together with the possibly accumulated interest. Collateral can be posted also in connection with various other financial transactions which expose one or both counterparties to the risk of non-payment (default risk). Since collateral thus provides certain security, a financial transaction with posted collateral is called secured, as opposed to an unsecured transaction which does not have collateral. The question of unsecured versus secured transactions in financial markets gained extreme importance in view of the previously discussed counterparty risk. Since it is by now generally understood that no financial institution is “too big to default”, various mechanisms of reducing the exposure to counterparty risk in OTC derivatives have been put forward.
In particular, a large number of OTC bilateral contracts is collateralized, i.e. the value of the contract is periodically marked-to-market and the party whose position has lost in value has to post collateral in the collateral account. The posted collateral remains the property of the collateral payer and is remunerated. In case no default occurs during the lifetime of the contract, the collateral provider receives it back, together with the accumulated interest. The details regarding the frequency of posting the collateral, eligible currencies and securities which can serve as collateral, specifications of close-out cash flows are all described in CSA (Credit Support Annex) agreements. The ISDA (International Swaps and Derivatives Association) nowadays provides standard CSAs for OTC derivatives, which “is part of ISDA’s continuing efforts to increase efficiency and improve standardization in the OTC derivatives markets”. These standard CSAs in particular promote adoption of the OIS discounting.
An alternative to collateral posting directly between two counterparties is “central clearing” of a bilateral contract, which is done via central counterparties (CCPs) or clearing houses. Nowadays many financial contracts are cleared in this way and a number of CCPs exist specializing in particular types of markets and products. For example, some of the main CCPs for interest rate swaps and credit default swaps are SwapClear, ICE Trust US and ICE Clear Europe. A short description of functioning of a CCP given here is based on the article by Heller and Vause (2011).
The role of a CCP is to act as an intermediary between two counterparties in a bilateral contract and to take on their own respective counterparty risks. The CCP thus becomes the new counterparty for both parties in the contract, which are consequently only exposed to the counterparty risk of the CCP. This risk should, however, be small as the CCP is well capitalized and thus well equipped to face possible defaults of its members. In order to manage the counterparty risk of the members, the CCP relies on several mechanisms. The first one are participation constraints, which exclude counterparties with default probability above a certain acceptable threshold from dealing with the CCP. Upon initiation of a bilateral contract through the CCP, each party is required to post an initial margin to the CCP, usually in form of cash or highly liquid securities. This initial margin serves to cover most possible losses in case of default of a counterparty, whose positions are then inherited by the CCP. In particular, this concerns the period between the last time the defaulting party’s position was valued and variation margins were paid and the close-out of the position. Variation margins represent a third mechanism of protection and concern the changes in the market value of counterparties’ positions in the contract. The counterparty whose position has lost in value is obliged to post the variation margin to the CCP (which is necessarily done in cash as opposed to collateral agreements). The CCP typically passes this margin to the other counterparty. The variation margins are usually calculated and collected daily. Finally, as the fourth line of defense against losses due to defaults of its members, the CCP also disposes of a non-margin collateral such as default funds which contain collateral posted by all members of the CCP.
Acknowledging the role of collateralization and central clearing in mitigation of counterparty risk in OTC derivatives (we refer in particular to Cont et al. (2011) who found significant differences in the values of cleared and uncleared interest rate swaps), one still has to be aware of the remaining liquidity risk that might even get more pronounced as a consequence of need to finance the collateral postings and variation margin calls in adverse mark-to-markets conditions.
1.2.3 Clean Prices Versus Global Prices
The term “clean price” was introduced in the post-crisis literature in Crépey (2015) and Crépey et al. (2014) and refers to a price of an OTC derivative in a hypothetical situation where default and liquidity risk of the two counterparties are assumed to be negligible. In particular, in case of interest rate derivatives, this means that the counterparty and liquidity risk of the two parties are ignored, whereas the counterparty and liquidity risk of the interbank market, which directly influence the reference rates in these contracts and create the multiple curve phenomenon, are still taken into account. Hence, in this sense the interest rate derivative prices studied in this book are clean, multiple curve prices. This is also supported by the fact that most market quotations of derivative prices reflect collateralized transactions, thus leading to clean prices.
On the other side, a global price of a derivative is a price including also the adjustments due to counterparty and liquidity risk. This can be done in two ways: either by developing a pricing framework which takes these issues into account from the beginning, or by computing firstly the clean prices and then “adjusting” them for these risks by CVA (credit valuation adjustment), DVA (debt valuation adjustment) and FVA (funding valuation adjustment), as well as other adjustments referred to simply by XVA. The computation and the interplay of these adjustments in order to obtain the global price of a derivative are far from trivial.
Even though both of these approaches have their advantages and disadvantages for OTC pricing in general, in our view the second approach seems to be well suited for the case of interest rate derivatives. Firstly, the “splitting” of the pricing procedure into two parts corresponds in fact to common practice, where the clean prices are computed on the case by case basis (derivative by derivative) and then used as an underlying to produce the valuation adjustments, for which the whole portfolio between two counterparties, and not only one specific derivative, plays a role. Secondly, in contrast to credit derivatives, the wrong-way risk and the gap risk in interest rate derivatives are rather small, hence disregarding counterparty risk when computing clean interest rate derivative prices seems to be a reasonable assumption. Finally, since already the clean pricing of interest rate derivatives requires complex models due to the multiple curve issue, we feel that the two-step approach in obtaining the global derivative prices should be preferred in this case. The treatment in this book therefore concerns only the clean pricing of interest rate derivatives. For a detailed overview of the global pricing of financial derivatives we refer the interested reader to Brigo et al. (2013), Pallavicini and Brigo (2013), Crépey et al. (2014) and the references therein.
1.3 The New Paradigm: Multiple Curves at All Levels
Let us fix a finite time horizon for all market activities, denoted by \(T^{*}>0\). Having seen in the previous sections that the key role in the post-crisis fixed-income markets is played by the tenor of the underlying interest rate, let us now formalize these discussions and introduce the notation and the needed probabilistic framework.
A discrete tenor structure \(\mathscr {T}^x\) with tenor x is a finite sequence of dates
We denote \(\delta ^x_k:= T^x_{k} - T^x_{k-1}\) the year fraction corresponding to the length of the interval \((T^x_{k-1}, T^x_{k}]\), for \(k=1, \ldots , M_x\). Typically, the distance between the dates in the tenor structure will be constant, i.e. \(\delta ^x_k = \delta ^x\), for all k.
Remark 1.1
In this book, for sake of clarity of the exposition, we put aside the practical issue of day count conventions. We acknowledge that in practice, however, there is a variety of day count conventions that have to be taken into account and refer to e.g. Ametrano and Bianchetti (2013) for more details on these conventions.
As already seen in the previous sections, in practice the tenor x ranges from one day (\(\delta ^x=\frac{1}{360}\)) to twelve months (\(\delta ^x=1\)). In the multi-curve setup one has to consider different possible tenor structures simultaneously. We shall thus denote by \(\mathscr {X}:=\{x_1<x_2<\cdots <x_n\}\) a collection of tenors and by \(\mathcal{T}^{x_i}=\{0\le T_0^{x_i}<\cdots <T_{M_{x_i}}^{x_i}\}\) the associated tenor structures for \(i=1, \ldots , n\), thereby assuming that \(\mathcal{T}^{x_n}\subset \mathcal{T}^{x_{n-1}}\subset \cdots \subset \mathcal{T}^{x_1} \subseteq \mathcal{T}\), where \(\mathcal{T}:= \{0 \le T_0 < T_1< \cdots < T_{M} \le T^{*}\}\) can be seen as a reference tenor structure containing all the others. Moreover, assume that \(T_{M_{x_i}}^{x_i}=T_M\), for all i, meaning that all tenor structures have a common terminal date. Typically, we have \(\mathscr {X}= \{1,3,6,9, 12\}\) months. As an example, Fig. 1.5, taken from Grbac et al. (2014), illustrates the relation between different tenor dates in the 1-month, 3-month and 6-month tenor structures, assuming that the 1-month tenor structure is the reference tenor structure.
Illustration of different tenor structures
Having in mind the discussion at the beginning of this chapter and the impact of the underlying tenor on the values of the Libor rates, after the crisis, instead of having Libor rates of different tenors connected by no-arbitrage relations, one has to associate to each tenor \(x \in \mathscr {X}\) a different curve. In other words, at time \(t=0\), for each x the following rates are observable, where here and below by “observable” we mean quantities that are either directly observable or can be computed from market data as will be explained further in Remark 1.2:
Hence, one can define the associated discount curve by imposing the following classical relation:
and deriving from it the x-bond prices \(p^x(0, T^x_{k})\), cf. Ametrano and Bianchetti (2013) and Miglietta (2015). Note, as pointed out also by Miglietta (2015), that there is no unique inverse relationship between the initial Libor curve \(L(0; T^x_{k-1}, T^x_{k})\) and the initial x-bond term structure, only the quotients \(\frac{p^x(0, T^x_{k-1})}{p^x(0, T^x_{k})}\) are uniquely determined. Clearly, the simultaneous presence of several, mutually “disconnected” Libor curves that cannot be associated to only one common discount curve \(T \mapsto p(0, T)\), as it was the case in the pre-crisis setup, gives rise to the first obvious question of choice of the discount curve (or curves), as well as to other questions related to the mathematically sound and practically reasonable modeling of multiple curves.
Remark 1.2
(Tradable quantities and market data) Regarding the tradable quantities and bootstrapping of the yield curves based on market quotes, we briefly summarize the most important points from the paper by Ametrano and Bianchetti (2013), which provides a very detailed overview of the procedure of calibration and yield curve construction for the instruments traded in the European market. According to this paper, in the European market the most important tenors that are considered are the following ones: 1 day, 1, 3, 6 and 12 months and the most liquidly traded instruments are based on these tenors as underlyings. Constructions of the yield curves corresponding to the mentioned tenors is done by bootstrapping both from available market prices of traded instruments, as well as from prices of synthetic instruments, which are used for replacing the missing market quotes. These instruments include deposits (depos), FRAs, interest rate swaps, overnight indexed swaps and basis swaps based on various tenors. The most important yield curve is the one related to the 1-day tenor. This curve is constructed on the basis of the market OIS rates for overnight indexed swaps and is therefore usually called the OIS yield curve. More precisely, analogously to what is done in the pre-crisis setting, starting from the market OIS rates one can construct by bootstrapping the OIS bond prices (see Eq. (1.32) and Remark 1.6 in Sect. 1.4.4). This gives the OIS discount curve that in turn leads in the usual way to the OIS yield curve. The importance of the OIS discount curve (or equivalently the OIS yield curve) lies in the fact that it is the most commonly used discount curve for pricing of other interest rate derivatives. We recall, as mentioned already at the end of Sect. 1.1, that the underlying overnight rate of overnight indexed swaps in the European market is the Eonia rate.
The procedure of constructing the yield curves from market data is based in general on two types of algorithms: the best-fit, where a functional form for the yield curves, such as Nelson-Siegel or Svensson, is assumed and then the parameters are calibrated, and the exact-fit, where a number of pre-selected market instruments is repriced exactly by bootstrapping and then the interpolation is used to obtain the remaining maturities. Ametrano and Bianchetti (2013) also provide details on the baskets of instruments used in the construction of each of the yield curves.
In the subsequent sections we define various probability measures used for pricing of interest rate derivatives in the sequel. In order to do so, we introduce a filtered probability space \((\varOmega , \mathscr {F}, (\mathscr {F}_t)_{0 \le t \le T^{*}}, Q)\), where the filtration \((\mathscr {F}_t)_{0 \le t \le T^{*}}\) is assumed to satisfy the usual conditions. All price processes introduced in the remainder of the chapter are defined on this probability space and adapted to the filtration \((\mathscr {F}_t)_{0 \le t \le T^{*}}\). We shall use the notation X, \((X_t)_{0 \le t \le T^*}\) or simply \(X_t\) to denote a stochastic process.
1.3.1 Choice of the Discount Curve
In the presence of multiple curves, the choice of the curve for discounting of the future cash flows, and a related choice of the standard martingale measure used for pricing (in other words, the question of absence of arbitrage), becomes non-trivial. One could possibly choose a different discounting curve depending on the tenor of the underlying interest rate and consider each x-tenor market as a separate market. However, note that this requires imposing in addition certain relations that ensure the absence of arbitrage between these markets that are interconnected by means of interest rate derivatives whose payments depend on more than one tenor simultaneously. The other possibility is to choose a common discounting curve that will apply to all future cash flows, regardless of their tenor. In fact, this is the choice that has been widely accepted and became practically standard, with the OIS discount curve (i.e. the discount curve stripped from the OIS rates) as the common discount curve, cf. also the comments in Remark 1.2. One of the main arguments justifying this choice, which is typically evoked, is the fact that in practice the majority of traded interest rate derivatives are nowadays being collateralized and the rate used for remuneration of the collateral is exactly the overnight rate, which is the rate the OIS are based on. Moreover, the overnight rate bears very little risk due to its short maturity and therefore can be considered relatively risk-free. For more detailed discussions on this issue we refer to Ametrano and Bianchetti (2013), Filipović and Trolle (2013) and Hull and White (2013).
A formal derivation of the OIS discount curve will be presented in Sect. 1.4.4. In the remainder of the book we shall assume that the discount curve is the OIS discount curve \(T \mapsto p^{OIS}(t,T)\) for any t and, in order to simplify the notation, we shall just use p(t, T). We shall call OIS bonds the, in general hypothetical, bonds with price \(p(t,T)=p^{OIS}(t,T)\). These bonds are not necessarily traded since they correspond to OIS rates that are based on an averaging procedure.Footnote 5 In the literature they are however often assumed to be tradable assets as e.g. in Mercurio (2010a).
From the OIS bonds p(t, T) we may formally derive corresponding instantaneous forward rates via the classical relationship
and from here then obtain the spot rate \(r_t=f(t,t)\) that we shall call the OIS short rate. In practice, this rate will be approximated by the overnight rate that corresponds to the shortest available tenor.
1.3.2 Standard Martingale Measure and Forward Measures Related to OIS Bonds
Given the OIS short rate \(r_t,\) we may define in the usual way the corresponding money market account as
We consider as standard martingale measure a probability measure Q, equivalent to the physical measure P, under which all traded assets, discounted by B as numéraire, are (local) martingales.
By analogy to the classical bond price formula we now postulate for the OIS bonds the relationship
which implies that the process \(\left( \frac{p(t,T)}{B_t}\right) _{t\le T}\) is, for each T, a Q-martingale. The meaningfulness of the above relationship between \(r_t\) and p(t, T) stems from the fact that, whenever the OIS bonds are actually traded, their prices should be arbitrage-free. Formula (1.13) for p(t, T), viewed as discount curve, can also be found in Kijima et al. (2009). Notice, furthermore, that formula (1.13) corresponds to formula (3), namely \(P_c(t,T)=E^Q\left\{ \exp \left[ -\int _t^Tr_c(s)ds\right] \mid \mathscr {F}_t\right\} \) in Filipović and Trolle (2013), which gives the price \(P_c(t,T)\) of a collateralized zero coupon bond when the collateral rate \(r_c\) is the overnight rate (see also Piterbarg 2010 and Fujii et al. 2011). This is typically the case in practice as mentioned in Sect. 1.3.1. In this sense the OIS bonds correspond to collateralized zero coupon bonds that may actually be traded.
Since the process \(\left( \frac{p(t,T)}{B_t}\right) _{t\le T}\) is a Q-martingale, we can use it as density process for an equivalent measure change. In fact, we may now introduce the standard forward martingale measures defining, for a generic \(T \in [0, T^*],\) the T-forward measure \(Q^T\) as given by
Note that the forward measure \(Q^{T}\) is associated to the OIS bond \(p(\cdot , T)\) as numéraire, hence the density process is a ratio of the two numéraires. Moreover, the link between two forward measures associated to the dates \(T, S \in [0, T^*]\) is given by
The forward martingale measures are particularly relevant in the context of interest rate derivative pricing, but also for the direct modeling of the forward interest rates; see the seminal paper by Geman et al. (1995), where the idea of changing a numéraire (and thus changing a measure) has been first proposed in financial modeling as a tool in asset pricing.
1.4 Interest Rate Derivatives
In this section an overview of the most standard interest rate derivatives is given with precise definitions and connections between different derivatives. Moreover, the quantities which may serve as building blocks for pricing models presented in the sequel are identified.
A. Linear Derivatives
We begin by presenting the linear interest rate derivatives such as coupon bonds, forward rate agreements and various types of interest rate swaps. A reader familiar with the basics of the interest rate theory will know that before the crisis the prices of these derivatives were given simply as linear combinations of zero coupon bond prices. Now they are functions of both OIS bond prices and forward Libor rates.
1.4.1 Forward Rate Agreements
Definition 1.3
A forward rate agreement (FRA) is an OTC derivative that allows the holder to lock in at any date \(0 \le t \le T\) the interest rate between the inception date T and the maturity \(S > T\) at a fixed value R. At maturity S, a payment based on R is made and the one based on the relevant floating rate (generally the spot Libor rate L(T; T, S)) is received. The notional amount is denoted by N.
As discussed at the beginning of the chapter, before the crisis the FRA rate was exactly the rate given by (1.4), where the last equality results from a no-arbitrage argument in which one “locks-up” a rate between T and S by buying and selling bonds of maturities T and S. Following a widely accepted practice in the post-crisis literature, we now define the post-crisis forward OIS rate, based on the OIS bond prices p(t, T), as the discretely compounded forward rate given by
Notice that, although (1.16) coincides with the last expression in (1.4), the bonds there are the pre-crisis zero coupon bonds, while here they are the OIS bonds. Sometimes, the forward OIS rates F(t; T, S) are also denoted by \(L^D(t;T,S)\) to make explicit the relation to the discount curve. In the pre-crisis framework, the forward Libor rate L(t; T, S) was assumed to be free of interbank risk and thus to coincide with the forward OIS rate, namely the following equality was supposed to hold
Coming back to the FRA rates implied by FRA contracts on the spot Libor rate L(T; T, S), recall that the spot Libor rate is no longer assumed to be free of various interbank risks and thus is no longer connected to the OIS bonds, i.e.
The payoff of the FRA with notional amount N at maturity S is equal to
where L(T; T, S) is the T-spot Libor rate for the time interval [T, S]. Thus, the value of the FRA at time \(t \le T\) is calculated as the conditional expectation with respect to the forward measure \(Q^{S}\) associated with the OIS bond with maturity S as numéraire and is given by
Hence, the key quantity is the conditional expectation of the spot Libor rate that we denote by L(t; T, S) and define by
As stated in Definition 1.2 we call this quantity the forward Libor rate, but we emphasize again the crucial difference with respect to the classical pre-crisis forward Libor rate, namely the connection to the bond prices which is now lost:
The value of the FRA at time t is then simply given by
and the forward rate \(R_t\) implied by this FRA at time \(t\le T\), i.e. the rate R such that \(P^{FRA}(t; T, S, R, N)\) = 0, is obviously equal to L(t; T, S).
Remark 1.3
We mention here that the traded FRA contracts are in fact defined in a slightly different way. More precisely, the payoff of the market FRA (as opposed to the standard, textbook FRA defined above) is given by
and is paid at the beginning of the reference interval, i.e. at time T (in contrast to the payment at time S in the case of the standard FRA). Intuitively speaking, the payoff of the market FRA equals the payoff of the standard FRA paid at T instead of S, where the amount is discounted by a discount factor coming from the Libor curve \(\frac{1}{1 + (S-T)L(T; T, S)}\) (and not from the OIS curve). Obviously, in the pre-crisis setup the market and the standard FRA definitions were equivalent, as it can be easily checked by a simple calculation.
In the sequel, when writing only FRA, we shall always mean the standard FRA. The difference in definitions should be kept in mind when calibrating a model to market data, although it has been pointed out by e.g. Mercurio (2010b) that the actual difference in value of the contract is small enough to be neglected.
Remark 1.4
The forward Libor rate together with the OIS bond prices are the building blocks for the prices of other linear interest rate derivatives such as various types of swaps. Many models in the recent literature consider either directly the forward Libor rate, or one of the related spreads, as a modeling object. These are the models in the spirit of the classical Libor market models, which are treated in Chap. 4. Another approach is to focus on the pre-crisis connection of the spot Libor rates and bond prices and introduce the following relation:
and set according to Definition 1.2
where \(\bar{p}(T, S)\) can be interpreted as price of a fictitious risky bond that is supposed to be affected by the same risk factors as the Libor rate. Here, we kept the classical formal relationship between the Libor rates and the bond prices, but replaced the prices p(T, S) in the classical relationship by the prices \(\bar{p}(T,S)\) of the fictitious bonds. Note that these fictitious bonds are not traded assets, but can be considered as being issued by an average Libor bank, see Ametrano and Bianchetti (2013) and Morini (2009). This is why these bonds are referred to as the Libor bonds by some authors. In Gallitschke et al. (2014) they are called interbank bonds, since interbank cash transactions can be represented as interbank bonds. The models for these bonds are then specified by specifying the dynamics of the process \((\bar{p}(t, S))_{0 \le t \le S}\) either directly (HJM approach, see Chap. 3), or via a suitable short-rate process (for the short-rate approach see Chap. 2).
1.4.2 Fixed and Floating Rate Bonds
Definition 1.4
A fixed rate bond (fixed rate note) is a financial instrument offering to its holder a stream of future payments called coupons. Denoting by \(0 \le T_0 < T_1 < \cdots < T_n\) a discrete tenor structure with \(\delta _k = T_k - T_{k-1}\) and by N the notional amount, the fixed rate bond pays out the amount \( N \delta _k c_k\) at date \(T_k\), for \(c_k \in (0, 1)\) and \(k = 1, \ldots n\). The notional amount N is paid in addition to the coupon payment at maturity \(T_n\). In a floating rate bond (floating rate note) the coupon payments are based on a floating rate (generally the spot Libor rate for a given period), i.e. the floating rate bond pays out the amount \(N \delta _k L(T_{k-1}; T_{k-1}, T_{k})\) at date \(T_k\), where \( L(T_{k-1}; T_{k-1}, T_{k})\) is the spot Libor rate fixed at \(T_{k-1}\) for the period \([T_{k-1}, T_k]\) with \(k = 1, \ldots n\). In the sequel we shall use the shorthand notation \(Q^{k}=Q^{T_k}\) for the forward measures.
The price at time \(t \le T_0\) of the fixed rate bond is given by
Similarly, the price of the floating rate bond at time \(t \le T_0\) can be expressed as
We recall that, before the crisis, the price of the floating rate bond was simply
due to the pre-crisis connection between the forward Libor rates and the bond prices as specified in (1.6). The third equality follows by cancellations in the telescopic sum. This means that the spot starting floating rate bond was worth par, i.e. \(p^{float}(T_0, T_n) = N\).
1.4.3 Interest Rate Swaps
In full generality, a swap is a financial contract between two parties to exchange one stream of future payments for another one.
Definition 1.5
An interest rate swap is a financial contract in which a stream of future interest rate payments linked to a pre-specified fixed rate denoted by R is exchanged for another one linked to a floating interest rate, based on a specified notional amount N. The floating rate is generally taken to be the Libor rate, with various possible conventions concerning the fixing and the payment dates. The swap is initiated at time \(T_0 \ge 0\) and \(T_1 < \cdots < T_n\), where \(T_1 > T_0\), denote a collection of the payment dates, with \(\delta _{k} := T_{k} - T_{k-1}\), for all \(k=1, \ldots , n\).
Note that in this book we shall always use the convention where the floating rates are fixed in advance and the payments are made in arrears. Moreover, note also that there exist other possible choices for the floating rate besides the Libor rates. One such example is treated below, namely the overnight indexed swap (OIS), in which the floating rate is obtained by compounding the overnight rates. Further examples include constant maturity swaps, in which the floating rates are market swap rates of Libor-indexed swaps.
We recall that if the fixed rate is paid and the floating rate is received, the swap is called a payer swap, as opposed to a receiver swap, where the fixed rate is received and the floating rate is paid. If not specified otherwise, we shall always consider a payer swap. The time-t value of the swap, where \(t \le T_0\), is given as a difference of the time-t values of the floating leg and the fixed leg and is equal to
where \(L(t; T_{k-1}, T_k)\) is given by (1.18), for every \(k=1, \ldots , n\). The swap rate \(R(t; T_0, T_n)\) is the rate that makes the time-t value \(P^{Sw}(t; T_0, T_n, R, N) \) of the swap equal to zero and it is easily seen that it is given by
i.e. the swap rate is a convex combination of the forward Libor rates, with weights \(w_{k} := \frac{\delta _{k} p(t, T_k)}{\sum _{i=1}^{n} \delta _{i} p(t, T_i)}\), \(k=1, \ldots , n\), which are functions of the OIS bond prices.
Remark 1.5
In practice the floating rate payments and the fixed rate payments of the swap defined above typically do not occur with the same frequency, as we have assumed to simplify the notation. For example, in the European markets, the fixed leg payments typically occur on a one-year tenor structure, whereas the floating rate payments adopt the tenor of the underlying Libor rate (from one month to three months and up to one year). In that case one has to work with two different tenor structures and modify the above formulas accordingly. In particular, if we denote by \(\mathscr {T}^x\) the tenor structure for the floating rate payments and by \(\mathscr {T}^y\) the tenor structure for the fixed rate payments, the time-t value of this interest rate swap is given, with some abuse of notation concerning the symbol \(P^{Sw}\) for the swap value, by
which follows exactly by the same reasoning as (1.27). The corresponding swap rate is given by
1.4.4 Overnight Indexed Swaps (OIS)
In an overnight indexed swap (OIS) the counterparties exchange a stream of fixed rate payments for a stream of floating rate payments linked to a compounded overnight rate. Let us assume the same tenor structure as in the previous subsection is given and denote again the fixed rate by R. The time-t value \(P^{OIS}(t; T_0, T_n, R, N)_{fix}\) of the fixed leg payments is given by
whereas to compute the value of the floating leg \(P^{OIS}(t; T_0, T_n, R, N)_{float}\) we proceed as follows: the floating rate for each interval \((T_{k-1}, T_k]\) is given by simply compounding the overnight rates between these two dates, i.e.
where \(T_{k-1} = t^{k}_{0} <t^{k}_{1} < \cdots < t^{k}_{n_k} = T_k\) is the subdivision into dates of the fixings of the overnight rate (i.e. working days) and \(\delta _{t^{k}_{j-1}, t^{k}_{j}}:= t^{k}_{j} - t^{k}_{j-1}\), with \(R^{ON}(t^{k}_{j-1}, t^{k}_{j})\) denoting thus the overnight rate for the period \((t^{k}_{j-1}, t^{k}_{j}]\). The payment based on the discretely compounded rate \(R^{ON}(T_{k-1}, T_{k})\) is made at \(T_k\). In order to find the value at time t of this payment, we proceed with a calculation inspired by the one in Ametrano and Bianchetti (2013). The overnight rate \(R^{ON}(t^{k}_{j-1}, t^{k}_{j})\) is assumed to be linked to the OIS bond prices via the classical pre-crisis forward rate formula
We refer to Filipović and Trolle (2013, Sect. 2.5) for a derivation of the above formula based on continuous compounding of the instantaneous rate approximating the overnight rate. Hence, the value of the floating leg is given by
where
The third equality follows from (1.29) and the fourth one is based on a sequence of subsequent measure changes from \(Q_{T_k}\) to \(Q_{t^{k}_{j}}\), for \(j= n_k-1, \ldots , 0\) (recall that \(T_k = t^{k}_{n_k}\)). To be more precise, in the first step, making use of the density between the forward measures \(Q_{t^{k}_{n_k}}\) and \(Q_{t^{k}_{n_k-1}}\) given in (1.15) and applying the abstract Bayes rule, we have
Repeating the same procedure, we obtain the following telescopic product
which concludes the derivation of (1.30). Consequently,
where the second equality follows by cancellations in the telescopic sum.
Therefore, the time-t value, for \(t \le T_0\), of the payer OIS (i.e. the OIS in which the floating rate is received and the fixed rate is paid) is given by
The OIS rate \(R^{OIS}(t; T_0, T_n)\), for \(t \le T_0\), is the rate R such that the value of the OIS at time t is equal to zero, i.e. \(P^{OIS}(t; T_0, T_n, R, N) =0.\) It is given by
and coincides with the classical pre-crisis swap rate, compare also Filipović and Trolle (2013, Sect. 2.5, Eq. 11).
Remark 1.6
Note that the OIS discount curve is obtained by stripping the OIS bond prices based on the expression (1.32) for the OIS rate, see Remark 1.2. More precisely, assuming a collection of market quotes for OIS rates of overnight indexed swaps with various lengths is given, the relationship (1.32) allows to obtain the OIS bond prices \(p(t, T_k)\) by solving a corresponding system of equations.
The additive spot Libor-OIS spread at time T, for the interval \([T, T + \varDelta ]\), where \(T \ge 0\) and \(\varDelta > 0\), is thus given by
where we have used (1.32) with a single payment date. Note that since a swap with a single payment date is in fact an FRA, we have \(R^{OIS}(T; T, T+\varDelta ) = F(T; T, T+\varDelta )\), where \(F(T; T, T+\varDelta )\) is a discretely compounded forward rate from Eq. (1.16). Even though it is not directly observable in the market, for modeling purposes, a multiplicative spot Libor-OIS spread sometimes turns out to be more convenient:
see Henrard (2014) and Cuchiero et al. (2015). Note here that the quantity in the numerator above is exactly the inverse of the Libor discount factor mentioned in connection to market FRAs in Remark 1.3. Therefore, multiplying the payoff of the market FRA \(P^{mFRA}(T; T, S, R, N)\) by the multiplicative Libor-OIS spread \(\varSigma (T;T, T+ \varDelta )\) one can express it as a payoff of the standard FRA discounted with an OIS discount factor.
Similarly, the additive forward Libor-OIS spread at time \(t \le T\), for the interval \([T, T + \varDelta ]\), is given by
and the multiplicative forward Libor-OIS spread is
The Libor-OIS swap spread at time \( t \in [0, T_0]\) is by definition the difference between the swap rate (1.28) of the Libor-indexed interest rate swap and the OIS rate (1.32) and is given by
Remark 1.7
Similarly to Remark 1.5, in practice the floating rate payments and the fixed rate payments of the OIS take place on different tenor structures. Hence, if we denote by \(\mathscr {T}^x\) the tenor structure for the floating rate payments and by \(\mathscr {T}^y\) the tenor structure for the fixed rate payments, the time-t value of this OIS is given by
which follows exactly by the same reasoning as (1.31). The corresponding swap rate is given by
1.4.5 Basis Swaps
A basis swap is an interest rate swap, where two floating payments linked to the Libor rates of different tenors are exchanged. For example, a buyer of such a swap receives semiannually a 6m-Libor and pays quarterly a 3m-Libor, both set in advance and paid in arrears. Note that there also exist other conventions regarding the payments on the two legs of a basis swap. Below we give a definition of a generic basis swap.
Definition 1.6
A basis swap is a financial contract to exchange two streams of payments based on the floating rates (typically Libor rates) linked to two different tenor structures denoted by \(\mathscr {T}^1 = \{T^1_0 < \cdots < T^1_{n_1} \} \) and \(\mathscr {T}^2 = \{T^2_0 < \cdots < T^2_{n_2}\}\), where \(T^1_0 = T^2_0 \ge 0\), \(T^1_{n_1} = T^2_{n_2}\), and \(\mathscr {T}^1 \subset \mathscr {T}^2\). The notional amount is denoted by N, \(T^1_0 = T^2_0\) is called the initiation date, \(T^1_{n_1} = T^2_{n_2}\) the maturity date of the basis swap and the first payments are due at \(T^1_1\) and \(T^2_1\), respectively.
The time-t value of the basis swap is, for \(t\le T_0^1= T^2_0,\) given by
Thus, we have
where \( L(t; T^x_{k-1}, T^x_{k})\) is given by (1.18), for each tenor structure \(\mathscr {T}^x\), \(x=1,2\) and \(k=1, \ldots , n_x\).
Note that in the classical one-curve setup the time-t value of such a swap is zero, whereas since the crisis markets quote positive basis swap spreads that have to be added to the payments made on the smaller tenor leg. More precisely, recalling that the smaller tenor leg corresponds to \(\mathscr {T}^2\), the floating interest rate \(L(T^2_{j-1}; T^2_{j-1}, T^2_j)\) at \(T^2_j\) is replaced by \(L(T^2_{j-1}; T^2_{j-1}, T^2_j) + S\), for every \(j=1, \ldots , n_2\), where S is the basis swap spread. The value of the basis swap with the added spread S is denoted by \(P^{BSw}(t; \mathscr {T}^1, \mathscr {T}^2, S, N)\) and is given by an expression analogous to (1.39), which follows from (1.38), where \(L(T^2_{j-1}; T^2_{j-1}, T^2_j)\) is replaced by \(L(T^2_{j-1}; T^2_{j-1}, T^2_j) + S\), for every j. The fair basis swap spread \(S^{BSw}(t; \mathscr {T}^1, \mathscr {T}^2)\) at the time t when the swap is contracted is the spread S which makes the t-value of the swap equal to zero, i.e. it results from solving \(P^{BSw}(t; \mathscr {T}^1, \mathscr {T}^2, S, N) = 0\) and is given by
We may check that the value of the basis swap in a pre-crisis one-curve setup is indeed zero. To this purpose recall first that in this setup the pre-crisis forward Libor rates, which in (1.6) were defined using the risk-free zero coupon bonds as \(\left( L(t; T, T+\varDelta ) = \frac{1}{\varDelta } \left( \frac{p(t, T)}{p(t, T+\varDelta )} - 1\right) \right) _{0 \le t\le T}\), are martingales under the corresponding forward measures. We thus have
by the assumptions \(T_0^1=T_0^2\) and \(T_{n_1}^1=T_{n_2}^2\).
In the multiple curve setup we cannot use the same calculation, since now the Libor rates are not connected to the bond prices as above. Hence, one ends up with formula (1.39), which in general yields a non-zero value of the basis swap and produces a non-zero basis swap spread (1.40). The market spreads are typically positive, hence multiple curve models are usually constructed in such a way that ensures this property of the model spreads (1.40).
Remark 1.8
Note that the price of a basis swap \(P^{BSw}(t; \mathscr {T}^1, \mathscr {T}^2)\) with tenor structures \(\mathscr {T}^1\) and \(\mathscr {T}^2\) can be expressed as a difference of prices of two interest rate swaps which share the same tenor structure \(\mathscr {T}^3\) for the fixed rate payments and the same fixed rate R, and the floating rate payments are made respectively on the two tenor structures \(\mathscr {T}^1\) and \(\mathscr {T}^2\) of the basis swap. More precisely, we have
where \(P^{Sw}(t; \mathscr {T}^1, \mathscr {T}^3, R, N)\) and \(P^{Sw}(t; \mathscr {T}^1, \mathscr {T}^3, R, N)\) are defined as in Remark 1.5. Clearly, one could take the tenor structure of the fixed leg to coincide with the tenor structure of one of the floating legs, but, as mentioned already, in practice the fixed leg is often paid on a one-year tenor structure (at least in the European markets) and is thus common for both interest rate swaps, but the floating payments are made on two different tenor structures in general. Finally, we want to point out that, based on the representation (1.41), we could follow a slightly different convention concerning the basis swap spread (see Ametrano and Bianchetti 2013), namely by defining
where \(R(t; \mathscr {T}^i, \mathscr {T}^3) \), \(i=1, 2,\) is the swap rate as defined in Remark 1.5. Notice that the difference with the definition in (1.40) consists in the denominator which, in the new formulation, stems from the fixed leg with tenor \(\mathscr {T}^3\).
B. Optional Derivatives
The most common nonlinear interest rate derivatives are caps, floors and swaptions. All these derivatives are of optional nature, therefore we refer to them as optional derivatives. Caps (floors) consist of a series of call (put) options on a floating interest rate and swaptions are options which allow to enter into an underlying interest rate swap of the various types described in the previous part A.
1.4.6 Caps and Floors
Recall that an interest rate cap, respectively floor, is a optional financial derivative defined on a pre-specified discrete tenor structure. The buyer of a cap, respectively floor, has a right to payments at the end of each sub-period in the tenor structure in which the interest rate exceeds, respectively falls below, a mutually agreed strike level K. These payments, made by the seller, are thus given as a positive part of the difference between the interest rate and the strike K for the cap, respectively a positive part of the difference between the strike K and the interest rate for the floor. Every cap, respectively floor, can be decomposed into a series of options applying to each sub-period, which are called caplets, respectively floorlets. Below we thus focus on a single caplet with generic inception date and maturity, whereas the reasoning for a floorlet is completely symmetric.
Definition 1.7
A caplet with strike K, inception date \(T\ge 0\) and maturity date \(T +\varDelta \), with \(\varDelta >0\), on a nominal N is a financial contract whose holder has the right to a payoff at maturity given by \(N \varDelta (L(T; T, T+\varDelta ) - K)^{+}\), where \((L(T; T, T+\varDelta )\) is the spot Libor rate fixed at time T for the time interval \([T, T+\varDelta ]\).
Note that the caplet can be seen as a call option with maturity \(T +\varDelta \) and strike K on the Libor rate, where the Libor rate is known already at time T, i.e. the caplet is said to be settled in arrears (again, similarly to Sect. 1.4.3, we adopt one of several possible settlement conventions). In the sequel we shall assume without loss of generality that the nominal is equal to one, i.e. \(N=1\).
The time-t price, for \(t\le T\), of the caplet is given by
In case of (1.22), we further have
where \(\bar{K}=1 +\varDelta K\).
It is worthwhile mentioning that, when using the representation (1.22), the classical transformation of a caplet into a put option on a bond does not work in the multiple curve setup. More precisely, the fact that the payoff \(\left( (1 + \varDelta L(T; T, T+\varDelta )) - \bar{K} \right) ^+\) at time \(T+\varDelta \) is equivalent to the payoff \(p(T, T+\varDelta )\left( (1 + \varDelta L(T; T, T+\varDelta )) - \bar{K} \right) ^+\) at time T is still valid, since the OIS discounting is used. However, this will not yield the desired cancellation of discount factors. Since the Libor rate depends on the fictitious bonds \(\bar{p}(T, T+\varDelta )\) and the OIS bonds \(p(T, T+\varDelta )\) are used for discounting, we have
which cannot be simplified further as in the one-curve case.
1.4.7 Swaptions
Definition 1.8
Consider a generic fixed-for-floating (payer) interest rate swap with inception date \(T_0\), maturity date \(T_n\) and nominal N. A swaption is an option to enter the underlying swap at a pre-specified swap rate R, called the swaption strike rate, and a pre-specified date \(T \le T_0\) called the maturity of the swaption.
Let us consider the Libor-indexed interest rate swap from Sect. 1.4.3 and assume that the notional amount is \(N=1\). Moreover, for simplicity we choose the maturity of the swaption to coincide with the starting date of the swap, i.e. \(T =T_0\). Therefore, the payoff of the swaption at maturity is given by \(\left( P^{Sw}(T_0; T_0, T_n, R) \right) ^{+} \) and we shall use the shorthand notation \( P^{Sw}(T_0; T_n, R) = P^{Sw}(T_0; T_0, T_n, R)\). The value \(P^{Swn}(t; T_0, T_n, R) \) of the swaption at time \(t \le T_0\) is
The second equality follows from (1.27) and the third one from (1.28), where \( R(T_0; T_0, T_n)\) is the swap rate of the underlying swap at time \(T_0\). Note that the last equality allows to perceive a swaption as a sequence of payments \(\delta _{k}\left( R(T_0; T_0, T_n) - R \right) ^{+}\), \(k=1, \ldots , n\), fixed at time \(T_0\), that are received at payment dates \(T_1, \ldots , T_n\). These payments are equivalent to the payments \( p(T_0, T_k) \delta _{k}\left( R(T_0; T_0, T_n) - R \right) ^{+}\), \(k=1, \ldots , n\), received at \(T_0\), cf. Musiela and Rutkowski (2005, Sect. 13.1.2, p. 482).
To price a swaption, it is convenient to introduce the following process
Because \(A_t\) is a linear combination of OIS prices, the process \(\left( \frac{A_t}{p(t, T_0)}\right) _{t\le T_0}\) is a (positive) martingale with respect to the \(Q^{T_0}\)-forward measure, see Sect. 1.3.2. Thus, \((A_t)_{t\le T_0}\) can be used as a numéraire to define the following change of measure
Changing the measure to the swap measure in (1.43), the price of the swaption can be expressed as a price of a call option with strike R on the swap rate \(R(T_0; T_0, T_n)\):
Remark 1.9
Recalling the definition of the OIS from Sect. 1.4.4, one could also consider an option with maturity \(T=T_0\) to enter in an OIS as an underlying swap. Assume again that the nominal of the underlying OIS is set to \(N=1\). In this case, the price at time t of the corresponding swaption is given by (see 1.31)
where \(c_k= R \delta _k\), for \(k=1, \ldots , n-1\), and \(c_n=1+ R \delta _n\), which can be recognized as a classical pre-crisis transformation of a swaption into a put option with strike 1 on a coupon bearing bond. Note that, as can be easily seen from Eqs. (1.27) and (1.19), such a pre-crisis transformation is no longer available in the post-crisis setup for Libor-indexed interest rate swaps. On the other hand, the expression for the OIS swaption price similar to (1.43)
remains valid.
Remark 1.10
Similarly, a basis swaption is an option to enter into a basis swap. Considering a basis swap defined in Sect. 1.4.5 with nominal \(N=1\) as an underlying basis swap and \(T_0 = T_0^1= T_0^2\) as a maturity date of the option, the price of the basis swaption with strike basis spread S at time t is given by
by Eq. (1.40). Hence, we can define, analogously as for swaptions, a numéraire process
and the corresponding (basis) swap measure
such that the price of the basis swaption can be expressed as a price of a call option on the basis swap spread \(S^{BSw}(T_0; \mathscr {T}^1, \mathscr {T}^2)\):
Notes
- 1.
The figure is taken from http://www.emmi-benchmarks.eu/euribor-org/about-euribor.html.
- 2.
The figures are taken from http://www.emmi-benchmarks.eu/euribor-org/about-euribor.html and http://www.emmi-benchmarks.eu/euribor-eonia-org/about-eonia.html.
- 3.
The figure is taken from Crépey et al. (2012).
- 4.
The figure is taken from Grbac et al. (2014).
- 5.
We would like to thank Darrell Duffie for having clarified to us some issues pertaining to OIS bonds in relation to the standard traded bonds.
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Grbac, Z., Runggaldier, W.J. (2015). Post-Crisis Fixed-Income Markets. In: Interest Rate Modeling: Post-Crisis Challenges and Approaches. SpringerBriefs in Quantitative Finance. Springer, Cham. https://doi.org/10.1007/978-3-319-25385-5_1
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