MSC. FINANCE MASTER THESIS AARHUS

DEP. OF ECONOMICS AND BUSINESS MARCH 2013

PRICING OF SWAPS AND SWAPTIONS

MSC. FINANCE STUDENT: SUPERVISOR:

MIKKEL KRONBORG OLESEN PETER LØCHTE JØRGENSEN

STUDENT NR.: 283755 DEPARTMENT OF ECONOMICS AND BUSINESS

AARHUS SCHOOL OF BUSINESS AND SOCIAL SCIENCES

AARHUS UNIVERSITY Indholdsfortegnelse 1 Preface ...... 4 2 Executive Summary ...... 4 3 Problem statement ...... 4 3.1 Structure ...... 6 3.2 Delimitations and Assumptions ...... 7 3.3 Methods ...... 7 4 Introduction ...... 8 4.1 The Expectation Hypothesis ...... 9 4.2 The liquidity preference hypothesis ...... 10 4.3 Market segmentation theory ...... 10 4.4 The interest reference rates ...... 11 5 The Theoretical basis ...... 12 5.1 Martingality ...... 12 5.2 The equivalent martingale measure ...... 12 5.3 Choice of numeraire ...... 14 5.3.1 The bond price as numeraire ...... 14 5.3.2 An annuity as numeraire ...... 15 5.4 The Pre-crisis pricing of interest rate swaps ...... 17 5.5 The Financial Crisis – How the financial landscape changed ...... 18 5.6 Arbitrage in the pre-crisis framework ...... 20 5.6.1 Liquidity crisis ...... 21 6 Multiple curve pricing of interest rate swaps ...... 22 6.1 The discount curve ...... 23 6.2 The forward curve ...... 23 6.2.1 Short term ...... 23 6.2.2 Medium term ...... 24 6.2.3 Long term ...... 27 6.2.4 Interpolation technique ...... 31 6.3 Pricing of an interest rate swap ...... 32 6.3.1 Eliminating arbitrage in the framework ...... 32 6.3.2 The relation between basis swaps and the FRA rate ...... 34 7 Pricing of options on bonds and interest rates ...... 38 7.1.1 Bond options ...... 38 7.1.2 Caps and Floors ...... 38 7.1.3 Swap options or swaptions ...... 39 7.2 Theoretical justification of the Black (1976) model ...... 39 7.2.1 Solving the partial differential equation ...... 42 7.3 Pricing of interest rate swaptions ...... 48 7.4 Issues related to the use of the Black 1976 model ...... 49 7.5 The Black 1976 model and multiple curves ...... 50 8 Conclusion ...... 51 9 Bibliography ...... 52

Page 2 of 53 List of Figures

FIGURE 1: GLOBAL OTC DERIVATIVES MARKET ...... 9 FIGURE 2: 3-MONTH EURIBOR DEVELOPMENT ...... 19

FIGURE 3: QUOTED FRA VS. IMPLIED FRA ...... 20 FIGURE 4: PLOT OF THE SPOT FORWARD CURVE ...... 30 FIGUR 5: SPOT FIXED POINTS AND SPOT FORWARD CURVE ...... 32

FIGURE 6: BASIS CONSISTENT FRA REPLICATION ...... 36

List of Tables TABLE 1: QUOTED EURIBOR RATES AS OF 10/01/2013 ...... 11

TABLE 2: 3-MONTH EURIBOR RATE QUOTED 10/01/2013 ...... 24 TABLE 3: 3-MONTH EURIBOR FRA'S ...... 25 TABLE 4: 3-MONTH EURIBOR FUTURES ...... 26 TABLE 5: QUOTED EURIBOR SWAP RATES AS OF 10/01/2013 ...... 29

TABLE 6: INSTRUMENTS IN THE SPOT FORWARD CURVE ...... 30

Page 3 of 53 1 Preface This thesis is written for an academic audience, where an understanding of economic terms and knowledge about standard pricing of interest rate derivatives prior to the financial crisis is assumed known. The main focus of this thesis is on new pricing methodologies, which appeared in the aftermath of the financial crisis of 2008-2010. Still economic terms and methodology will be shortly explained as the reader is introduced to it. In order to ease the reading of this thesis the structure have been visualised in appendix A at the end of the thesis. Along with the thesis is a cd containing the most important spread sheets as well as a copy of the thesis. The length of the thesis is within the limits given by Aarhus University - Aarhus School of Business and Social Sciences for a thesis written by 1 person (max. of 60 pages).

2 Executive Summary This thesis has investigated the issues in pricing interest rate swaps and swaptions that arose as a consequence of the financial crisis. The thesis shows how the pricing of swaps has adapted to this new situation using multiple curves instead of a single curve when pricing interest rate swaps. Pricing of swaptions is introduced using Black’s 1976 model. The model is shown to consistent but only with respect to the risk neutral world that it is pricing. The issues’ regarding its use is highlighted. Finally pricing of interest rate swaps and swaptions is discussed highlighting the market trends as discussed in the literature.

3 Problem statement This thesis will investigate the issues relating to pricing of interest rate swaps and swaptions. The main focus of this thesis will be on the problems in pricing interest rate swaps and swaptions that arose as a consequence of the financial crisis and ensuing liquidity crisis. The pricing of interest rate swaps have been complicated by a breakdown of assumptions in the pricing of these products. The assumptions collapsed as a result of raising basis point spreads between Interbank offered rates (Libor, Euribor etc.) of different tenors. Prior to the financial crisis spreads had been largely insignificant. By comparing Libor or Euribor

Page 4 of 53 interest rates with their government counterparties, one would see that there was no significant spread and these rates were therefore considered risk free or at least good proxies for the risk free rate. However, as the financial crisis changed the financial landscape spreads between interbank offered rates and the government rates began to appear suggesting that interbank offered rates where not risk free. This has influenced the pricing of interest rate products significantly and interest rate swaps especially as pricing of such products no longer could be done on just one curve. Previously the interbank offered rates were used to calculate expected future spot rates but also as discounting factor, hence the one curve pricing. This pricing on one curve was based on the assumption that the interbank offered rates were risk free and as that was no longer true multiple curves have to be used. The multiple curves seem to gain more and more popularity in the pricing of interest rate swap after the financial crisis. It was not only the spread between interbank offered rates and government bonds that rose during the crisis, also interbank offered rates with different tenors e.g. 3-month Euribor against the 6-month Euribor began to appear in the markets. There had not been any significant spread between these before the crisis. This tenor spread is important as it distorts the expectation hypothesis. The expectation hypothesis states that bonds are priced so that forward rates are equal to the expected spot rates. This means that the return from holding a long bond should equal the return from investing repeatedly in short-term bonds. These increased spreads in the market have led to a paradigm shift in the pricing of interest rate derivatives such as swaps. Before the crisis pricing was done on a single curve of interest rates. Today, pricing is done using multiple curves in order to capture the larger spreads, which are seen in the market. The forward estimation and discounting is no longer based on the same curve. Problems arise when constructing these multiple curves, as they have to be constructed while taking account of arbitrage, this has led the construction of a multiple curve-pricing framework. Pricing of European swaptions can be conducted by means of an extended Black-Scholes model, which was proposed by Fisher Black in 1976. The model is commonly referred to as Black’s model or Blacks 1976 model. This thesis will be using these names interchangeably. The nice feature of this model is that it gives an analytical formula for pricing European swaptions that is very close to quoted prices even though the model has some very restrictive assumption build into it. The model has become a very important model, as

Page 5 of 53 practitioners often prefers it over more sophisticated, since these sophisticated models can be quite time consuming to calibrate to market prices. The main question of this thesis is:

How are interest rate swaps and swaptions priced in the aftermath of the financial crisis and can the pricing on multiple curves be incorporated into the pricing of swaptions?

The following sub questions will help in answering the main question: How has reference interest rates such as Libor and Euribor, developed during the financial crisis? How has this development influenced pricing of interest rate products and derivatives? Which models can be used to value swaptions? How are interest rate swaps and swaptions affected by changes in the underlying interest rates? How is the multiple curves constructed? How is pricing affected by multiple curve pricing? What are the issues relating to the use of Blacks 1976 model?

3.1 Structure This thesis will examine the impacts of the financial crisis on the pricing of interest rate swaps and swaptions in order to do so the thesis is outlined in the following way. The thesis will start with an introduction to the markets for interest rate swaps and how it has evolved. Furthermore, the reader will be introduced to the different expectation hypothesis’ that underlines the pricing of interest rate swaptions. The next chapter “The theoretical basis” will be explaining the important concepts such as martingality and choices of numeraire. The chapter then moves on to explain the pricing of interest rate swaps in the pre-crisis framework i.e. pricing on a single curve and shows how arbitrage possibilities arose in the single curve framework during the financial crisis. The next chapter “Multiple curve pricing of interest rate swaps” tries to illustrate how the pricing of interest rate swaps is done in the current markets. The chapter will illustrates how a forward curve for the 3-month Euribor rate is constructed, noting that the procedure can be used to construct forward curves for other tenors than the 3-month.

Page 6 of 53 The chapter “Pricing of options on bonds and interest rates” introduces the reader to option theory and especially the theory used to price swaptions. The chapter builds on some of the concepts outlined in the theoretical basis chapter. The chapter will derive the Black 1976 model and show that it can be used to price swaptions. The final part of this chapter will discuss how Black’s 1976 model can be combined with a multiple curve setting. The final chapter will conclude the thesis highlighting the main points of the thesis. Each section will start with a small introduction to that section which will include a review of literature when needed. This should give the reader a good understanding of what he is about to read and open up for the possibility of reading single sections alone. The structure of the thesis is visualised in appendix A.

3.2 Delimitations and Assumptions The thesis is written for an audience with knowledge about pricing interest rate derivatives prior to the financial crisis. However, much of the new methodologies are adaptations of older techniques and will be introduced if needed. The focus of the thesis is however, the new adaptations and their implications for pricing interest rate swaps and swaptions. The thesis will make use of the Euribor interest rate as the main reference rate. However, other reference rates have shown similar developments during the financial crisis and credit crunch.

3.3 Methods The focus of this method is the practical issues relating to pricing interest rate swaps and swaptions after the financial crisis. For the pricing of interest rate swaps the thesis will focus on giving the reader the tools for conducting pricing on multiple curves from a practical point of. For the pricing of swaptions the Black 1976 model was chosen as it has proved itself as a very important market model used extensively by practitioners. The author is aware that the model has its drawbacks and these will be highlighted in order to give the reader a good understanding of the models capabilities and disabilities.

Page 7 of 53 4 Introduction Swaps was introduced in the early 1980’s to restructure assets, obligations and to mitigate and transfer risk between parties that wants to reduce it, to those who want to increase it (Homaifar, 2004). Interest rate swaps are derivatives, this means that they derive value from another asset, in this case an interest rate. The name swap refers to what the product is all about, namely exchanging/swapping, e.g. interest rates. Several other types of swaps do exist though. An interest rate swap contract is a contract wherein two parties agree to exchange interest rate payments based on a notional principal. The initial swaps were currency swaps designed to avoid the control of foreign currency exercised by the British government. Interest rate swaps followed in early 1980’s with the first contract between the World Bank and the company IBM. Since then the international market for swaps has increased significantly as the products have become popular tools in risk management, not only for financial institutions but also among other types of companies (Homaifar, 2004; Kolb, 2003). Kolb, 2003 defines a swap contract as: “A swap is an agreement between two or more parties to exchange a sequence of cash flows over a period in the future”. The bank of international settlements, which is conducting market surveillance on the world wide use of swap agreements defines a swap contract as: “Swaps are transactions in which two parties agree to exchange payments streams based on a specified notional amount for a specified period” (von Kleist & Pêtre, 2012). The market for interest rate swaps has grown significantly since their first introduction. The bank for international settlements is monitoring the global Over-the-counter (OTC) market and publishes semi-annual statistical data on the development in the OTC market, which includes data on swaps. Figure 1 below shows the size of the global swap market and its expansion since 1998 until December 2011. The dark column represent the total market for OTC interest rate contracts, this includes forward rate agreements (FRA), Swaps and options. The light column represents the total amount of notional principal on interest rate swaps. The line indicates how much swap contracts represents out of the total interest rate contracts. The figure shows the extraordinary growth of the global swap markets and how swap contracts have increased in popularity.

Page 8 of 53 Figure 1: Global OTC derivatives market

Global OTC derivatives market Interest rate contracts and swap contracts of notional principal

600.000 82,0% Swap contracts in % 500.000 80,0% 400.000 78,0% 76,0% 300.000 74,0% 200.000 72,0% billion US$ 100.000 70,0%

Notional Principal - 68,0%

Interest rate contracts Swap contracts % Swap Contracts of Notional

Source: Own representation of data from the bank of international settlements

4.1 The Expectation Hypothesis The expectation hypothesis stand for numerous statements that links yields, bond returns and forward rates with different maturities and periods (Sangvinatsos, 2008). The literature is operating with two different terms of expectation hypothesis – Pure Expectation Hypothesis (PEH) and Expectation Hypothesis (EH). The PEH postulates that a) expected excess return on long-term over short-term bonds are zero, b) yield term premia is zero and c) forward term premia is zero(Sangvinatsos, 2008). The EH postulates that aa) expected excess returns are constant over time, bb) yield term premia is constant and cc) forward term premia is constant over time(Sangvinatsos, 2008). The expectation hypothesis emphasises the expected values of future forward rates. By means of the expectation hypothesis the long-term return on a bond can be determined by the return from continuously rolling over short-term bonds throughout the holding period. (Campbell, Lo, & MacKinlay, 1997) defines two types of the expectation hypothesis. The pure expectations hypothesis (PEH) states that: “expected excess returns on long-term over short- term bonds are zero”. The pure expectation hypothesis is a strong form expectation hypothesis. The long rate is determined by the short rate development in the holding period. The expectation hypothesis (EH) states that: “Expected excess returns are constant over time”. This is a weaker form of the expectations hypothesis as it allows for a risk premium for holding long-term bonds. The premium in EH is not a time varying risk premium but is

Page 9 of 53 dependent on the maturity of the bond i.e. a constant premium which size is determined by the maturity of the bond. Criticism of the expectation hypothesis is that empirical research has rejected the hypothesis, as market data does not support the theory. The reason for rejecting the expectation hypothesis has not been fully explained in the literature and recent research suggest that the failure might lie in models ability to predict the development in the short- term interest rates (Guidolin & Thornton, 2008). The expectation theory is an important part of the swap-pricing framework as it allows us to assume that calculated forward rates will equal future spot rates.

Eq. 1. � �, �! = � �, �!!! ∗ � �!!!, �!

4.2 The liquidity preference hypothesis J.R. Hicks advocated the liquidity preference hypothesis, which focuses on the effects of the risk preferences of the market participants (Cox, Ingersoll, & Ross, 1985). Though it does not deny the importance of future spot rates its focus is on understanding the term premium that is required in order for investors to hold longer-term and thereby riskier assets. The hypothesis postulates that this risk aversion between holding short-term and long-term securities will lead to forward rates that are systematically higher than expected future spot rates. (Hull, 2009) advocates this theory, as it is consistent with empirical results as yield curves are more often than not upward sloping.

4.3 Market segmentation theory The market segmentation theory has an alternative approach to the bond market than the above-mentioned theories. This theory states that there may not be a link between short-, medium- and long-term bond markets as investors are operating with a fixed investment horizon and therefore are not likely to shift from a long bond to a short bond or vice versa. The interest rates in short-term markets are thereby determined by supply and demand of that market irrespective of medium- and long-term markets (Hull, 2009). The same goes for the other markets. (Gibson, Lhabitant, & Talay, 2001) give an overview of the different hypothesis’ that are used for understanding the term structure of interest rates.

Page 10 of 53 4.4 The interest reference rates It is the European Banking Federation (EBF) that is calculating and publishing the Euribor rates. The federation consists of members from the national banks of the countries participating in the Eurozone collaboration. The Euribor rates are defined as: “Euribor is the rate at which euro interbank term deposits are offered by one prime bank to another prime bank within the EMU zone and it is published at 11 AM CET for a spot value (T+2).”1 The Euribor Overnight Index Average or Eonia is defined as: “The effective overnight reference rate for the euro”2. Table 1 below presents the daily listings of Euribor interest rates as of 10th of January 2013.

Table 1: Quoted Euribor rates as of 10/01/2013

Name Start date End date Quote, % EONIA 10/01/13 11/01/13 0,069% EURIBOR 1 WEEK 10/01/13 17/01/13 0,080% EURIBOR 2 WEEK 10/01/13 24/01/13 0,088% EURIBOR 3 WEEK 10/01/13 31/01/13 0,098% EURIBOR 1 MONTH 10/01/13 08/02/13 0,110% EURIBOR 2 MONTH 10/01/13 08/03/13 0,155% EURIBOR 3 MONTH 10/01/13 10/04/13 0,190% EURIBOR 4 MONTH 10/01/13 10/05/13 0,234% EURIBOR 5 MONTH 10/01/13 10/06/13 0,283% EURIBOR 6 MONTH 10/01/13 10/07/13 0,325% EURIBOR 7 MONTH 10/01/13 09/08/13 0,364% EURIBOR 8 MONTH 10/01/13 10/09/13 0,404% EURIBOR 9 MONTH 10/01/13 10/10/13 0,437% EURIBOR 10 MONTH 10/01/13 08/11/13 0,478% EURIBOR 11 MONTH 10/01/13 10/12/13 0,517% EURIBOR 12 MONTH 10/01/13 10/01/14 0,550% Source: datastream 28/01/2013 There are similar procedures for the calculation of the Libor rates, which is governed by the British Bankers Association (BBA). The US counterpart is the Federal Funds rate.

1 Euribor homepage 2-2-2013: http://www.euribor-ebf.eu/euribor-org/euribor-rates.html 2 Euribor homepage 2-2-2013: http://www.euribor-ebf.eu/euribor-eonia-org/about- eonia.html

Page 11 of 53 5 The Theoretical basis In order to price interest rate swaps and interest rate options an understanding of numeraires must be present. The numeraire is used to price a product in terms of another. In this section the theoretical basis on, which interest rate, products as swaps and options on interest rates are priced, will be explained. This section will explain the concepts of Martingality, the equivalent martingale measure and choices of numeraire. These concepts are important theories, which underlines the pricing of swaps and interest rate options. The section is mainly based on Hull’s Options, Futures and other derivatives (Hull, 2009) but other literature will be used if found necessary. The development of models for pricing interest rate products and their derivatives is dependent on some sort of expectation about how the world in which they are sold will develop. In order to form a common ground martingales and the equivalent martingale measure is often used. The equivalent martingale measure allows us to value the price of a financial product in terms of another financial product. This fact will come in handy, when deriving the value of interest rate swaps and interest rate options.

5.1 Martingality A variable is a martingale if the expectation of the value of the variable tomorrow is solely determined by the value of the variable today, regardless of the previous development in the variable. This can be expressed as.

Eq. 2. �[�! �!!!, �!!!, … , �!] = �!!!

Where E is an expectations operator. This can be seen as the expected value X at any future time is the value of the variable today. From this result it can be understood that a martingale process is a stochastic process, which has no drift.

5.2 The equivalent martingale measure Consider two financial products, which are traded in the market, pay no dividend and share a single source of uncertainty. We denote them by f & g. If a variable η = f/g then the price of f can be expressed in terms of the prices of g, that is what the equation shows. The product g

Page 12 of 53 is also known as the numeraire in this example. If the processes for f and g are given as below it is possible to show that the ratio η is a martingale for all prices of f.

Eq. 3. �(�) = � + �!�! � �� + �!� �� and

! �(�) = � + �! � �� + �!� ��

These processes are Ito processes where Z is coming from a wiener process. The expected

2 value, µ, of f and g is given by (r+σgσf) and (r+σ g) respectively. By finding the process for η one can show that η is a martingale and thereby proving the equivalent martingale measure. Taking the logarithm of the above processes proves the equivalent martingale measure. By taking the log it should be noted that the processes of f and g transforms into a lognormal process, which is characterised by.

Eq. 4. �! � ln � = � − �� + � �� 2

Using equation 4 and Ito’s lemma to transform the processes for f and g gives.

Eq. 5. �! � ln � = � + � � − ! �� + � ! ! 2 ! and �! � ln (�) = � + ! �� + � �� 2 !

Combining these logarithmic processes and again using Ito’s lemma it can be shown that the process for η is given by.

Eq. 6. � �(�) = (� − � ) �� ! ! �

Page 13 of 53 This shows that η is a stochastic process with a constant volatility and no drift, it is in other words, a martingale. This is an important finding as allows us to price different securities under an assumption that the world is forward risk neutral with respect to g, the numeraire. Since η is a martingale the expected value of f and g at maturity is their value today. This can be expressed mathematically in the following way.

Eq. 7. �! �! = �!�! �!

This expression states that the value today of f is equal to the value of g today multiplied by the expectation, under the forward risk neutral measure g, of fT/gT.

5.3 Choice of numeraire Depending on the product that you want to price it can be very useful to change the numeraire. There are several different types of products that can be used as numeraire, commonly used are money market accounts, zero coupon bond prices and bond prices. This section will focus on the bond price and annuity bond as numeraires, since they will be used in the pricing of swaps and swaptions, respectively.

5.3.1 The bond price as numeraire A forward interest rate can be defined in terms of zero coupon bonds. The forward interest rate for the period between time Ti-1 and Ti can be determined at time t as the following relation by the relation between two zero coupon bonds that mature in time Ti-1 and Ti respectively. Defining the forward as F(t; Ti-1, Ti).

Eq. 8. �(�, �!) F t; T!!!, T! = �(�, �!!!)

This can be rewritten in the following way as.

Eq. 9. 1 � �, �!!! − �(�, �!) F t; T!!!, T! = �! − �!!! �(�, �!)

Page 14 of 53

By letting f and g be defined as

Eq. 10. 1 � = � �, �!!! − � �, �! �! − �!!!

� = �(�, �!)

By inserting this into the equation for η it can be shown that the forward rate F(t;Ti-1,Ti) is a martingale with respect to the forward risk neutral measure P(t,Ti). This finding is an important finding in the pricing of interest rate swaps. The findings allow us to make the crucial assumption that calculated forward rates equal future spot rates.

5.3.2 An annuity as numeraire The annuity as a numeraire is an important part of the understanding of Blacks 1976 model for pricing interest rate options. Many interest rate options involve a string of payments, which is why the annuity factor is so important. Especially, the valuation of swaptions is dependent of this annuity factor. In order to prove that the annuity is useful as a numeraire it is useful to think of an interest rate swap starting in the future where the principal it set to 1. Considering the fixed leg of such a swap and define the forward swap rate as the rate that sets the value of the forward swap equal to zero. The forward swap rate will be defined as SF. The value of the fixed leg is the sum of discount factors multiplied by the forward swap rate. The value of the fixed leg can then be calculated as.

Eq. 11. !!! ����� ��� = �! �!!!, �! �(�, �!!!) !!!

Defining the annuity, A(t), as

Eq. 12. !!! � � = �!!!, �! �(�, �!!!) !!!

Page 15 of 53 It is worth noting that the notation is set such that time t is today and time T0 is the start of the forward swap. The value of the floating leg at time t is given by the following equation.

Eq. 13. �������� ��� = � �, �! − �(�, �!)

When the value of the principal is added to an interest rate swap, here the principal = 1, then the value of the floating leg is set equal to the principal. This comes from the fact that the discount bonds used for calculating the futures value of underlying are the same that are used for discounting(Hull, 2009). Knowing that the value of a swap is found by setting the floating leg = fixed leg the forward swap rate can be found as.

Eq. 14. � �, �! − �(�, �!) �! = !!! !!! �!!!, �! �(�, �!!!)

By following the same steps as when defining the bond price as numeraire it can be shown that the expected future swap rate is the swap rate today. This is done again by defining η as f/g and setting g equal to A(t) and f equal to the floating leg.

Eq. 15. � �, �! − � �, �! � = !!! !!! �!!!, �! � �, �!!!

From this it is straightforward to see that in a world that is forward risk neutral with respect to the annuity, A(t), the expected forward swap rate equals the swap rate today.

Eq. 16. �! = �! �!

Where St is the swap rate observed in the market today, at time t.

The findings in section will be used in the rest of this thesis when valuating swaps and interest rate products. In the rest of the thesis the expectancy operator with respect to the numeraire is left out of the notation for simplicity.

Page 16 of 53 5.4 The Pre-crisis pricing of interest rate swaps The main aspect of pricing interest rate swaps in the pre-crisis framework was construction of a single interest rate curve, which was used for calculating future forward rates and discounting calculated cash flows. (Bianchetti & Carlicchi, 2012) lists the main differences between the old- and new framework, as well as, 4 steps in calculating the value of an interest rate swap. This section is based in a world that is forward risk neutral with respect to the discount bond shown above. The section uses discount bond and zero coupon bonds interchangeably. The 4 steps are elaborated below highlighting the main assumptions behind. 1) The underlying interest reference rate is risk free.

1 1 Eq. 17. �!,!! = − 1 �! �!,!!

Here R(t, Tk) is the risk free spot rate. P(t, Tk) is the time t value of a default free zero coupon bond. τk is the day count for the period t to Tk. In the pre-crisis framework R(t, Tk) equals the reference rate underlying the interest rate swap agreement. For the euro market this is equivalent to say that the Euribor rate is risk free. This is an important assumption as it allows for pricing on 1 curve.

2) Replication of forward rates and calculation of individual cash flows. The calculation of individual cash flows is based on an assumption that forward rates will equal future spot rates. By rearranging eq. 17 in terms of default free discount bond the formula for replicating forward rates is expressed below following(Bianchetti & Carlicchi, 2012).

1 � �, �!!! Eq. 18. �! � = � �; �!!!, �! = − 1 �! � �, �!

By Eq. 18 future cash flows are calculated assuming that forward rates equal future spot rates of the underlying reference rate. The assumption is expressed in Eq. 19.

Page 17 of 53 � �; � , � = � Eq. 19. !!! ! !!!!, !!

3) Discounting individual cash flows

The discount factor equals the price at time t of the risk free zero coupon bond, P(t, Tk). The formula for calculating the different discount factors are derived by isolating P(t, Tk) in Eq. 17.

4) Calculation of the value of the interest rate swap The value of an interest rate swap is then calculated as the present value of expected future cash flows.

Eq. 20. ! !

� � = � � �, �! �! � �! − � � �, �! �! !!! !!! where ! !!! � �, �! �! � �! � = ! !!! � �, �! �!

K is the swap rate, which makes the swap agreement fair at initiation or approximately zero in value.

5.5 The Financial Crisis – How the financial landscape changed In the summer of 2007 the first initial signs of trouble in the financial systems was beginning to show as there were a number of mortgage backed securities that defaulted on their obligations (French, 2010). This was translated into interbank offered rates, such as Euribor, Libor etc. Until this point the Libor and Euribor rates had shown no significant spread between the different tenors and their respective overnight rates.

Prior to the financial crisis reference rates such as Euribor, Libor and the US Fed Funds rate, were considered good proxies for a risk free rate. There was no significant spread between e.g. Euribor rates with different tenors and their corresponding over night rates. Overnight

Page 18 of 53 rates are considered the best proxies for a risk free rate as they are based on lending between prime banks and with a maturity of 1 day (Bianchetti & Carlicchi, 2012). This made pricing of interest rate swaps prior to the financial crisis a straightforward task as calculating future forward rates and discounting could be done based on the same term structure of interest rates. In figure 2 below the 3-month Euribor rate is depicted along with its corresponding Eonia OIS.

Figure 2: 3-Month Euribor development

Source: Datastream 28/1/2013. Own representation.

From figure 2 it can be seen that prior to the financial crisis 6-month Euribor rates exhibited no significant spread towards OIS. Given that the OIS rate, is the best possible proxy for a risk free interest rate the assumption of risk free Euribor rates was in practicality valid. In the summer of 2007 with the emerging troubles in the American mortgage securities a spread between the two appears. The spread has not since then returned to its previous level and the assumption of risk free Euribor rates is therefore not valid any longer. The figure above shows that the assumption implicit in equation 1 no longer holds and the old pricing framework is not valid.

Page 19 of 53 5.6 Arbitrage in the pre-crisis framework Given the assumptions of the previous section replicated forward rates should in theory be very close to market quoted FRA’s as there should be no significant spread among different tenors of Euribor rates and thereby also no significant spread in basis swaps. Looking at Euribor forward rates a significant spread opened up during the crisis. This is important for the pricing of swaps, as a FRA can be considered a very simple interest rate swap as well as pricing of swaps can be conducted by considering a swap as a portfolio of FRA agreements. With the increased spread in Euribor-OIS rates the forward rates saw a similar discrepancy between implied forward rates and market quoted forward rates. This results in a significant violation of the pre-crisis framework because it opens for arbitraging forward rates. In figure 3 below the development in market quoted FRA’s and implied FRA’s calculated from Euribor 3- and 6 month rates using equation 18.

Figure 3: Quoted FRA vs. Implied FRA

Quoted FRA vs implied FRA -base 3- & 6 month Euribor

6,0000%

5,0000%

4,0000%

3,0000%

2,0000%

1,0000%

0,0000% 01/01/07 01/01/08 01/01/09 01/01/10 01/01/11 01/01/12

Market Quoted FRA 3X6 FRA 3X6 Replicated

Source: Datastream 28/1/2013. Own representation.

Figure 3 above shows that replicated FRAs followed market quoted FRAs until the initial turmoil in the summer of 2007 with beginning troubles in US sub-prime loans. The spread is further increased through the financial crisis and credit crunch years 2008-2010. It is important to note that the spread does not seem to decrease or return to pre-crisis levels.

Page 20 of 53 5.6.1 Liquidity crisis With the explosion in spreads on basis swaps and increased spread in Euribor rates over OIS rates several key issues has arisen. As can be seen from figures 2 and 3 above the impact of the financial crisis has been significant on rates such as Euribor. With the financial crisis and the ensuing credit crunch liquidity has become a key issue in maintaining a well functioning market. (Morini, 2009) discusses the development in forward rates and basis swaps during the financial crisis. He argues that short-term lender have an advantage over longer-term lenders because. 1) They are exposed to a lower risk of default, as they have already had one coupon payment if the borrower defaults in the second half of a lending year as opposed to a longer-term lender who will loose the entire coupon. 2) If the lenders credit worthiness decreases they can discontinue lending to them faster than a long-term lender can. This is some of the reasons why a longer-term lender has received positive basis spread when entering a basis swap with a short-term lender.

Page 21 of 53 6 Multiple curve pricing of interest rate swaps This section focuses on pricing of interest rate swaps using the pricing framework that arose as a consequence of the financial crisis and the credit crunch. Several researchers have focused on these changes to the pricing framework. (Morini, 2009) dealt with the gap that opened in interest rates and basis swap quotes in the aftermath of the financial crisis. He shows that FRA’s can be replicated in a basis consistent way by means of basis swaps, in order to ensure the non-arbitrage in the pricing framework. (Marco Bianchetti, 2012) focused on decoupling forward curves from discounting curves like (Morini, 2009) he uses basis swap spreads in his FRA replication but goes further to discuss hedging strategies and pricing plain vanilla interest rate products such as swaps and swaptions. (Fujii, Shimada, & Takahashi, 2010) derives formulas for multiple curves construction introducing cross- currency swaps and collateral into the formulas.

This section will be working with the construction of a multiple curve framework to be used in pricing of interest rate swap. The procedure described here will be based on the article “Two Curves, One Price” by Marco Bianchetti. Other important articles within this field are (Morini, 2009), (Kirikos & Novak, 1997) which will be drawn into the section when needed. The section will start with a derivation of the discounting curve used to discount future cash flows to present values. Second the forward curves will be constructed. The forward curves are used to determine the value of forward rates in order to calculate the amount cash payments in the future. Finally, combining the curves into an interest rate swap pricing formula will conclude this section. (Marco Bianchetti, 2012) defines a 7-step working procedure for calculating the price and hedge ratios for interest rate derivatives using multiple curves. 1) Build one discounting curve 2) Build multiple distinct forwarding curves based on instruments carrying the same underlined reference rate e.g. 3M Euribor. 3) Using the forwarding curves of 2) calculate the relevant forward rates. 4) Calculate the value of the coupon payments on the interest rate derivative. 5) Compute relevant discount factors based on the discounting curve from 1). 6) Compute the price at time t of the derivative based on the discounted cash flows from 5). 7) Compute delta sensitivities in order to hedge the position.

Page 22 of 53 Hedging is beyond the scope of this thesis and will therefore not be dealt with any further.

6.1 The discount curve There can be different approaches to the construction of the discount curve. According to (Marco Bianchetti, 2012) there seem to be two prominent approaches. One approach is to construct the curve from the same instruments as prior to the financial crisis. This will result in a swap curve similar to the pre-crisis swap curve. (Marco Bianchetti, 2012) argues that though this will result in a swap curve based on the most liquid instruments it is also a case of inertia and perhaps expresses a lack of willingness to change to post crisis pricing. The construction based on the most liquid products, will ensure that pricing is very accurate. The second approach is to construct a discount curve based on OIS rates. The arguments for using a discount curve constructed from the OIS rates is that they are the best proxies for a risk free rate and many of the swap contracts being made today are consisting of marking- to-market agreements. This makes the swap contracts very low risk and the OIS rates can therefore be used as discounting.

6.2 The forward curve This section derives the forward curve, which can be used for calculating forward rates. The forward curve will be constructed on the basis of instruments that have the same underlying rate. The example presented here will be based on a forward curve with the 3-month Euribor reference rate as the underlying interest rate. The forward curve will be constructed based on quoted products that are all having the same underlying interest rate. The forward curve is consisting of spot rates and it is these spot rates that are used to calculate forward rates, hence the name forward curve or forward spot curve.

6.2.1 Short term This part of the curve covers the first 3 months. The short-term part of the forward curve is constructed on the basis of normal quotes in the market. Euribor rates are quoted on a daily basis for maturities up to 12 months. Table 2 below presents the rates, which have been used in the construction of the forward curve.

Page 23 of 53 Table 2: 3-month Euribor rate quoted 10/01/2013

Name Start date End date Quote, %

EURIBOR 3 MONTH 10/01/13 10/04/13 0,190% Source: Datastream 28/1/2013

6.2.2 Medium term The medium term is normally defined as the time between 3 months and up to 3 years. The medium term of the forward curve is normally extrapolated from FRA’s and/or futures contracts depending, which contracts are the most liquid. FRA contracts are sold in the OTC market, which gives them flexibility as compared to a futures contract. A FRA contract fixes an interest rate between two different times in the future. The medium term of the forward curve is constructed based on a mixture of FRA’s and futures with the 3-month Euribor as the underlying rate. FRA’s are forward starting contracts and can therefore not be directly used in the forward curve without a transformation. Following (Kirikos & Novak, 1997) a FRA can be transformed into a price of a ZCB maturing in time Ti by the following equation given the previous time price.

�!!(�!, �!!!) Eq. 21. �!! �!, �! = 1 + �!!(�!; �!!!, �!)�!(�!!!, �!)

Here 3M refer to the underlying, which is the 3-month rate; τF is the day count convention for the period between Ti-1 and Ti; F3M is the 3-month forward rate covering the same period. From equation 21 spot rates can be extracted by using equation 17. Table 3 below shows the quoted FRA’s as of 10th of January 2013 and their corresponding spot rates.

Page 24 of 53 Table 3: 3-month Euribor FRA's

Name Start End Quoted Spot rate rate EURO 3 MTH (ICAP/TR) FRA 1X4 08/02/13 10/05/13 0,212% 0,1874% EURO 3 MTH (ICAP/TR) FRA 2X5 08/03/13 10/06/13 0,217% 0,1936% EURO 3 MTH (ICAP/TR) FRA 3X6 10/04/13 10/07/13 0,222% 0,2061% EURO 3 MTH (ICAP/TR) FRA 4X7 10/05/13 09/08/13 0,231% 0,2062% EURO 3 MTH (ICAP/TR) FRA 5X8 10/06/13 10/09/13 0,243% 0,2124% EURO 3 MTH (ICAP/TR) FRA 6X9 10/07/13 10/10/13 0,252% 0,2217% EURO 3 MTH (ICAP/TR) FRA 7X10 09/08/13 08/11/13 0,262% 0,2231% EURO 3 MTH (ICAP/TR) FRA 8X11 10/09/13 10/12/13 0,274% 0,2293% EURO 3 MTH (ICAP/TR) FRA 9X12 10/10/13 10/01/14 0,280% 0,2365% EURO 3 MTH (ICAP/TR) FRA 12X15 10/01/14 10/04/14 0,328% 0,2548% EURO 6 MTH (ICAP/TR) FRA 12X18 10/01/14 10/07/14 0,493% 0,3218% EURO 1 YR (ICAP/TR) FRA 12X24 10/01/14 09/01/15 0,692% 0,4648% Source: Datastream 28/1/2013

Futures contracts differ from FRA’s in that they are sold on exchanges and are highly standardised contracts compared to FRA’s. One significant difference between forward rates and futures are that they are markt-to-marked daily. This eliminates the counterparty risk of a futures contract. 3-month futures are settled and expire 4 times a year in March, June, September and December. The futures contracts are fixed the third Wednesday in each of the four mentioned months. The fact that futures are highly standardised products, which are mark-to-market daily, has ensured that the futures market is a very liquid market (Kirikos & Novak, 1997). This liquidity makes them preferable to FRA contracts. The table below shows the quoted futures as of 10th of January 2013.

Page 25 of 53 Table 4: 3-month Euribor futures

Name Start End Price Conv. Spot Date Date Adj. rate

LIFFE 3 MTH EURIBOR March 2013 20/03/13 19/06/13 99,7050 0,0002 0,275% LIFFE 3 MTH EURIBOR June 2013 19/06/13 18/09/13 99,6150 0,0009 0,315% LIFFE 3 MTH EURIBOR September2013 18/09/13 19/12/13 99,5300 0,0018 0,358% LIFFE 3 MTH EURIBOR December 2013 19/12/13 19/03/14 99,4450 0,0032 0,400% LIFFE 3 MTH EURIBOR March 2014 19/03/14 18/06/14 99,3700 0,0048 0,441% LIFFE 3 MTH EURIBOR June 2014 18/06/14 17/09/14 99,2950 0,0069 0,482% LIFFE 3 MTH EURIBOR September2014 17/09/14 17/12/14 99,2200 0,0093 0,522% LIFFE 3 MTH EURIBOR December 2014 17/12/14 18/03/15 99,1450 0,0120 0,563% LIFFE 3 MTH EURIBOR March 2015 18/03/15 17/06/15 99,0700 0,0151 0,603% LIFFE 3 MTH EURIBOR June 2015 17/06/15 16/09/15 98,9850 0,0186 0,644% LIFFE 3 MTH EURIBOR September2015 16/09/15 16/12/15 98,9000 0,0224 0,687% LIFFE 3 MTH EURIBOR December 2015 16/12/15 16/03/16 98,8000 0,0265 0,731% LIFFE 3 MTH EURIBOR March 2016 16/03/16 15/06/16 98,7050 0,0310 0,777% Source: Datastream 28/1/2013

From table 4 above it can be seen that futures are not quoted based on their interest rate but instead by their price. The forward interest rate implied by a futures contract can be derived by the following equation.

!"# !"# Eq. 22. �!! �!, �!!!, �! = 100 − �!! (�!, �!!!, �!)

The equation defines the forward interest rate, implied by a future at time zero, for the period starting in time Ti-1 and maturing in time Ti. Again the subscript 3M denote the underlying 3-month Euribor interest rate. Marking-to-market complicates usage of futures in estimating the forward curve, as their convexity is different. A drop in the price of an interest rate future result in a gain for an investor long in that contract, as rates will increase. This gain creates a correlation miss- match between the implied forward rate of the futures contract and the standard forward rates, as the correlation to spot rates are different in the two types of contracts. This can be corrected by applying a convexity adjustment. Calculation of a convexity adjustment is not a straightforward task it requires knowledge, about the development in interest rates and especially the volatility of these developments. There are two approaches that can be used in the calculation of the convexity adjustment. (Kirikos & Novak, 1997) calculate the convexity adjustment based on short-term interest

Page 26 of 53 rate models. This requires an iterative procedure were the convexity adjustment is backed out of current market observed prices on interest rate options. (Hull, 2009) advocates a simple method were the correction is calculated using the following equation.

Eq. 23. 1 ���. ���. �; � , � = − �!� � !!! ! 2 !!! !

This requires an estimate of σ, which can be obtained by either estimating σ from historical prices on options or by computing the implied volatility from option prices today. For this example an implied volatility from a 5 year interest rate cap as quoted in the market on the 10th of January 2013 was used. Having calculated the convexity adjustment the forward rate implied by a futures contract is then given by.

Eq. 24. !"# �!! �!, �!!!, �! !"# = 100 − �!! �!, �!!!, �! + ���. ���. �; �!!!, �!

This forward rate can then be transformed into a spot rate by using equation 21 & 17 just as was done with FRA’s. The convexity adjustment and final spot rates can be seen from table 4 above.

6.2.3 Long term The long end of the forward curve covering the period from 3 years up to 30 years is constructed by using swaps quoted on the underlying interest rate. For Euribor interest rates swaps are available for maturity up to 30 years. From these swap rates it is possible to extrapolate a spot rate from time zero to maturity of the swap. Quoted interest rate swaps are based on the 6-month Euribor with one leg paying floating 6-month Euribor semi- annually and one fixed leg paying a fixed rate annually. This fixed rate is quoted in the market. When constructing a forward curve with the 3-month Euribor interest rate as the underlying rate, a problem arises, as interest rate swaps on 3-month Euribor is not quoted in the market. This problem is overcome by introducing basis swaps (Ametrano & Bianchetti, 2009). A basis-swap is that exchanges floating payments of different maturities. A basis-

Page 27 of 53 swap on the 3- and 6-month Euribor is thus an interest rate swap, which pays the 3-month Euribor quarterly on one leg and 6-month Euribor semi-annually on the other leg. Basis- swaps are quoted in the market by the amount of basis points that is needed to make the swap “fair” at initiation. Basis-swaps can be combined with the quoted 6-month swap in order to find the swap rate used for extrapolating a spot rate.

Eq. 25. �!! �!, �! = �!! �!, �! + Δ!! �!, �!

The swap is determined at initiation as the rate that sets the two payment legs equal in value to each other. Knowing that a swap at initiation has zero value and assuming that the payments dates on the two legs are identical it is easy to show that the swap rate S3M.

Eq. 26. ! !!! �!! �!, �! �! �!!!, �! �!! �!; �!!!, �! �!! �!, �! = ! !!! �!! �, �! �! �!!!, �!

For K = S3M(t0,Ti). The swap rate S3M can be analysed further in order to derive the discount factor P(t0, Ti). The derivation requires that the summation be split, separating the last expression from the summation.

�!! �!, �! !!!

= �!! �!, �! �! �!!!, �! �!! �!; �!!!, �! + �!! �!, �!!! !!!

1 − �!! �!, �! !!! !!! �!! �, �!!! �! �!!!, �! + �! �!!!, �! �!! �!, �!

Rearranging equation 22 and isolating �!! �!, �! gives the formula for determining the discount rates associated with these interest rate swaps.

Page 28 of 53 Eq. 27. �!! �!, �! !!!

= �!! �!, �! �! �!!!, �! �!! �!; �!!!, �! + �!! �!, �!!! !!!

!!! 1 − �!! �!, �! �!! �, �!!! �! �!!!, �! 1 + �!! �!, �! �! �!!!, �! !!!

The calculation requires that the discount factors are known for all individual payments, which it is not. Interpolation is therefore required in order to calculate them. In order to calculate the individual discount factors standard linear interpolation was used following. In table 5 below the market quoted swaps as of 10/1/2013 is presented and their corresponding spot rates.

Table 5: Quoted Euribor swap rates as of 10/01/2013

Name Start date End date Quoted rate Spot rate ICAP EURO VS EURIBOR 3M IRS 1Y 10/01/13 10/01/14 0,370% 0,049% ICAP EURO VS EURIBOR 6M IRS 2Y 10/01/13 09/01/15 0,465% 0,567% ICAP EURO VS EURIBOR 6M IRS 3Y 10/01/13 08/01/16 0,585% 0,755% ICAP EURO VS EURIBOR 6M IRS 4Y 10/01/13 10/01/17 0,747% 0,947% ICAP EURO VS EURIBOR 6M IRS 5Y 10/01/13 10/01/18 0,938% 1,140% ICAP EURO VS EURIBOR 6M IRS 6Y 10/01/13 10/01/19 1,131% 1,336% ICAP EURO VS EURIBOR 6M IRS 7Y 10/01/13 10/01/20 1,319% 1,538% ICAP EURO VS EURIBOR 6M IRS 8Y 10/01/13 08/01/21 1,490% 1,744% ICAP EURO VS EURIBOR 6M IRS 9Y 10/01/13 10/01/22 1,635% 1,959% ICAP EURO VS EURIBOR 6M IRS 10Y 10/01/13 10/01/23 1,778% 2,181% ICAP EURO VS EURIBOR 6M IRS 12Y 10/01/13 10/01/25 1,998% 2,460% ICAP EURO VS EURIBOR 6M IRS 15Y 10/01/13 10/01/28 2,224% 2,957% ICAP EURO VS EURIBOR 6M IRS 20Y 10/01/13 10/01/33 2,377% 3,336% ICAP EURO VS EURIBOR 6M IRS 25Y 10/01/13 08/01/38 2,430% 3,662% ICAP EURO VS EURIBOR 6M IRS 30Y 10/01/13 09/01/43 2,450% 3,929% Source: Datastream 28/1/2014

The short-, medium- and long-term of the term structure is presented in table 6, below, showing the chosen instruments that will be used for the construction of the spot forward curve. In figure 4 below is further shown the indicated spot forward curve, which will be finished by using an interpolation scheme.

Page 29 of 53 Table 6: Instruments in the spot forward curve

Name Start End Time to Quoted Spot date date maturity rate rate EURIBOR 3 MONTH 10/01/13 10/04/13 0,3 0,190% 0,190% EURO 3 MTH (ICAP/TR) FRA 3X6 10/04/13 10/07/13 0,5 0,222% 0,206% EURO 3 MTH (ICAP/TR) FRA 6X9 10/07/13 10/10/13 0,8 0,252% 0,222% EURO 3 MTH (ICAP/TR) FRA 9X12 10/10/13 10/01/14 1,0 0,280% 0,237% LIFFE 3 MTH EUR Sep. 2014 17/09/14 17/12/14 2,0 0,789% 0,522% LIFFE 3 MTH EUR Dec. 2014 17/12/14 18/03/15 2,2 0,867% 0,563% LIFFE 3 MTH EUR Mar. 2015 18/03/15 17/06/15 2,5 0,945% 0,603% LIFFE 3 MTH EUR Jun. 2015 17/06/15 16/09/15 2,7 1,034% 0,644% LIFFE 3 MTH EUR Sep. 2015 16/09/15 16/12/15 3,0 1,122% 0,687% ICAP EURO VS EURIBOR 6M IRS 5Y 10/01/13 10/01/18 5,1 0,938% 1,140% ICAP EURO VS EURIBOR 6M IRS 6Y 10/01/13 10/01/19 6,1 1,131% 1,336% ICAP EURO VS EURIBOR 6M IRS 7Y 10/01/13 10/01/20 7,1 1,319% 1,538% ICAP EURO VS EURIBOR 6M IRS 8Y 10/01/13 08/01/21 8,1 1,490% 1,744% ICAP EURO VS EURIBOR 6M IRS 9Y 10/01/13 10/01/22 9,1 1,635% 1,959% ICAP EURO VS EURIBOR 6M IRS 10Y 10/01/13 10/01/23 10,1 1,778% 2,181% ICAP EURO VS EURIBOR 6M IRS 12Y 10/01/13 10/01/25 12,2 1,998% 2,460% ICAP EURO VS EURIBOR 6M IRS 15Y 10/01/13 10/01/28 15,2 2,224% 2,957% ICAP EURO VS EURIBOR 6M IRS 20Y 10/01/13 10/01/33 20,3 2,377% 3,336% ICAP EURO VS EURIBOR 6M IRS 25Y 10/01/13 08/01/38 25,4 2,430% 3,662% ICAP EURO VS EURIBOR 6M IRS 30Y 10/01/13 09/01/43 30,4 2,450% 3,929% Source: Bloomberg/ Datastream 28/01/2013

Figure 4: Plot of the spot forward curve

Spot forward rate

5,000%

4,000%

3,000%

2,000%

1,000%

0,000% 0,3 0,5 0,8 1,0 2,0 2,2 2,5 2,7 3,0 5,1 6,1 7,1 8,1 9,1 10,1 12,2 15,2 20,3 25,4 30,4

Spot forward curve

Source: Bloomberg/Datastream 28/01/2013

Page 30 of 53 6.2.4 Interpolation technique The plotted term structure in figure 4 above is not yet complete, as it can be seen, there are several gaps where it is not possible to find FRA’s, futures or swaps that are liquid enough for the spot curve so these parts of the term structure will have to be interpolated in order to get a smooth curve. There are several different methodologies for interpolating between different points on a curve. One approach is to use linear interpolation like was done when interpolating the missing swap rates. This approach does however, have some significant drawbacks. (Ametrano & Bianchetti, 2009; Marco Bianchetti, 2012) shows that using linear interpolation for creating the spot forward curve will result in a forward curve that is jagged. A jagged forward curve leads to inaccurate result for the individual cash flows in the pricing of an interest rate swap. A simple approach that creates a smooth forward curve is the piecewise cubic polynomial. The piecewise cubic polynomial consist of combing three different cubic splines. This is done in order to ensure that the curve is giving meaningful results throughout the entire length of the term-structure. (Tuckman, 2002) introduces piecewise cubic polynomials. In the construction of the final forward curve piecewise cubic polynomials was used in the following way. The cubic spline was split in three according to maturity. For the time to maturity, t, the three sections were t < 10, 10 ≤ t < 20 and 20 ≤ t ≤ 30. The function repeated in each section was then given by.

! ! Eq. 28. �! = �! + �� + �� + ��

In order to ensure continuity in the curve when the sections changes the first term, �!, is set equal to the last term in the last section. This ensures a continuous smooth forward curve. Figure 5 below depicts the final forward curve.

Page 31 of 53 Figur 5: Spot fixed points and spot forward curve

Spot ixed points and Interpolated spot forward curve

4,500% 4,000% 3,500% 3,000% 2,500% 2,000% 1,500% 1,000% 0,500% 0,000%

Spot ixed points Interpolated spot forward curve

Source: Bloomberg & Datastrean 28/1/2013

6.3 Pricing of an interest rate swap Given the forward curve derived in the previous section it is possible to calculate the forward rates needed in order to calculate the future payments involved in an interest rate swap. In order to calculate the forward rates in a manor that is consistent with the new developments in the financial markets a correction is needed.

6.3.1 Eliminating arbitrage in the framework From figure 3 in chapter 5.6 above it can be seen that FRA’s calculated on 3- and 6 month Euribor rates are significantly higher than market quoted FRA’s, which allows for arbitrage if interest rate swaps are priced according to these new market situations. Inspired by (Mercurio, 2009) it will be shown that in theory it is possible to construct an arbitrage position based on the spread between replicated FRA’s and market quoted FRA’s.

Having two different forward rates where one is higher priced than the other Fd > Fx. This corresponds to the situation in figure 3 above where Fd equals the replicated forward rate and Fx is the market quoted forward rate for the same period. The strategy will then be.

1) buy a position of (1+τ1,2Fd)D(0,T2) = D(0, T1)

Page 32 of 53 where τ1,2 is the day count between time T1 and T2 and D(t, T) is the price of the position in time t with respect to the maturity T. At time T D(0, T1) = 1.

2) Sell a position with maturity T1 = D(0,T1)

3) Enter into a FRA paying out at time T1

� (� � , � − � ) Eq. 29. !,! ! ! ! 1 + �!,!�(�!, �!)

Where E(T1,,T2) is the forward Euribor rate for the period between T1 and T2. This creates an investment position with an initial price of zero.

At time T1 step 2 + 3 gives.

� (� � , � − � ) 1 + � � Eq. 30. !,! ! ! ! − 1 = − !,! ! 1 + �!,!�(�!, �!) 1 + �!,!�(�!, �!)

This will be negative when one assumes that interest rates are positive. In order to cover the debt that has been created the position in 1) is sold giving the following situation.

1 + � � 1 + � � � � − � Eq. 31. !,! ! − !,! ! = !,! ! ! > 0 1 + �!,!�(�!, �!) 1 + �!,!� �!, �! 1 + �!,!� �!, �!

This is a possible way of constructing an arbitrage position, as there is a clear positive gain from a zero investment. Even though it has been theoretically possible to construct these arbitrage strategies during the financial crisis arbitrage did not take place, which indicates that markets was not functioning very well (Mercurio, 2009; Morini, 2009). (Marco Bianchetti, 2012) argues that liquidity and counter party risk played a vital role in explaining why such arbitrage opportunities were not exploited. (Morini, 2009) further analyses the situation arguing that liquidity or lack of it was the main reason for such in-effective markets. (Morini, 2009) shows that it is possible to replicate quoted FRAs by using replicated FRA with an added spread derived from basis swaps. Inspired by (Morini, 2009) it will be shown how to construct implied forward rates that follow quoted FRA prices very well.

Page 33 of 53 Using 3- and 6 month Euribor rates and quoted basis spreads from a 1-year basis swap of 3 for 6 month Euribor it is possible to construct a framework, which follows the market quoted FRA’s on a satisfactory level.

6.3.2 The relation between basis swaps and the FRA rate The following is inspired by the article by (Morini, 2009), in his article Morini shows that the gap, displayed in figure 3, between market quoted FRA’s and replicated FRA’s is highly dependent on quoted basis swaps. It is therefore important to understand basis swaps in order to be able to consistently model FRA’s rates. A basis swap or tenor swap is a swap, which exchanges floating payments. It is constructed by combining two swaps that have similar fixed legs but pays different floating legs. The calculations in this section are based on a basis swap with 3 month Euribor against a 6 month Euribor. A basis swap is priced as the expectation of the Euribor dependent payoff discounted by the risk free rate. Basis swaps are quoted in basis points such that the price at time 0, with a maturity of 2α and a spread of Z is given by:

Eq. 32. ����� 0; 2�; � =

�! � 0, � ��! 0, � + � 0,2� ��! �, 2� − � 0,2� 2�(�! 0,2� − �)

Where the expectation at time 0 of E0[D(0,α)] = Pr(0,α) is the price of a discount bond based on a risk free interest rate. The time operator, α, is set to be 3 months and it is assumed that the 3 month period is exactly half of a 6 month period such that 2α equals a 6 month swap. This might not be the case in reality as holidays etc. can influence the amount of days in each period but it is ignored here for the sake of simplicity. EM(0,α) is the market quoted Euribor rate for the period [0;α]. The only uncertain part of equation 32 is the second term in the expectations operator, which allows for the following rewriting.

Page 34 of 53 Eq. 33. ����� 0; 2�; � =

�! � 0,2� ��! �, 2�

1 �!(0, �) 1 − �!(0,2�) − 1 − 2�� − − 1 �!(0,2�) �!(0,2�) �!(0, �)

Where the expectation is the standard market replication of a FRA. Again Pr(…) defines the price of a default free discount bond and PE(…) defines the price of a zero coupon bond calculated on the basis of Euribor rates. By defining � � , the correction to the standard FRA replica, as

Eq. 34. 1 � (0, �) 1 � � = − 1 − 2�� − ! − 1 � �!(0,2�) �!(0,2�) �!(0, �)

Rewriting equation 34 gives a better understanding of how the correction is calculated.

Eq. 35. 1 � 0, � 1 � 0, � � � = − 1 − 2�� − ! + ! � �! 0,2� �! 0,2� �! 0, � �! 0,2� 1 � 0, � 1 � 0, � = − ! + ! − 1 − 2�� � �! 0,2� �! 0,2� �! 0, � �! 0,2� 1 � 0, � � 0, � � 0, � = ! − ! + ! − 1 − 2�� � �! 0, � �! 0,2� �! 0,2� �! 0,2� 1 = �!"# 0; �; 2� + − 1 �!"#(0; �; 2�) − �!"#$(0; �; 2�) − 2� �!(0, �)

The final formula in Eq. 8 can be used to estimate basis consistent forward rates by setting Z equal to market quoted basis spreads. By combining equations 34 and 35 we can produce the price of a basis swap as.

Eq. 36. ����� 0; 2�; � = �! � 0,2� ��! �, 2� − �!(0,2�)�(�)�

If one compares equation 36 to the payoff derived from a FRA it is possible to see that the price of a FRA is equal to the price of a basis swap. The price of a FRA as the one presented in equation 18 is given by.

Page 35 of 53

Eq. 37. ��� 0; �; 2�; � = �! � 0,2� ��! �, 2� − � 0,2� ��

Remembering that the expectation of D(0,2α) equals Pr(0,2α) it is straightforward to see that equation 36 and 37 are identical when K = �(�). �(�) is the equilibrium spread in a basis swap. This allows us to consistently replicate forward rates when there are significant spreads observed in the market using only market quoted basis spreads, Euribor rates and Euribor OIS. This an important finding for pricing of swaps in the post crisis framework as it avoids the possibility of arbitrage in the pricing framework of interest rate swaps. The ability of Eq. 37 to construct basis consistent FRA’s is shown in the following figure.

Figure 6: Basis consistent FRA replication

Basis consistent FRA replication

6,0000%

5,0000%

4,0000%

3,0000%

2,0000%

1,0000%

0,0000%

Market Quoted FRA 3X6 FRA 3X6 Replicated FRA 3X6 Basis consistent replication

Source: Bloomberg/datastream 28/1/2013. Own representation

Figure 6 above shows that basis consistent FRA’s are lining up against market quoted FRA’s.

With these corrections to the forward curve it is now possible to define the equations used to value interest rate swaps.

Page 36 of 53 ! !

Eq. 38. � � = � �! �, �! �! � �! − �!! �! �, �! �! !!! !!!

Equation 38 shows the value of an interest rate swap at time t depending on the pricing on multiple curves. We have that N is the notional principal and the subscripts d and f denotes discount curve and forward spot curve respectively. τk denotes the day count between the periods k-1 and k and similar for τj, which denotes the time interval between j-1 and j. S3M is the swap rate for this interest rate swap with 3M denoting that the underlying interest rate is the 3-month Euribor interest rate.

One could correct the pricing shown in this chapter even further by taking into consideration multiple currencies. This would protect the pricing against arbitrage from arbitrageurs moving to exploit the exposure to foreign interest rates. This issue is not considered in this thesis as practice has shown that these kinds of imperfections have not been corrected in the market. The literature gives to possible explanations for this either because there has not been the financial instruments to exploit the arbitrage possibility or because liquidity has not been sufficient (Marco Bianchetti, 2012; Morini, 2009).

Page 37 of 53 7 Pricing of options on bonds and interest rates This section will cover the pricing of interest rate swap options. The chapter is based on the finding of (Black & Scholes, 1973) and (Black, 1976). The chapter will start by an introduction to swaptions. The chapter will then move on to introduce the model by Fisher Black (1976), which gave an analytical solution to a wide range of European options on bonds and interest rates. Like swaps, interest rate options are dependent on the expected development in interest rates but also the volatility or fluctuations in these interest rates. Valuation of interest rate products such as caps/floors, bond options and swaptions is difficult when compared to valuation of derivatives dependent on stocks or currencies (Hull, 2009). The behaviour of interest rates is considerably more complex than the behaviour of stocks and currencies. Interest rates are often significantly impacted by policy changes, macro economic shocks etc. Furthermore it is often necessary to know the development in yield curves in order to price options on bonds and interest rates. This also requires knowledge about the volatility of these yields, which can be different dependent the point in time the yield is covering. Interest rate caps/floors, bond options and swaptions are some of the most popular OTC interest rate products (Hull, 2009). In the following a brief description of them is given.

7.1.1 Bond options A bond option is a simple option on a bond. A bond option gives the holder the right, but not the obligation to buy or sell a bond at pre-specified price and at a specified price (Hull, 2009). These options are often embedded in bonds in order to make them more appealing to potential investors. The majority of sold bond options are European options, meaning that they can only be exercised at maturity.

7.1.2 Caps and Floors Caps and floors are instruments, which, as their names suggest, limits the movements in interest rates. A cap ensures that the interest rate on a floating rate loan will not exceed a certain level. For interest rates higher than the given level the interest rate is capped. A floor is the opposite of a cap ensuring that the level of interest cannot fall below a certain floor.

Page 38 of 53 7.1.3 Swap options or swaptions A swap option or swaption is the option to enter into a swap contract. A swaption gives the holder the possibility to lock in a specified rate, should the development in interest rates make it favourable. A standard call swaption gives the holder the right to pay a fixed interest rate and receive a floating rate. The swaption consists of two parts an interest rate swap and an option. The two products are placed in time chronological order, since when the option matures the holder either choses to enter into an swap contract starting at option maturity or not. A forward swap is a contract, much like the well-known forward rate contracts of interest rates, where the contract does not start until a pre-specified point and at a pre- specified swap rate. An At-the-money (ATM) swaption is therefore a swaption where the swaption swap rate equals the forward swap rate for a swap starting at the same time.

7.2 Theoretical justification of the Black (1976) model In his 1976 article “The Pricing of Commodity Contracts” Fisher Black proposed a model that could be used to calculate an analytic solution for valuing interest rate options. This model has become a well-established model within the financial community and is often used by practitioners. The model has come to be known as the Black model. The model is an extension of the Black-Scholes model (Black & Scholes, 1973), which has revolutionised the pricing of options on stocks and currencies. The Black model can be used to find the value of interest rate derivatives, such as those described above. The model though is static, which means that it cannot ensure no-arbitrage between different products (Hull, 2009). The model is adapted to the specific product that it values under some forward expectation, which changes according to the different products. This will be further explained in valuing the different products. In order to fully understand the Black model it is important to understand the dynamics in the derivation of the Black-Scholes model. The Black and the Black-Scholes models both have an underlying which have the following characteristics. The process followed by the underlying, S, is.

Eq. 39. �� = ���� + ����

Page 39 of 53 Furthermore S is log-normally distributed. The second term on the right hand side of the equation is the stochastic term with a constant volatility, σ, and Z, which is a Wiener process. In order to ensure no-arbitrage in the model, Black-Scholes derived a partial differential equation, which all products must fulfil in order to be correctly priced i.e. in order to rule out arbitrage. The partial differential equation is found by determining a function, which is dependent on the underlying and time, f(S, t). This function could be an option. From Ito’s lemma it is known that the change in the function f is given by the following partial differential equation.

�� �� 1 �!� �� Eq. 40. �� = �� + + �!�! �� + ���� �� �� 2 ��! ��

Then a portfolio consisting of a short option and long !" amounts of the underlying S is !" constructed. The portfolio is constructed in such a way that it is riskless. The value of such a portfolio can be expressed as.

�� Eq. 41. Π = � − � ��

The change in value of such a portfolio over small period of time, Δ, is given by.

�� Eq. 42. ΔΠ = �Δ − �Δ ��

By discretising equation 39 and 40 and inserting into equation 42 it can be shown that the portfolio is riskless.

Eq. 43. �� ΔΠ = �� Δ� + �� Δ� �� �� �� 1 �!� �� − �� + + �!�! Δ� + �� Δ� �� �� 2 ��! ��

Page 40 of 53 �� �� �� �� 1 �!� = �� Δ� + �� Δ� − �� − − �!�! Δ� �� �� �� �� 2 ��! �� − �� Δ� �� �� 1 �!� = − − �!�! Δ� �� 2 ��!

Equation 43 shows that the portfolio is risk free because the terms affected by the stochastic wiener process Z has been eliminated. As the portfolio is risk free it must also earn the risk free rate of return. This can be expressed in the following way in order to find the partial differential equation of Black-Scholes.

Eq. 44. ΔΠ = �ΠΔ�

Equation 44 is the mathematical expression of the change in value in the portfolio over the period Δt. When we insert equation 43 and 41 into equation 44 we will end up with the partial differential equation, which all Black-Scholes priced options must follow.

Eq. 45. �� 1 �!� �� − − �!�! Δ� = � � − � Δ� �� 2 ��! �� �� 1 �!� �� − − �!�! Δ� = �� − �� Δt �� 2 ��! �� �� 1 �!� �� + �!�! + �� = �� �� 2 ��! ��

This is the partial differential equation, which must be solved in order to find the price of a derivative under the Black-Scholes model.

Page 41 of 53 7.2.1 Solving the partial differential equation There are different ways of solving the partial differential equation of Black (1976) and Black-Scholes (1973). The derivation of Black-Scholes and Blacks 1976 model follow along the same lines. The main difference between the two models is the expectation operator and thereby the choice of numeraire. A general result in the derivation of the two models is very handy as it makes for an easy derivation of prices for other products. This section will focus on the derivation of this general result and then move on to prove that it can be used in the derivation the Black 1976 model. The derivation follows that of (Hull, 2009). The reasoning in this derivation is based on the assumptions underlying the Black-Scholes model. Black and Scholes defined the model assumptions as (Black & Scholes, 1973): I. The short term interest rate is known and constant through time II. The stock price follows a random walk in continuous time with a variance rate proportional to the square of the stock price. Thus the distribution of possible stock prices is log-normal III. The stock pays no dividend or other kinds of payments IV. The option is an European option V. There are no transaction costs in buying and selling the option and underlying VI. It is possible to borrow any fraction of the price of a security to buy it or hold it at the short term interest rate VII. There are no penalties to short selling. By these assumptions the price of the option is solely dependent on the underlying stock and time.

Following the notation of the derivation of the partial differential equation, one must start with defining S as a lognormal distributed variable. The payoff from a European call option at maturity is given by.

Eq. 46. ������ ���� = � max � − �; 0

By defining g(S) as the probability density function of S. This gives a space to quantify the probability of different outcomes, which gives the option its value. This follows from a lognormal density function and is.

Page 42 of 53

(!" ! !!) 1 ! Eq. 47. � � = � !!! � 2��!

This is the lognormal density function for the variable S, the underlying. By using the density function above the payoff from the option can be defined in terms of an integral.

! Eq. 48. � max � − �; 0 = � − � � � �� !

Notice the lower bound of the integral on the RHS of equation 48 it stems from K, which is the strike price of the call option. The reason for defining the integral for area from K to ∞ is that for values of S < K the value of the option is zero. It is therefore only relevant to define the area as the area where S > K as that will give the option a positive value. Taking the logarithm of S transforms the variable from a lognormal distributed variable to a standard normal distributed variable. The standard deviation of ln(S) = w. From the properties of the lognormal distribution it is known that the mean of S, m is given by �! � = ln � − 2. Knowing the mean and standard deviation. The next step is to define a variable that is standard normally distributed. This is done by defining the variable N as.

ln � − � Eq. 49. � = �

The variable N has a probability density function following the standard normal distribution. The standard normal probability density function is defined as.

1 !! Eq. 50. ℎ � = �! ! 2�

Using this transformation of the variable S and the features of the new distribution allows us to define the integral for the payoff on a European call as.

Page 43 of 53 ! Eq. 51. � max � − �; 0 = �(!"!!) − � � � �� (!" ! !!)/!

The expression eNw-m = ln(S), which is a standard normal distributed variable. The transformation has also altered the lower bound in the equation, as K was also transformed. The transformation of K has been done following the defined procedure for calculating N in equation 49. S in equation 51 is given by isolating ln(S) in equation 49. Using the summation qualities of integrals equation 51 can be extended. This is done below.

Eq. 52. � max � − �; 0 = ! ! �(!"!!)ℎ � �� − � ℎ � �� (!" ! !!)/! (!" ! !!)/!

The first integral on the RHS of equation 52 can be rewritten such that one ends up with an exponential and the density function of the standard normal distribution with a correction. This can be shown in the following way.

Eq. 53. � !"!! ℎ � =

1 !! � ! ! � !"!! 2� ! 1 !! !!"!! � ! 2� 1 !!!!!!"!!! � ! 2�

Note that by adding +w2 to the exponential and remembering that

–(N2-w)2 = -N2 – w2 + 2Nw it is possible to rewrite the equation as.

1 ! !!! !!!!!!! � ! 2�

Page 44 of 53

Splitting the exponential we get a constant and the probability density function with the correction, w.

! 1 ! ! !!! �(!!!! )/!� ! 2� ! �!!! /!ℎ(� − �)

By this transformation equation can be written as.

Eq. 54. � max � − �; 0 = ! ! ! �(!!! /!) ℎ � − � �� − � ℎ � �� (!" ! !!)/! (!" ! !!)/!

Equation 54 above shows that the expected payoff from a European call options can be calculated by solving this integral. The change to the standard normal distribution gives the nice feature that the area under the normal distribution is known and equal to 1. This can be done by defining a variable φ(x) as the probability that a standard normal variable is less then x. Solving this will lead to the general result, which can be used to derive both the Black 1976 model and Black-Scholes model. The fact that the area under a standard normal distribution equals 1 can be used to convert the probabilities such that 1-φ(x) is the probability that a standard normal variable is larger than x. Focusing on the first integral of the RHS in equation 54, x can be defined by inserting the lower bound of the integral into the following equation.

ln � − � Eq. 55. 1 − �( − �) �

Inserting m into equation X then gives

�! ln � − ln � − Eq. 56. 2 1 − �( − �) �

Page 45 of 53

Note that w can also be expressed as w = w2/w. Substituting w by w2/w then gives.

�! ln � − ln � − 2 �! Eq. 57. 1 − � − � �

Using the conversion noticed above gand reducing the expression gives.

Eq. 58. �! −ln � + ln � − 2 �! � − � �

� �! ln + � � 2 = �(�1) �

This is the general result for φ(d1) in the Black-Scholes equation. Notice that S in these equations is an expected value with respect to the forward risk neutral world that the model is pricing in. Following a similar approach the N(d2) of the general result can be found to be.

� �! ln − Eq. 59. � � 2 = �(�2) �

We have now found a solution to the integrals on the RHS of equation 54, which can then be expressed as.

! Eq. 60. �[max � − �; 0) = �!!! /!� �1 − ��(�2)

This is the general result, which can be used to price European call options on several different underlying assets by making small changes to the model and using the correct forward risk neutral expectation and numeraire.

Page 46 of 53 7.2.1.1 The Black 1976 model The above result can be modified to find the price of a European option on both assets and consumption goods e.g. Commodities and interest rate products, using Blacks 1976 model(Black, 1976) when the underlying interest rate is stochastic. This is done by considering a world, which is forward risk neutral with respect to a zero coupon bond, P(t, T). Considering a European call option on an asset, S. The forward price of this asset F follows a lognormal distribution with a standard deviation of � �. This is a process where the volatility is dependent on time i.e. the underlying is stochastic. It has been shown previously that in a world that is forward risk neutral with respect to P(t, T), the bond numeraire, a forward price of S at maturity equals the spot rate at maturity, FT = ST. The payoff from a call option can therefore be written as.

Eq. 61. � = � 0, � �![max �! − �; 0 ]

From the general results presented above the price of this European call option can be calculated.

Eq. 62. � = � 0, � [�!� �1 − � �(�2)]

Where φ refers to the standard normal probability. Notice that since the world is forward risk neutral with respect to P(t, T) the expected value of FT is F0, as shown in equation 7. Noting that the standard deviation to the ln(FT) is � �, the variance can be calculated as ( � �)2 = �!�. Substituting � � and �!�, for w and w2, respectively, and inserting them into the formula for d1 and d2 in equation 58 and 59 gives.

Eq. 63. � �!� ln ! + �1 = � 2 � � and � �!� ln ! − �2 = � 2 � �

Page 47 of 53 Where σ is the volatility in the forward prices of the asset. This is the Black 1976 model, which is the basis for pricing interest rate swaptions in this thesis.

7.3 Pricing of interest rate swaptions An interest swaption can be priced following Blacks 1976 model in a world that is forward risk neutral with respect to an annuity. The reason for this can be explained by analysing the payoff structure of an interest rate swaption. The payoff structure of a swaption can be seen as a series of payoffs coming from the difference between the swaption swap rate and the swap rate from a forward starting swap (Hull, 2009). This can be formulated as.

� Eq. 64. max � − � ; 0 � ! !

Where L is the notional and m is the number of payments a year in the swap. The understanding of equation 64 comes from the knowledge that SK is fixed and Sw can vary.

The option is therefore exercised if the value of Sw at maturity of the option is higher than SK.

At initiation the expected value of Sw at option maturity is set equal to the forward swap rate at time zero. If the swaption is exercised the holder will enter into a swap contract paying a fixed rate of SK. The value today of an annuity with a principal of 1 can be defined as the sum of discount bond prices, which can be expressed in the following way.

!" 1 Eq. 65. � = �(0, � ) � ! !!!

Notice that this is similar to the case with an annuity as the numeraire in section 5.3.2. The summation goes from i=1 to mn, where m is the number of payments in the swap a year and n is the number of years the swap is running. P(0,Ti) is the discount bond used to discount payments. Having defined A, the annuity, and using the equivalent martingale measure it is easy to see that the expected value of Sw in a world that is forward risk neutral with respect to the annuity is Sw. Pricing a swaption as just described with a swap contract running n years can be done using Black’s 1976 model in a world that is forward risk neutral with respect to the annuity.

Page 48 of 53

Eq. 66. �! = �� �!� �1 − �!� �2

Where d1 and d2 are defined as just shown in equation 64, with the exception that the rate

Sw is given by its value today.

Eq. 67. � �!� ln ! + �1 = � 2 � � and � �!� ln ! − �2 = � 2 � �

Notice that the formula uses S0 instead of Sw, which is also a result of the equivalent martingale measure. Since the expected value of Sw in this forward risk neutral world is a martingale its expected value equals its value today and we can therefore use S0.

7.4 Issues related to the use of the Black 1976 model Though the Black 1976 model is highly appreciated among practitioners for its fast pricing and relative accuracy, one must be careful when using it. The model is not capable of telling anything about how underlying assets evolve through the time and thus not able to price American or Bermudan style options. One of the assumptions of the Black model as shown is that interest rates are constant or that their volatility is. This is a very restrictive assumption especially for a model, which is supposed to price products which price is dependent on the volatility of interest rates. This is a problem, which one should be aware when using this model and perhaps consider using multiple models in order to verify ones result. The model is well suited for situations were speed and fast delivery of quotes is important, but models of higher accuracy can be found. Though the model can be used to price many different financial products the prices it produces are not necessarily consistent (i.e. arbitrage free) with each other. This means that if you are using the model price an interest rate cap and then want to price a payer European swaption these two prices would not be consistent with each other. The reason for this is the

Page 49 of 53 assumptions underlying the pricing of these two different products. The pricing of the cap is taken place in a world that is forward risk neutral with respect to a zero coupon whereas the pricing of the payer swaption is conducted in a world that is forward risk neutral to an annuity.

7.5 The Black 1976 model and multiple curves This section will discuss the use of multiple curves and the approach used by Blacks 1976 model. The essence of multiple curves is the separation of forward spot curve, which is used to calculate future cash flows, and the discount curve, which as the name implies, discounts the cash flows back to present time. The Black model cannot in its present state take account of these two different curves. The assumption in the Black model is that it is the same curve that is used for calculating future cash flows and for discounting them i.e. a single curve framework. This can be seen from the model as it ignores the payment stream on the floating leg of the swap. The only reason why it can ignore the payments in the floating leg is that when the principal is added to the last payment the value of the floating leg equals the principal. This can be illustrated by a small example. When you forward €100 for 1 period at rate r and discounted 1 period at rate r the value is still €100 (Hull, 2009, p. 160). This is a direct violation of the multiple curve-pricing framework. (Pallavicini & Tarenghi, 2010) investigated the use of multiple curves in pricing of interest rate swaps and swaptions. They found that though multiple curves were used in practice for pricing interest rate swaps this was not the case for the pricing of European swaptions. The market evidence suggested that the European swaptions were still being priced according to the old pricing framework.

Page 50 of 53 8 Conclusion This thesis has shown the difficulties in pricing interest rate swaps and swaptions given the influences of the financial crisis. The financial crisis impacted the interbank reference rates significantly during the financial crisis. It was shown that prior to the financial crisis interbank reference rates followed each other very well with few basis point deviation, after the financial crisis a spread was shown which was significant and does not seem to revert to levels prior to the crisis. These changes in the financial landscape of interbank reference rates was shown to have a significant impact on the pricing of interest rate derivatives, such as swaps and swaptions. The crucial assumptions underlining the pricing framework prior to the crisis was shown to brake down as a consequence of high basis point spreads interbank offered rates and their “risk free” OIS counterparty. This breakdown in assumptions was discussed and a possible arbitrage strategy was shown. The literature though, indicates that such arbitrage was not possible in practice as there was not enough liquidity in the market combined with a lack of financial instruments to exploit this arbitrage possibility. The new pricing framework for interest rate swaps after the financial crisis, was then introduced. This is known as multiple curve pricing and consists of creating a discount curve and then multiple forward spot curves. The forward spot curve is constructed using only financial products, which are dependent on the underlying rate. This means that if you want to price an interest rate swap on the 3-month Euribor then the forward spot curve is constructed from observations of spot, FRA’s, futures and swaps, which all have the 3-month Euribor as underlying. The procedure for creating such a forward spot curve was shown. Then option pricing of interest rate bonds was introduced with a focus on Black’s 1976 model. The model was highlighted, as it has been a very important for the pricing of interest rate options since its introduction in 1976. The derivation of the model was shown giving the reader a full understanding of the advantages and disadvantages of using this model. The important finding being that it has been a very popular model among practitioners: Though the model has some very restrictive assumptions about the development in the underlying asset it has been able to price options with a fair precision when compared to quoted market prices. The market today seem to have adapted the pricing of interest rate swaps to this multiple curve setting, however, the pricing of swaption is a different story. It still seems that the Black model is still widely used when pricing swaptions.

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