Yield curve estimation models with real market data implementation and performance observation

School of Education, Culture and Communication

Division of Applied Mathematics

MASTER THESIS IN MATHEMATICS / APPLIED MATHEMATICS

Yield curve estimation models with real market data implementation and

performance observation

by

Penny Andersson

Masterarbete i matematik / tillämpad matematik

DIVISION OF MATHEMATICS AND PHYSICS

MÄLARDALEN UNIVERSITY SE-721 23 VÄSTERÅS, SWEDEN

School of Education, Culture and Communication

Division of Applied Mathematics

Master thesis in mathematics / applied mathematics

Date:

2020-02-10

Project name:

Yield curve estimation models with real market data implementation and performance obser-vation Author: Penny Andersson Supervisor(s): Jan Röman Co-supervisor(s): Milica Ranˇci´c Reviewer: Examiner: Anatoliy Malyarenko Comprising: 30 ECTS credits

Abstract

It always exists different methods/models to build a yield curve from a set of observed market rates even when the curve completely reproduces the price of the given instruments. To create an accurate and smooth interest rate curve has been a challenging all the time. The purpose of this thesis is to use the real market data to construct the yield curves by the bootstrapping method and the Smith Wilson model in order to observe and compare the performance ability between the models. Furthermore, the extended Nelson Siegel model is introduced without implementation. Instead of implementation I compare the ENS model and the traditional bootstrapping method from a more theoretical perspective in order to perceive the performance capabilities of them.

Contents

1 Introduction 6 1.1 Structure. . . 7 1.1.1 Purpose . . . 7 1.1.2 Target Audience . . . 7 2 Theoretical background 8 2.1 Relevant terminology . . . 8

2.1.1 Forward rates from zero-coupon bonds and zero-coupon rates . . . . 8

2.1.2 Forward rates from coupon bonds . . . 11

2.1.3 Par rate and par yield . . . 13

2.1.4 Market instruments . . . 14

2.1.5 Interbank rate . . . 18

3 Implementation with market data(excl. the ENS model) 20 3.1 Bootstrapping method. . . 20

3.1.1 Bootstrapping the OIS Eonia curve . . . 20

3.1.2 Bootstrapping 3-months Euribor and 6-months Euribor . . . 23

3.2 The Smith–Wilson model . . . 31

3.2.1 Smith-Wilson technique for zero coupon bond prices as input . . . . 38

3.2.2 Smith-Wilson technique for a set of general inputs . . . 40

3.2.3 Important parameters . . . 42

3.3 The extended Nelson-Siegel model . . . 53

3.3.1 The original model . . . 57

3.3.2 The extended version of Svensson . . . 59

4 Conclusion 63 4.1 Comparing the bootstrapping method with the Smith Wilson technique . . . . 63

4.2 Comparing the bootstrapping method with the extended Nelson and Siegel model . . . 64

Appendices 68

B The calculation of 3-months Euribor with the bootstrapping method in Python 81 C The calculation of 6-months Euribor with the bootstrapping method in Python 104 D The calculation of Eonia with the Smith Wilson model in Python 126 E The calculation of 3-months Euribor with the Smith Wilson model in Python 150 F The calculation of 6-months Euribor with the Smith Wilson model in Python 191

G Criteria for a Masters Thesis 224

G.1 Objective 1: Knowledge and understanding . . . 224 G.2 Objective 2: Methodological knowledge . . . 224 G.3 Objective 3: Critically and Systematically Integrate Knowledge. . . 224 G.4 Objective 4: Ability to Critically, Independently and Creatively Identify and

Carry out Advanced Tasks . . . 225 G.5 Objective 5: Ability in both national and international contexts, Present and

Discuss Conclusions and Knowledge . . . 225 G.6 Objective 6: Scientific, Social and Ethical Aspects. . . 225

List of Figures

2.1 The par rate rparis the constant rate that equalizes the value of the floating leg (dotted arrows) to the fixed leg over the lifetime of the swap [19, p.23] . . . . 13 2.2 The par yield is the yield that equals the coupon rate cpar so that the price of

the bond is equal to its face value, nominal amount, here set to 100 [19, p.23] 14 2.3 Microsoft and Intel use the swap to transform a liability [13, p.155]) . . . 18 3.1 The bootstrapped discount curve for Eonia at 2020-04-14 (Table 3.2) . . . 26 3.2 The bootstrapped yield curve for Eonia at 2020-04-14 (Table 3.2). . . 26 3.3 The bootstrapped discount curve for 3-months Euribor at 2020-04-14 (Table

3.4) . . . 33 3.4 The bootstrapped yield curve for 3-months Euribor at 2020-04-14 (Table 3.4) 33 3.5 The bootstrapped discount curve for 6-months Euribor at 2020-04-14 (Table

3.6) . . . 36 3.6 The bootstrapped yield curve for 6-months Euribor at 2020-04-14 (Table 3.6) 36 3.7 The yield curve for Eonia by Smith-Wilson at 2020-04-14 (Table 3.8) . . . . 52 3.8 The discount factors for Eonia by Smith-Wilson technique at 2020-04-14 (Table

3.8) . . . 52 3.9 The yield curve for 3-months Euribor by Smith-Wilson at 2020-04-14 (Table

3.9) . . . 53 3.10 The discount factors for 3-months Euribor by Smith-Wilson at 2020-04-14

(Table 3.9) . . . 55 3.11 The yield curve for 6-months Euribor by Smith-Wilson at 2020-04-14 (Table

3.10) . . . 55 3.12 The yield curve for 6-months Euribor by Smith-Wilson at 2020-04-14 (Table

3.10) . . . 57 3.13 Yield curve shapes, parameter a spans from_{−6 to 12 with equal increments}

[16, p.476]. . . 59 3.14 Components of the forward rate curve [16, p.477] . . . 60 3.15 Forward rate curve and components [21] . . . 61 4.1 Plot of fitted yield curves for 3M Euribor rates, together with actual market rates 64 4.2 3M Forward Curves for EONIA rates by linear interpolation and NSS model 65 4.3 Discount Curves for 6M Euribor rates built by linear interpolation and NSS

List of Tables

3.1 EONIA rates at 2020-04-14. . . 24

3.2 The discounted OIS curve at 2020-04-14 . . . 25

3.3 3-months Euribor rates at 2020-04-14 . . . 28

3.4 The bootstrapped 3-months Euribor curve at 2020-04-14 . . . 32

3.5 The 6-months Euribor rates at 2020-04-14 . . . 34

3.6 The bootstrapped 6-months Euribor curve at 2020-04-14 . . . 35

3.7 Application of Smith–Wilson method for different input instruments . . . 43

3.8 The yield curves and discount factors for Eonia by Smith–Wilson technique at 2020-04-14 . . . 51

3.9 The yield curves and discount factors for 3-months Euribor by Smith–Wilson technique at 2020-04-14 . . . 54

3.10 The yield curves and discount factors for 6-months Euribor by Smith–Wilson technique at 2020-04-14 . . . 56

Chapter 1

Introduction

The values of interest rate derivatives are always difficult to estimate, and the pricing approach relies on modelling the future dynamics of the yield curves. Yield curves are used to estimate cash-flows, discount rates and to calculate the theoretical values of interest rate derivatives. But there are many different methods to construct a yield curve. Every method gives a slightly different yield curve and no one of them is perfect.

Banks need algorithms as accurate as possible since e.g. only a small change in a dis-count curve can have drastically changes in values of interest rate instruments. There are two main methods to estimate yield curves. Both methods, bootstrapping and parameterization use traded (or quoted) instruments on the market to predict the future interest rate. There are mainly two kinds of yield curves; curves from bonds (usually Government bonds) represent-ing the bond market and swap curves, representrepresent-ing the interbank market. Corporate bonds are valued as a spread above the Government curve.

Bootstrapping is a method where we use liquid, trades instrument by starting with the shortest to maturity and moving forward in time. If we use bonds, bills and notes, we need to strip their coupons (if any) by using linear interpolation and/or extrapolation. The swap curve includes in bootstrapped from deposits, future/forwards and swaps with different maturities.

In parameterization I use the same instruments as for the bootstrapping. But in this method I try to optimize a best fit of the market prices with a pre-determined function with some unknown constants. This function can be a stochastic process or a specified function. The two most common parameter-models are Smith Wilson (SW) and Nelson Siegel (NS). Many central banks prefer one of these models since they result in smooth forward rate curves. This is often not the case in bootstrapping.

In this master thesis I will compare the SW model and the extended Nelson-Siegel (ENS) model with Bootstrapping. Advantages and disadvantages of analyzed methods are discussed based on the obtained results.

1.1

Structure

1.1.1

Purpose

There are two main purposes in this thesis. Firstly, to find a better method to construct yield curves as accurate as possible as well as smooth as possible, in order to derive interest rate instruments and predict the future interest rate. Since a slight changing of yield curves can cause a huge changing of values of interest rate instruments. Furthermore, if the current term structure is incorrect it will indeed lead to mispricing. Hence, to construct the right current term structure for quoted market data is extremely important.

Secondly, to discover how yield curves influence interest rate risk. The interest rate risk is often calculated by shifting the curve and studying how the price of financial instruments is affected. Usually, such shifts are made in "time buckets" in order to calculate the risk between two time nodes on the curve. Most often people divide the curve into several "time buckets" and shift them one by one. The most common problem with parameterized curves is desire of smoothness. If we make such shifts (in "time buckets") the risk is affected along the whole or most parts of the curve, which is not desirable as we want. Thus I want to study the risk only on the selected "time buckets". In addition, when we use linear interpolation or extrapolation, only the part of the curve what we shift will be affected. Unfortunately, linear interpolation or extrapolation makes curves bumpy or zigzag shape, especially for the forward rate curves [14]. Anyway what we actually want is the curve without arbitrage.

To achieve these purposes I will use two methods, namely the bootstrapping method and the SW model on swap curves which consists of Deposits, FRA (Forward Rate Agreement) and IRS (Interest Rate Swaps). Furthermore, a theoretical introduction of the ENS model will be included. At the end I will analyze the result of those case studies and make a conclusion.

1.1.2

Target Audience

The target audience of this thesis are people who work or study within financial mathemat-ics and/or financial engineering areas on advanced level. Therefore knowledge of advanced mathematics and finance are required.

Chapter 2

Theoretical background

To estimate values of complex interest rate derivatives, it usually requires modelling the fu-ture dynamics of the yield curve term strucfu-ture. There are two approaches to achieve this, bootstrapping and parameterization. Sometimes they are called as "exact fit" and "best fit" respectively [19].

Forward interest rate has been widely used and its use has been standard long time in financial analysis, especially for pricing new financial instruments and in discovering arbitrage possibilities. In followed section, I will interpret some topic relevant terminologies, especially related to forward rates, yields to maturity and spot rates.

There is a number of methods to estimate forward rates both for financial analysis and monetary police analysis. For financial analysis, some of those methods are quite complex in order to achieve sufficient precision. Whereas the requirement of precision is a less disputable in analysis of monetary police. In this section, theoretical background of those three meth-ods, the SW, the ENS model and the bootstrapping method, will be presented. This includes histories, definitions and models which will be used in practical section later.

2.1

Relevant terminology

2.1.1

Forward rates from zero-coupon bonds and zero-coupon rates

In some literature, forward rate has been described as a risk-less interest rate which is related to a forward contract. A forward contract which is traded in the over-the-counter market (OTC) is an agreement between a buyer and a seller for trading an asset with agreed price at a certain future time [13]. At this certain future time an interest rate will be adopted. In other words, a future interest rate which is calculated from the zero-coupon rates (spot rates) or a yield curve and will be used between two future dates. In this subsection all forward rates what I talk about are forward rates from yields to maturity on zero-coupon bonds, namely spot rates or zero-coupon rates. Forward rates from yields to maturity on coupon bonds will be presented in the next subsection.

Volumne II, to describe the relationship between forward rate and spot rate. (1 + r_tspot₁ )t1_{(1 + r}f orward t2−t1 ) t2−t1 _{= (1 + r}spot t2 ) t2_{, (2.1)} =_⇒ r_tf orward_2−t1 = (1 + r spot t2 ) t2 (1 + rspot_t1 )t1 ! 1 t2−t1 − 1. (2.2)

Equation (2.1) and (2.2) tell us that at time t, t< t1< t2, we have made a contract guaran-teeing a risk-less rate of interest over the future interval [t1,t2]. Such an interest rate which is over a future period is called a forward rate. In other words, we can discount a future value with the future spot rate r for period [t1,t2] to obtain the current price for purchasing a zero-coupon bond at time t1with maturity t2, t2> t1. We call this future spot rate as forward rate as well. Therefore, there exists an easy way to represent the forward rate, namely via the discount function [19, p.25]: p(0,t1)· p(t1,t2) = p(0,t2), (2.3) =⇒ p(t1,t2) = p(0,t2) p(0,t1)≡ p(t2) p(t1) , (2.4)

where p is discount factor, also called as zero-coupon bond’s price. In terms of continuous compounding the discount function can be displayed as:

e−r(t1)·t1_{· e}− f (t1,t2)·(t2−t1)_{= e}−r(t2)·t2_{, (2.5)} where f(t1,t2) is forward rate over future period [t1,t2], r(t1) and r(t2) are spot rates at t1and t2respectively.

Moreover, from eq.(2.3) we can obtain: 1 p(t1,t2) = 1 + r_simpf orward(t2−t1) = p(0,t1) p(0,t2) . (2.6)

Thus, we can define simple forward rate over future period[t1,t2] with a contract time at t (t< t1< t2) as:

r_simpf orward(t,t1,t2) =−

p(t,t2)− p(t,t1) p(t,t2)(t2−t1)

. (2.7)

And for continuous compounding, according to eq.(2.5) we can have: ef(t1,t2)·(t2−t1)₌e r(t2)·t2 er(t1)·t1 = p(t,t1) p(t,t2) . (2.8)

Thereby we can define continuously compounded forward rate over future period[t1,t2] with a contract time at t (t< t1< t2) as:

r_compf orward(t,t1,t2) =− ln(p(t,t2))− ln(p(t,t1)) t_2−t1 =₋ln(p(t1,t2)) t_2−t1 . (2.9)

The instantaneous forward rate with start at S and maturity at T contracted at t (t< S < T ) can be defined as [19, p.294]: fT(t) = f (t, T ) = lim S_→Tr(t, S, T ) =− ∂[ln p(t, T )] ∂ T (2.10) , =_⇒ p(t, T ) = exp  − Z T t f(t, u)du  . (2.11)

Eq.(2.11) can be interpreted as, at contract time t as agreed to pay 1 cash unit (CU) at maturity T is equivalent to receive ef(t,T )_{· ∆T at time t. Or simply it is the price of a pure} discount bond as the final cash flow discounted by the instantaneous forward rate.

However zero-coupon rate are called as spot rate or short rate due to that it is defined as the theoretical profit obtained by a zero-coupon bond. Usually this rate is used to estimate how much we can gain in the future (time t1) if we invest X amount CU today (time t0). The following equation explains this relationship (Röman, 2017, P.24):

X_t1 = (1 + rspot))t1Xt0. (2.12) If we invert this equation we can get the discounted present value as [19, p.24]:

PV(Xt1) =

1 (1 + rspot)t1

X_t1, (2.13)

where_(1+r1

spot)t1 is called as discount factor which shows the relation between the spot rate and the discount function. It is easy to see that this rate is the same as the annual effective rate, denoted as r_annual.

For the term of continuous compounding, the discount function becomes:

p(t) = e−rspot(t)·t_{, (2.14)} =⇒

p(t, T ) = exp{−r(t,T ) · (T −t)}. (2.15) According to the previous equation and eq.(2.11) we can also define the spot rate as the continuous average of forward rate:

rspot(t, T ) = 1 T_−t Z T t f(t, u)du  . (2.16)

Therefore, the instantaneous spot rate at t is given as:

2.1.2

Forward rates from coupon bonds

In the previous subsection I interpreted implied forward interest rates based on yields to matur-ity on zero-coupon bonds, spot rate. Because a coupon bond can be considered as a portfolio which is consisted by zero-coupon bonds with different maturities (each zero-coupon bond corresponding to a particular coupon payment). We notice that yields to maturity on coupon bonds are not same as yields to maturity on zero-coupon bonds of the same maturity.

Furthermore, to compute implied forward interest rates from yields to maturity on zero-coupon bonds is easier than to compute implied forward interest rates from yield to maturity on coupon bonds. This is because that "yields to maturity on coupon bonds are a kind of average of yields to maturity on zero-coupon bonds of maturities from the time of the first coupon payment to the time of the payment of the face value and last coupon" [21, p.2]. Whereas, we have to discover the approach of estimating forward rates from coupon bonds since almost all bonds with time to maturity over 12 months are coupon bonds rather than zero-coupon bonds. Lars Svensson [21] proposed a two steps approach of estimating forward rates from coupon bonds. First step: to estimate implied spot rate from yields to maturity on coupon bonds. Second step: to calculate implied forward rates from implied spot rates.

To understand this more precisely, we can start with discount functions with different interest rate types. Especially, for annual rates, if we receive interest, we have to ask us how often we get payments. That is why we have [19, p.19]:

(1 + rannual)t=  1+rf f f·t . (2.18)

In continuous compounding this will turn to [19, p.19]:

lim f→∞  1+rf f f·t , =_{⇒ (1 + r}annual)t = erc·t, (2.19) where f denotes the number of annual payments, rf is interest rate paid f times every year and r_c is continuous compounding interest rate. Actually the right side of the equation above is same as the discount function eq.(2.14). We find that there is conversion between the annually compounded interest rate and continuously compounded interest rate. According to eq.(2.19) we can also obtain [19, p.20]:

r_c= f· ln  1+rf f  , (2.20) and r_f = f·  exp  r_c f  − 1  . (2.21)

The relation between the annually compounded interest rate and continuously compoun-ded interest rate showed above is the same as Lars Svensson [21] in his research paper ex-plained1. Thus we can present discount function as [21, p.3]:

1_{The expressions from a research paper of Frage [21, p.2] to explain the relation between the continuously}

d(t, T ) = exp  −i(t, T ) 100 (T−t)  , (2.24)

where d(t, T ) denotes the price at time t of a zero-coupon bond that pays 1 UC at the maturity date T , i(t, T ) (measured in percent per year) is the continuously compounded spot rate for a zero coupon bond traded at t and matured at T , T > t.

Regarding to coupon bonds, if we consider a coupon bond with coupon rate c percent per year, and assume the time to maturity m= T_{−t is integer, unit in year. Then the present value} P(t,t + m) at the trade date t of a coupon payment made in year k, k = 1, 2, ..., m will equal to [21, p.3]: P(t,t + m) = m

∑

k=1 cd(t,t + k) + 100d(t,t + m), (2.25) where d(·) is discount functions and 100 is face value.

Since yields to maturities are often used to discount cash flows. We reform the previous equation in term of yield to maturity [21, p.3]:

P(t,t + m) = m

∑

k=1 cexp  −y(t,t + m) 100 k  + 100 exp  −y(t,t + m) 100 m  , (2.26) where y(t,t + m) denotes yield to maturity. Here yield to maturity which is considered as internal rate of return for the coupon bond is constant interest rate that makes the present value of the coupon payments and the face value equal to the price of the bond.

Nevertheless, yield curves based on the yield to maturity on coupon bonds for different maturities are an inaccurate expression of the term structure of interest rates. Svensson [21] listed two reasons for this. One is that a given yield to maturity can be seen as an average of the spot rates up to the time to maturity. The corresponded spot rates in eq.(2.24) which also affect eq.(2.25) are generally vary with the maturity. On the other hand, the yield to maturity, y(t,t + m), in eq.(2.26) is constant and can be seen as a somewhat complex average of the spot rates. The second reason is two coupon bonds with same maturity date usually have different yields to maturity if they have different coupon rates. This is easy to see if we take everything else equal, that is, P(t,t + m) doesn’t change, when coupon rate c increases, y(t,t + m) will decrease. Hence, we shouldn’t use yield curves for coupon bonds as direct expression of term structure of interest rates. Instead, we should use spot rates.

It is easy to compute implied forward rates from spot rates if we readjust a forward in-vestment by a sale and a buy of zero-coupon bonds. A better explanation is that to sell a

i= 100  exp  i 100  − 1  , (2.22) and i= 100 ln  1+ i 100  , (2.23)

where i is continuously compounded spot rate, i is annually compounded spot rate and both measured in percent-age. Here all rates are assumed to be continuously compounded.

zero-coupon bond with maturity at t0 which is also the settlement date of a forward contract and to buy a same market value zero-coupon bond with same maturity date as the forward contract’s maturity date, T . Let f(t,t0, T ) (measured in percent per year) be the continuously compounded (implied) forward rate for the forward contract with the trade date, t. Then in relation with the spot rate, the forward rate equals to [21, p.4]:

f(t,t0, T ) = (T−t)i(t,T ) − (t0−t)i(t,t0)

T_−t0 , (2.27)

where t < t0< T , t,t0 and T denote the trade date, the settlement date and the maturity date respectively. i(_{·) is spot rate. For example, if this investment is 1-year forward contract with} settlement in 4 years from now (t0_{−t = 4) and mature in 5 years (T −t = 5). Then the forward} rate for this investment equals to 5 times 5-year spot rate subtract 4 times 4-year spot rate. This is exactly as eq.(2.27) expressed.

Then the instantaneous forward rate can be presented as [21, p.4]: f(t,t0) = lim

T_→t0f(t,t

0_{, T ). (2.28)}

And according to the definition of spot rate and eq.(2.16), we obtain the spot rate i(t, T ) is hence defined as the continuous average of forward rate [21, p.4]:

i(t, T )_≡

RT

t f(t, τ)dτ

T_−t (2.29)

2.1.3

Par rate and par yield

The par rate rpar is the fixed rate which makes value of its payments same as a number of opposite floating rate payments. This means that their total values sum up to zero as Figure 2.1below shows. The typical instrument with the par rate is a plain vanilla IRS.

Figure 2.1: The par rate rpar is the constant rate that equalizes the value of the floating leg (dotted arrows) to the fixed leg over the lifetime of the swap [19, p.23]

The par yield cparis the coupon rate which values the total (discounted) value of an instru-ment, included the nominal value, equals to its nominal value as Figure2.2shows.

Figure 2.2: The par yield is the yield that equals the coupon rate cpar so that the price of the bond is equal to its face value, nominal amount, here set to 100 [19, p.23]

Thereby the par rate for a swap can be calculated as [19, p.23]:

∑

i (p(0,ti)· rpar) =

∑

i (p(0,ti)· r ti−ti−1 f orward) =⇒ rpar= ∑i(p(0,ti)· rt_{f orward}i−ti−1 ) ∑i(p(0,ti) , (2.30) where p(0,ti) what is the discount factor at time tiis the price of a zero-coupon bond with maturity at ti. The rate rt_{f orward}i−ti−1 expresses the floating rate and is given by the forward rate between time tiand ti₋₁.

Subsequently, the par rate of a bond can be calculated as [19, p.24]:

100= n

∑

i (p(0,ti)· cpar) + (p(0,tn)· 100) =⇒ cpar= 100_{· (1 − p(0,t}n)) ∑ni(p(0,ti) . (2.31)

2.1.4

Market instruments

In this section, I will briefly introduce some market instruments which will be used in practical section later.

In the current market situation, due to similar instruments may present very different price levels, liquidity and even may also cause unreliable forward rates. Thus it is necessary to select the corresponding sets of instruments carefully in order to construct multiple yield curve. Market instruments nearly cover different maturities and overlap in major areas. That is why we should choose those with more liquid ones, namely ones with a tighter bid/ask spread. Bonds

As fixed-income securities bonds are issued by the seller who will pay a series of cash flows in the future in order to receive bond price. There are two kinds of bonds, namely zero coupon bonds and coupon bonds.

Zero coupon bonds what are also called as pure discount bonds only have one single cash flow at maturity. The payment equals to one unit of currency (also called principal or notional amount).

On the contrary, coupon bonds have a sequence of coupons in special coupon payment dates in addition to the principal.

Deposits

A deposit is zero-coupon contract which is a standard loan arrangement between two banks and traded in over-the-counter (OTC) market. In more details, the party that place the deposit has a credit that can be used in various ways, such as for purchases or to transfer it to another party. The rate for a loan is referred as a deposit rate [19]. Deposits start at a reference date t0 (today or spot) and pay a given fixed interest rate accrued until maturity Ti. Let RDepox (t0, Ti) be the quoted rate associated to the i:th deposit with maturity Ti and underlying rate tenor x= t0months. Then implied discount factor at time Tiis displayed as [19, p.547]:

p(t0, Ti) =

1

1+ RDepox (t0, Ti)· τF(t0, Ti)

, t₀< Ti, (2.32) where τF is time interval.

Forward rate agreements

FRAs are forward starting deposit contracts. In more details, FRA is a contract between two counter parties that agree during a time t to exchange interest rate payments in a future time T (where T > t).

FRA contracts have many different expressions. For example in Euribor, the 3x9 FRA is a 6 months deposit starting 3 months forward. In some markets such as in Sweden FRAs are quoted between IMM days2. Usually the underlying forward rate is determined two working days before the forward start date.

Together with deposits market FRAs can be used to construct the short-term yield curve. For instance, let FX(t, Ti₋₁, Ti) be the i:th Euribor forward rate resetting at time Ti₋₁with tenor x= Ti− Ti−1months related to the i:th FRA with maturity Ti. Then according to eq.(2.6) the implied discount factor at time Tican be expressed as [19, p.548]:

p_x(t0, Ti) =

p_x(t0, Ti₋₁)

1+ Fx(t0, Ti−1,ti)· τF(Ti−1, Ti)

, t0< Ti₋₁< Ti. (2.33) According to the definition of the instantaneous forward rate as eq.(2.10) showed, when the forward starting date of the FRA Ti−1gets closer to the current date t0, each FRA rate gets closer to the matching spot deposit rate as the following equation shows [19, p.548]:

lim Ti−1→t0

F_x(t0, Ti−1, Ti) = Rxdepo(t0, Ti). (2.34)

2_{IMM is abbreviation of International Monetary Market days that is the third Wednesday in March, June,}

Interest rate futures

There is another instrument which is often traded in stock market. It is called as interest rate future which is a futures contract with an interest-bearing instrument as the underlying asset. A common misconception as many believe that buying an interest rate futures contract allows the buyer of the contract to lock in a future borrowing rate. Actually buying the contract is equivalent to lending money. Conversely selling equivalent to borrowing money. For instance, being long involved in an interest rate future indicates that the buyer has agreed to receive a rate at a certain period in the future [19].

The underlying asset of interest rate futures is an underlying security which is a debt obligation and its value changes as interest rates change. Typical interest rate futures are EuroDollar3futures and Euribor futures.

A single future is closer to a FRA, which means to borrow or lend a nominal amount for a time, typcally 3 months, in the future. The future contracts are contracts with delivery at IMM4 and settled in cash5 on the second London business day before the IMM days. This indicates that its interest rate underlying is the interest rate for 91-day period [19].

Furthermore, the future contracts what are quoted in price6 are based upon LIBOR rate7 at maturity and can be used to hedge future interest rate exposures. The futures price what depends on the final marking to market equals to 100_{−R, where R is the interest rate expressed} with quarterly compounding and an actual/360 day-count convention [19].

Swaps

Swaps are OTC agreements and are one of the most popular fixed-income derivatives which are contractual agreements that two counter parties agree to exchange cash flows during a specific period in the future, for instance a floating interest rate cash flows, usually a Libor rate cash flows, against a fixed rate cash flows. In a swap agreement the dates when the cash flows are to be paid and the way in which they are to be calculated will be defined. It is common that the future value of an interest rate, an exchange rate, or other market variable will be used for calculating the cash flows. A forward contract what has the exchange of cash flows on just one future date can be seen as a simple example of a swap. However, swaps typically lead to cash flow exchanges on several future dates.

3_{EuroDollars are USD deposited in banks outside the United States, which implies they are not under the}

jurisdiction of the Federal Reserve. Euribor futures are similar contracts in Euro [19].

4_{The actual interest rate for the contract period is known on IMM [19].}

5_{"The settlement price of a contract is defined to be 100.00 minus the official British Banker’s Association}

(BBA) fixing of 3-month LIBOR/Euribor on the day the contract is settled" [19, p.73]

6_{"A quoted price of 95.00 means an interest rate of 100.00–95.00, or 5 percent" [19, p.73].}

7_{"LIBOR reflects the average rate at which banks can obtain unsecured funding in the London inter-bank}

market for a particular currency and a particular time period" [19, p.32]. Before 1 February 2014 the administrator of LIBOR was the BBA. Regarding to the finding of the attempted manipulation of LIBOR, a new administrator of LIBOR, Intercontinental Exchange (ICE), took over LIBOR. However its era of influence is slated to end by 2022 [12].

Plain vanilla interest-rate swap is an agreement that two counter parties agree to exchange fixed against floating rate cash flows. In this agreement, notional amount or face value of the swap, the payment frequency, maturity, day count convention, the coupon, that is the fixed rate, and the floating rate have to be specified. In most cases, swaps are arranged so that their net value is zero at the starting date. In other words, the value of the fixed leg should equal to the value of the floating leg. Generally for IRSs, two counter parties only exchange interest rate payments not the face value.

There are several purposes to be involved in a swap for counter parties. One of the most common application is to hedge against interest rate risk. The theory of comparative ad-vantages can explain motivation of this application. The comparative adad-vantages means that counter parties often has different abilities for borrowing in capital markets. This could hap-pen in regard to differences in credit rating or tax treatment, or for accounting reasons. One explanation of the popularity of swaps concerns comparative advantage can be considered as the use of an IRS to transform a liability. Some companies might have a comparative ad-vantage when borrowing in fixed-rate markets, whereas other companies have a comparative advantage when borrowing in floating-rate markets. To achieve this goal, it makes sense for companies to search a new opportunity in the market where it has a comparative advantage. Swaps become good choice for those companies, for example, to transform a fixed-rate loan into a floating-rate loan, and vice versa.

To understand aforementioned contents well I take a simple example. I assume Microsoft and Intel both want to enter in a swap. Suppose that Microsoft has arranged to borrow $100 million at LIBOR plus 10 basis points8. After Microsoft has entered into the swap, it has the following three sets of cash flows [13, p.155]:

1. It pays LIBOR plus 0.1% to its outside lenders. 2. It receives LIBOR under the terms of the swap. 3. It pays 5% under the terms of the swap.

The net of these three cash flows will be an interest rate of 5.1 percent.

For Intel, the swap could have the effect of transforming a fixed-rate loan into a floating-rate loan. Suppose that Intel has a 3-year $100 million loan remaining which it pays 5.2 percent. After it has entered into the swap, it has the following three sets of cash flows [13, p.156]):

1. It pays 5.2% to its outside lenders.

2. It pays LIBOR under the terms of the swap. 3. It receives 5% under the terms of the swap.

The net of these three cash flows will be an interest rate of Libor plus 20 basis points. We also can use one figure2.3to explain those transforms.

Figure 2.3: Microsoft and Intel use the swap to transform a liability [13, p.155])

In this swap, Microsoft pays a fixed rate at 5% to Intel while Intel pays a Libor rate to Microsoft. The swap helps Microsoft transform borrowings at a floating rate of Libor plus 10 basis points into borrowings at a fixed rate of 5.1%. Meanwhile, the swap helps Intel transform borrowings at a fixed rate of 5.2% into borrowings at a floating rate of Libor plus 20 basis points.

Regarding to the comparative advantages there is another use of the swap, namely using the swap to transform an asset. But this is not the concern in this writing.

Overnight indexed Swap (OIS) are IRSs based on a specific currency that a fixed rate (OIS rate) for a period (e.g., 1 month or 3 months) is swapped for the geometric average of the overnight rates during the period [13]. However, it is not perfectly risk-free since a default on an overnight loan or the swap is always possible. For swaps based on the Euro (EUR), the referenced floating rate is termed the euro overnight index average (EONIA) which will be discuss more later. In the US, the weighted average of the rates in brokered transactions (with weights proportional to transaction size) is termed the effective federal funds rate and in the UK the average of brokered overnight rates is termed the sterling overnight index average (SONIA). Other popular overnight indices are TONAR (Tokyo Over-Night average rate) and SARON (Swiss Average Rate Overnight).

The choice of a risk-free discount rate has become one of the most important issues in derivatives markets since the credit crisis of 2007. Prior to the credit crisis, LIBOR/swap rates were used as proxies for risk-free rates. After the credit crisis, most banks have changed their risk-free proxy from the LIBOR rate to the OIS rate—at least for collateralized derivatives transactions. This change indicates that the discount rate used by a bank for a collateralized derivative is not a true risk-free rate since it should correspond to the average funding costs for this collateralized derivative. For a non-collateralized derivative, its average funding costs should be equal or higher than Libor [13]. OIS rates have been considered as a suitable rate which can provide an estimate of the funding costs for these transactions.

2.1.5

Interbank rate

The interbank rate is the average rate which banks can borrow from each other. Some of the in-terbank rates are LIBOR-London Inin-terbank Offer Rate, Euribor-Euro Inin-terbank Offered Rate and STIBOR-Stockholm Interbank Offer rate, etc. The following contents will be focused on two interbank rates which are needed in practice sections.

Euribor

is a daily reference rate which is published by the European Money Markets Institute (EMMI) and is based on the averaged interest rates at which euro-zone banks offer to lend and borrow unsecured funds from each in the euro interbank market [19]. It is the rate which is contributed by the panel of banks. Prior to September 2012, the panel of banks contributing Euribor consisted of 44 banks. Whereas nowadays the panel of banks contributing to Euribor consists of 18 banks which provide daily quotes of the rate that each panel bank believes one prime bank is quoting to another prime bank for interbank term deposits within the Euro zone [3]. The Euribor rates is used as a reference rate for euro-denominated FRAs, short-term interest rate futures contracts and IRSs.

For IRSs, Euribor rate quoted as x-months Euribor is standard plain vanilla swap which begins at spot date with annual fixed leg against floating leg. Here index x-months indicates interests are paid with x-months frequency. For example, 3-months Euribor and 6-months Euribor. Such swaps can be seen as portfolios of FRA contracts (the first one being actually a deposit) [19]. The day count convention for this swap is act/360. Swaps are selected to contribute medium-long term structure section of the yield curve.

Eonia

is the other widely used reference rate in the euro-zone and it refers to Euro Over-Night Index Average which is computed as a daily weighted average of all overnight unsecured lending transactions in the interbank market within euro-zone [19]. It is necessary to point out that Eonia differs with Euribor as it is an average of actual transactions that has taken place between banks. Moreover, the panel of banks contributing to Eonia consists of 28 banks [2](EMMI, 2020). However, it has the same day count convention as Euribor, namely act/360. It is used by most banks as OIS discount rate.

Note that "Overnight" means from 1 day to next bussiness day, until the interbank payment system TARGET9 closes. On each day when the TARGET system is open, each Panel Bank should send the total volume of unsecured lending transactions that day and the weighted average lending rate for these transactions calculated by the respective Panel bank itself to the European Central Bank (ECB) no later than 6:30 p.m.. Based on those information contributed by panel banks ECB calculates Eonia and publish it between 6:45 p.m. and 7:00 p.m. on the same evening.

Chapter 3

Implementation with market data(excl.

the ENS model)

3.1

Bootstrapping method

In financial markets there are two types of yield curves, i.e., market curves and implicit curves. While the market curves are constructed from the market quotations (e.g., swap curves), the implicit curves are constructed through transformations (e.g., zero-coupon yield curves).

As we all know that without the yield curves we are not able to value financial instrument such as cash-flow instruments, equities, commodities and derivatives. For this reason, many researchers have developed methods and models to construct and forecast yield curves from prices of traded assets. One of the most popular methods is bootstrap method with linear interpolation to create the term structure of interest rate.

Bootstrap method in finance is a method for obtaining zero-coupon rates from a series of market prices of coupon bearing instruments such as bonds and swaps in order to build a yield curve. This method involves starting with short-term instruments and moving progressively to longer-term instruments. The generic methodology to construct a curve of zero-coupon instruments is the following:

1. During the first step, a set of yielding products (e.g., coupon-bearing bonds). 2. In the second step, discount factors are derived.

3. In the third step, the zero-coupon curve is bootstrapped.

In the following subsections I will explain algorithm and procedure of bootstrap method for some instruments in detail.

3.1.1

Bootstrapping the OIS Eonia curve

Different kind of curves created by banks are used for different kinds of trades. The discount rate should reveal the cost of funding for a bank, which we have discussed in the aforemen-tioned section of OIS. The market standard discount curve now consists of OIS rates.

In euro-zone, Eonia represents OIS curve. Hull [13] argued that it is necessary for the rates being bootstrapped to be the same as the rates being used for discounting in order to define a series of par yield bonds for the swap rates. To get OIS discounting we have to bootstrap Eonia curve first. However, in some systems of market participants annually compounded discount function, p(t) = 1

(1+r(t))t, is adopted to calculate OIS discount factors.

The data of Eonia rates was obtained from Bloomberg on 2020-04-14. Note that here we are not going to discuss how the market quotes are estimated. The quotes are set by the market-makers as prices (some instruments are quoted with yield) they agree to be used when buying or selling the specific swaps on the interbank market.

In this section I aim to describe the basics of discount curve bootstrapping. The compre-hensive description of the standard techniques will be introduced when extracting discount factors from market data.

Now we will focus on the extraction of discount factors from European OIS, called Eonia swaps. Notes that here we assume that the collateral and the pay currency are the same, both are Euro.

Let us consider a "riskless" asset B with value process_{B(t)}t≥0, B(0) = 1, then we get [9, p.2]: B(t) = nt

∏

j=1 1+r OIS j dj 360 ! , t_{≥ 0,} (3.1)

where ntis the number of business days in the considered period from today 0 until t and rjthe reference rate fixed on day j relevant for the next djdays (usually one day as it is an overnight rate, except for weekends and holidays). The existence of such an asset is always implicitly assumed. However, it is central assumption of bootstrap. Since the discount factors what we need are based on the assumption that one is able to borrow/lend at the given reference rate on every date. For this reason we also can interpret why using Euribor rates with longer tenor is difficult as it does not seem reasonable to receive those rates on a riskless, respectively default free, asset [9].

"For an OIS the floating rate is a daily compounded O/N rate and the market quotes" [19, p.563]. The Eonia curve we have typically quotes given by O/N, 1W, 1M, 2M, 3M for short-term of the curve. 6M, 9M, 1Y...30Y for medium- and long-short-term of the curve. The maturities of Eonia is a set, Ti∈ O/N,1W,1M,2M,3M..., and T1< T2< ... < Tn, respectively, the cor-responding par swap rates si. We assume that a swap with maturity Tiexchanges payments at the dates T1< T2< ... < Ti, which at Tj is a fixed payment of si(Tj− Tj−1) versus a floating payment of f(Tj₋₁, Tj)(Tj− Tj₋₁). The floating payment is linked to the reference rate by [9, p.3]: f(Tj₋₁, Tj)(Tj− Tj₋₁) = n_Tj

∏

j=n_Tj−1+1 1+r OIS j dj 360 ! − 1, (3.2) where rOIS_j is OIS rate (fixed rate) at time Tjand f(Tj₋₁, Tj) is floating rate at interval (Tj₋₁, Tj). Then the fixed leg will be [9, p.3]:

FixedLeg_i= si i

∑

j=1

D(0, Tj)(Tj− Tj₋₁), (3.3) where siis quoted fixed rate, namely swap rate.

There is an obvious connection between both aforementioned properties of swaps that the value of the fixed leg equals to the value of the floating leg, we obtain [9, p.3]:

f(Tj₋₁, Tj)(Tj− Tj₋₁) =

B(Tj) B(Tj₋₁)− 1.

(3.4) This reflects the property of swaps that the swaps are structured in a way that the floating payments equal the interest earned on the reference rate. Eq.(3.4) indicates that an investment of 1 in B at t=0 can yield all the required floating payments plus an additional fixed payment of 1 at the last payment date. According to eq.(2.6) we get

f(Tj₋₁, Tj)(Tj− Tj₋₁) = 1 B(Tj−1) 1 B(Tj) − 1 =D(0, Tj−1) D(0, Tj) − 1 =D(0, Tj−1)− D(0,Tj) D(0, Tj) . (3.5)

Hence, the floating leg will be:

FloatingLegi= i

∑

j=1

D(0, Tj) f (Tj−1, Tj)(Tj− Tj−1) = 1− D(0,Ti). (3.6) Similarly, the fixed leg turns to:

FixedLeg_i= si i

∑

j=1

D(0, Tj)(Tj− Tj−1) = 1− D(0,Ti). (3.7) Once again both the aforementioned equations prove that the property of swaps that the value of the floating leg equals the value of the fixed leg. Consequently, solving for D(0, Ti) by [9, p.3]:

D(0, Ti) =

1_{− s}i∑i_j=1−1(Tj− Tj₋₁)D(0, Tj)

1+ si(Ti− Ti−1) , 1 < i < n, (3.8) is commonly known as bootstrapping the discount factors.

Nevertheless, Röman [19] argued that discount function should be different between ma-turities up to one year and mama-turities over one year for STINA1. Thus accordingly we infer [19, p.7]:

D_T = D_{O/N− si∑}i_j=1−1Tj−T₃₆₀j−1D(0, Tj) 1+ si  Ti−Ti−1 360  , (3.9)

for one year tenors and

D_T = D_{O/N− si∑}i_j=1−1Tj−T₃₆₀j−1D(0, Tj) (1 + si) Ti−Ti−1 360 , (3.10)

for tenors longer than a year. And over-night discount factor will be: D_O/N= 1

1+ sO/N·360d

. (3.11)

Table3.1 shows the data of Eonia rates at 2020-04-14. The calculations of OIS discount factors and OIS forward rates are carried out in Python. The process of implementation is included in AppendixA.

According to eq.(2.7) and eq.(3.5) we can obtain forward rates as: f(Tj₋₁, Tj) = 1 T_{j− Tj−1}· D(0, Tj₋₁)− D(0,Tj) D(0, Tj) . (3.12)

As aforementioned the floating rate of OIS is a daily compounded O/N rate and in accord-ance with eq.(2.9) we can interpret the continuously compounded zero rate as:

ZOIS_T =−100 · ln(DT) (Tj− Tj₋₁)/365

. (3.13)

Consequently, we get the following table, Table 3.2. The discount curve and the yield curves are illustrated in Figure3.8and figure3.2.

3.1.2

Bootstrapping 3-months Euribor and 6-months Euribor

Before and after the financial crisis

Before the financial crisis in 2007, people never doubted those sophisticated investment banks’ abilities to value a plain vanilla IRS. The credit crunch was the main reason of the crisis which was based on increase of basis spreads between similar interest rate instruments with different underlying rate tenors, especially for swaps. For instance, basis spreads between Euribor and Eonia OIS swaps diverged significantly from zero and were not negligible anymore [14].

What we have learned from the crisis was that the risk related to unsecured bank lending became apparent and consequently, the longer the tenor of Euribor rates, the higher the risk investors associate with those rates and thus, the higher the rates. Hence, It is not appropriate to value a swap with a different tenor based on the previously bootstrapped discount factors. That’s why there is another procedure so called dual curve stripping or curve cooking which we will present later on.

Name Quote From date To date Days EONIA O/N -0.451 2020-04-14 2020-04-15 1 EUR 1W OIS -0.4583 2020-04-14 2020-04-23 9 EUR 1M OIS -0.4606 2020-04-14 2020-05-18 34 EUR 2M OIS -0.4645 2020-04-14 2020-06-16 63 EUR 3M OIS -0.4696 2020-04-14 2020-07-16 93 EUR 6M OIS -0.4841 2020-04-14 2020-10-16 185 EUR 9M OIS -0.4978 2020-04-14 2021-01-18 279 EUR 1Y OIS -0.5077 2020-04-14 2021-04-16 367 EUR 15M OIS -0.5169 2020-04-14 2021-07-16 458 EUR 18M OIS -0.5217 2020-04-14 2021-10-18 552 EUR 21M OIS -0.5242 2020-04-14 2022-01-17 643 EUR 2Y OIS -0.5232 2020-04-14 2022-04-18 734 EUR 27M OIS -0.5261 2020-04-14 2022-07-18 825 EUR 30M OIS -0.5245 2020-04-14 2022-10-17 916 EUR 2Y 9M OIS -0.5205 2020-04-14 2023-01-16 1007 EUR 3Y OIS -0.5149 2020-04-14 2023-04-17 1098 EUR 3Y 3M OIS -0.5095 2020-04-14 2023-07-17 1189 EUR 3Y 6M OIS -0.5037 2020-04-14 2023-10-16 1280 EUR 3Y 9M OIS -0.4973 2020-04-14 2024-01-16 1372 EUR 4Y OIS -0.4902 2020-04-14 2024-04-16 1463 EUR 4Y 3M OIS -0.4823 2020-04-14 2024-07-16 1554 EUR 4Y 6M OIS -0.4736 2020-04-14 2024-10-16 1646 EUR 4Y 9M OIS -0.4645 2020-04-14 2025-01-16 1738 EUR 5Y OIS -0.4549 2020-04-14 2025-04-16 1828 EUR 5Y 3M OIS -0.4453 2020-04-14 2025-07-16 1919 EUR 5Y 6M OIS -0.4353 2020-04-14 2025-10-16 2011 EUR 5Y 9M OIS -0.4251 2020-04-14 2026-01-16 2103 EUR 6Y OIS -0.4147 2020-04-14 2026-04-16 2193 EUR 6Y 3M OIS -0.4042 2020-04-14 2026-07-16 2284 EUR 6Y 6M OIS -0.3934 2020-04-14 2026-10-16 2376 EUR 6Y 9M OIS -0.3821 2020-04-14 2027-01-18 2470 EUR 7Y OIS -0.3714 2020-04-14 2027-04-16 2558 EUR 7Y 3M OIS -0.3602 2020-04-14 2027-07-16 2649 EUR 7Y 6M OIS -0.3486 2020-04-14 2027-10-18 2743 EUR 7Y 9M OIS -0.3373 2020-04-14 2028-01-17 2834 EUR 8Y OIS -0.3258 2020-04-14 2028-04-17 2925 EUR 8Y 6M OIS -0.3032 2020-04-14 2028-10-16 3107 EUR 9Y OIS -0.2806 2020-04-14 2029-04-16 3289 EUR 9Y 6M OIS -0.2584 2020-04-14 2029-10-16 3472

EUR 10Y OIS -0.2361 2020-04-14 2030-04-16 3654

EUR 12Y OIS -0.1477 2020-04-14 2032-04-16 4385

EUR 15Y OIS -0.0444 2020-04-14 2035-04-16 5480

EUR 20Y OIS 0.0225 2020-04-14 2040-04-16 7307

EUR 25Y OIS 0.0071 2020-04-14 2045-04-17 9134

EUR 30Y OIS -0.0368 2020-04-14 2050-04-18 10961

Name Quote From date To date Days Discount Forward Spot EONIA O/N -0.451 2020-04-14 2020-04-15 1 1.00001252793472-0.450999999999585-0.457266753162583 EUR 1W OIS -0.4583 2020-04-14 2020-04-23 9 1.00012711604027-0.515580936367857-0.515493400291399 EUR 1M OIS -0.4606 2020-04-14 2020-05-18 34 1.00044769541751-0.461427724142001-0.480506645922278 EUR 2M OIS -0.4645 2020-04-14 2020-06-16 63 1.0008258696548-0.469070282961994-0.478282565742515 EUR 3M OIS -0.4696 2020-04-14 2020-07-16 93 1.00122661312711-0.480303020784519-0.48111766658352 EUR 6M OIS -0.4841 2020-04-14 2020-10-16 185 1.00250434353045-0.498732464755508-0.493482542749158 EUR 9M OIS -0.4978 2020-04-14 2021-01-18 279 1.00387971576616-0.52470260606523-0.506579323897994 EUR 1Y OIS -0.5077 2020-04-14 2021-04-16 367 1.00520412748153-0.538999769137221-0.516234598122799 EUR 15M OIS -0.5169 2020-04-14 2021-07-16 458 1.00661600467764-0.554873777331133-0.525521498946875 EUR 18M OIS -0.5217 2020-04-14 2021-10-18 552 1.00805059208214-0.545028649458078-0.530199459381391 EUR 21M OIS -0.5242 2020-04-14 2022-01-17 643 1.00942662282304-0.539280218371716-0.532597411889658 EUR 2Y OIS -0.5232 2020-04-14 2022-04-18 734 1.01074540674105-0.516170255472784-0.531491971317548 EUR 27M OIS -0.5261 2020-04-14 2022-07-18 825 1.0121509384075-0.54935927469716-0.534347097481486 EUR 30M OIS -0.5245 2020-04-14 2022-10-17 916 1.01345767973635-0.51008801250395-0.532674046186954 EUR 2Y 9M OIS -0.5205 2020-04-14 2023-01-16 1007 1.01469016224648-0.480516631238024-0.528590572308455 EUR 3Y OIS -0.5149 2020-04-14 2023-04-17 1098 1.01585441051151-0.453393445413074-0.522902129681962 EUR 3Y 3M OIS -0.5095 2020-04-14 2023-07-17 1189 1.01699804005016-0.444863072114309-0.517421724564593 EUR 3Y 6M OIS -0.5037 2020-04-14 2023-10-16 1280 1.01810094488266-0.428556718139346-0.511543884644207 EUR 3Y 9M OIS -0.4973 2020-04-14 2024-01-16 1372 1.0191663387116-0.409053185504009-0.505066837835035 EUR 4Y OIS -0.4902 2020-04-14 2024-04-16 1463 1.02015692291336-0.384136455512101-0.49788847789851 EUR 4Y 3M OIS -0.4823 2020-04-14 2024-07-16 1554 1.02107708684452-0.356506771656568-0.489908863706539 EUR 4Y 6M OIS -0.4736 2020-04-14 2024-10-16 1646 1.02193394798843-0.328097027948742-0.481127201322867 EUR 4Y 9M OIS -0.4645 2020-04-14 2025-01-16 1738 1.02272673203556-0.303326230567583-0.471944730672365 EUR 5Y OIS -0.4549 2020-04-14 2025-04-16 1828 1.02342110401107-0.27139247873125-0.462260891980387 EUR 5Y 3M OIS -0.4453 2020-04-14 2025-07-16 1919 1.02407979838881-0.254455162191572-0.452578158870841 EUR 5Y 6M OIS -0.4353 2020-04-14 2025-10-16 2011 1.02467918453885-0.228893501572751-0.442493491309696 EUR 5Y 9M OIS -0.4251 2020-04-14 2026-01-16 2103 1.02521491033727-0.204475990398409-0.432207542589984 EUR 6Y OIS -0.4147 2020-04-14 2026-04-16 2193 1.02566162717677-0.174216068015335-0.42172053863055 EUR 6Y 3M OIS -0.4042 2020-04-14 2026-07-16 2284 1.02606053035214-0.153799746629003-0.411132251765344 EUR 6Y 6M OIS -0.3934 2020-04-14 2026-10-16 2376 1.02639640548717-0.128049357893781-0.400240815360121 EUR 6Y 9M OIS -0.3821 2020-04-14 2027-01-18 2470 1.02666284184119-0.0993894495549893-0.388844442645402 EUR 7Y OIS -0.3714 2020-04-14 2027-04-16 2558 1.02684869331318-0.0740422110185474-0.378050266554918 EUR 7Y 3M OIS -0.3602 2020-04-14 2027-07-16 2649 1.02697447311381-0.0484520728687219-0.366750937068524 EUR 7Y 6M OIS -0.3486 2020-04-14 2027-10-18 2743 1.02704101468862-0.0248130376571004-0.355044888532497 EUR 7Y 9M OIS -0.3373 2020-04-14 2028-01-17 2834 1.027040489208320.000202409073543519-0.343637774947475 EUR 8Y OIS -0.3258 2020-04-14 2028-04-17 2925 1.026964710610070.029191213927227-0.332026071331823 EUR 8Y 6M OIS -0.3032 2020-04-14 2028-10-16 3107 1.026669044881010.0569641515096021-0.309194175615984 EUR 9Y OIS -0.2806 2020-04-14 2029-04-16 3289 1.026137699933330.102424068862824-0.286339650186743 EUR 9Y 6M OIS -0.2584 2020-04-14 2029-10-16 3472 1.025417621854730.138143426609163-0.263867725571704 EUR 10Y OIS -0.2361 2020-04-14 2030-04-16 3654 1.024447884255190.187238639883609-0.24127376998384 EUR 12Y OIS -0.1477 2020-04-14 2032-04-16 4385 1.018350250246630.294882705856612-0.151359849541542 EUR 15Y OIS -0.0444 2020-04-14 2035-04-16 5480 1.006884899150320.374365580474891-0.0457003076200959 EUR 20Y OIS 0.0225 2020-04-14 2040-04-16 7307 0.9953924601538210.2275002162423280.0230688155985353 EUR 25Y OIS 0.0071 2020-04-14 2045-04-17 9134 0.998195077666152-0.05532384563534050.00721909247445399 EUR 30Y OIS -0.0368 2020-04-14 2050-04-18 10961 1.01131960701282-0.255716801992705-0.0374824119929784

Figure 3.1: The bootstrapped discount curve for Eonia at 2020-04-14 (Table 3.2)

Bootstrapping Euribor

Now, we move to Euribor which is considered as a set of swaps with floating payments linked to its rates with a given tenor. Note that it is not possible to invest at this rate, especially not riskless. However, if we replicate an investment with an asset at this rate or to model the dynamics of such a system then we also need to consider the related default risk [9].

Those swaps are contracts that two counter parties exchange fixed payments against float-ing payments linked to an exogenous index. Such contracts are usually collateralized swap contracts, which means that the exchanged payments themselves are not exposed to credit risk, especially for the credit risk causing the basis spread. Moreover, based on previous sub-section discount factor from OIS discounting will be used here to price those collateralized contracts.

A swap curve is usually consisted of Deposits, FRA and IRS . The basis of this bootstrap-ping method is the assumption that the theoretical price of a bond is equal to the sum of the cash flow discounted at the zero-coupon rate of each flow []Borodovsky.

Deposit I have introduced deposits contracts generally in 1.4.1. In our case, cash deposits include maturities in S/W2, one month, two months and three months. Table3.3is 3-months Euribor rates at 2020-04-14.

We can begin with calculating the discount factor and zero rate for short-term of the curve.

D_i= 1 1+ qi₃₆₀di ; i={1W, 1M, 2M, 3M}, (3.14) Z_i=−100ln(D_di i) 365 ; i={1W, 1M, 2M, 3M}, (3.15) where Diis discount factors, Ziis zero rates, qiis quoted rates in our case it is Euribor rates or swap rates and di is number of days. Note that day count convention for Euribor is ACT/360 but zero rates are commonly given as continuous compounding, Act/365. In addition, here we don’t need to calculate discount factor for overnight (O/N) and tomorrow next (T/N) since 3-months Euribor doesn’t include them. The table of Euribor rates viewed at 2020-04-14 will show that forward rates started at 2020-04-14 but the measure date was 2020-04-14. This is because 3-months Euribor starts with S/W which is settled in 2 days after the spot.

FRAs The general description of FRAs have been included in section 1.4.2. I will directly go forward to algorithms. There is a so-called stub rate which shall have Maturity the same date as the Start date of the first FRA contract. However we need stub rate only if we have IMM FRA contracts. FRAs within Euribor are not IMM FRA contracts. Thus we neglect stub rate here.

Now we can handle the FRA rates as given below [19, p.201]:

Name Quote Measured date From date To date Days 1Days EURIBOR S/W -0.518 2020-04-14 2020-04-16 2020-04-23 9 EURIBOR 1M -0.404 2020-04-14 2020-04-16 2020-05-18 34 EURIBOR 2M 0 2020-04-14 2020-04-16 2020-06-16 63 EURIBOR 3M -0.248 2020-04-14 2020-04-16 2020-07-16 93 EUR 1X4 FRA -0.27 2020-04-14 2020-04-16 2020-08-17 125 EUR 2X5 FRA -0.299 2020-04-14 2020-04-16 2020-09-16 155 EUR 3X6 FRA -0.322 2020-04-14 2020-04-16 2020-10-16 185 EUR 4X7 FRA -0.347 2020-04-14 2020-04-16 2020-11-16 216 EUR 1Y IRS 3M -0.347 2020-04-14 2020-04-16 2021-04-16 367 EUR 15M IRS 3M -0.363 2020-04-14 2020-04-16 2021-07-16 458 EUR 18M IRS 3M -0.373 2020-04-14 2020-04-16 2021-10-18 552 EUR 21M IRS 3M -0.377 2020-04-14 2020-04-16 2022-01-17 643 EUR 2Y IRS 3M -0.38 2020-04-14 2020-04-16 2022-04-18 734 EUR 3Y IRS 3M -0.371 2020-04-14 2020-04-16 2023-04-17 1098 EUR 4Y IRS 3M -0.346 2020-04-14 2020-04-16 2024-04-16 1463 EUR 5Y IRS 3M -0.31 2020-04-14 2020-04-16 2025-04-16 1828 EUR 6Y IRS 3M -0.271 2020-04-14 2020-04-16 2026-04-16 2193 EUR 7Y IRS 3M -0.229 2020-04-14 2020-04-16 2027-04-16 2558 EUR 8Y IRS 3M -0.184 2020-04-14 2020-04-16 2028-04-17 2925 EUR 9Y IRS 3M -0.139 2020-04-14 2020-04-16 2029-04-16 3289 EUR 10Y IRS 3M -0.094 2020-04-14 2020-04-16 2030-04-16 3654 EUR 11Y IRS 3M -0.05 2020-04-14 2020-04-16 2031-04-16 4019 EUR 12Y IRS 3M -0.008 2020-04-14 2020-04-16 2032-04-16 4385 EUR 15Y IRS 3M 0.094 2020-04-14 2020-04-16 2035-04-16 5480 EUR 20Y IRS 3M 0.157 2020-04-14 2020-04-16 2040-04-16 7307 EUR 25Y IRS 3M 0.14 2020-04-14 2020-04-16 2045-04-17 9134 EUR 30Y IRS 3M 0.092 2020-04-14 2020-04-16 2050-04-18 10961 EUR 40Y IRS 3M 0.016 2020-04-14 2020-04-16 2060-04-16 14612 EUR 50Y IRS 3M -0.053 2020-04-14 2020-04-16 2070-04-16 18264

Di_FRA= D i−1 FRA 1+ qi FRA di FRA 360

; i=_{{1x4 FRA, 2x5 FRA, 3x6 FRA, 4x7 FRA},} (3.16)

where qi_FRAis quoted FRA rates and d_FRAi is FRA contracts periods, in this case it is 3 months. Now we can get the zero rate as [19, p.201]:

Z_FRAi (T ) =−100ln(D i FRA) d_FRAi

365

; i={1x4 FRA, 2x5 FRA, 3x6 FRA,4x7 FRA}. (3.17)

In our case, tenor 1x4FRA means that a forward contract starts in one month from meas-ured date (2020-04-14) and has 3 months contract period. In this sense, tenor 1x4FRA starts at 2020-05-18 and ends up at 2020-08-17. Therefore, according to (3.16), Di_FRA−1 should equals to 1.000382. As we see in table3.3, there is no quoted rate at tenor Euribor 2M, in this situation we have to use linear interpolation method to obtain Euribor 2M discount factor. We can use this case to calculate and show that how linear interpolation works.

D(2020_{− 06 − 16) =}D(2020− 07 − 16) − D(2020 − 05 − 18)

93_{− 34} · (93 − 63) =1.000509.

(3.18)

Other tenors of FRAs will have same logical algorithm as we just explained above. The completed process of implementation in Python will be showed in AppendixB.

Swaps Euribor is typical standard plain vanilla swaps. Swaps contribute the medium-long term of the yield curve. Since the swaps have no value at the starting date of the contracts. And there is one-to-one relationship between the fixed rates and the floating rates. In some literature researchers recommended to exact the implied value of the [9]. For Euribor with specified tenor, we try to extract a "description" of the additional swap curve/the related Euribor rate which allows us to value arbitrary products linked to those floating payments accordingly [9]. Let’s explain it in a better way.

Let L3M(Tj−1, Tj) denotes Euribor rate for the time period betweem Tj−1 and Tj (in our example, the tenor is assumed to be 3 month), which is fixed at Tj₋₁ and payed at Tj in a swap. As aforementioned the idea would be to iteratively construct portfolios using the given instruments, which exchange a fixed payment today, PV(0), against a payment of L3M(Tj₋₁, Tj)(Tj− Tj₋₁) at Tj. In financial mathematics this can be expressed as [9, p.7]:

PV(0) =”Today0s value o f a f uture payment o f L3M(Tj₋₁, Tj)(Tj− Tj₋₁)” =D(0, Tj)EQTj[L3M(Tj₋₁, Tj)(Tj− Tj₋₁)]

=D(0, Tj)F3M(0, Tj₋₁, Tj)(Tj− Tj₋₁),

where QTj denotes the Tj forward measure using risk-neutral valuation. We can use a related forward rate F3M(0, Tj₋₁, Tj) presents PV at Tj. In other words, the contract is about a ex-changing F3M(0, Tj−1, Tj) against L3M(Tj−1, Tj) at Tj. In this sense, we can easily find out that F3M(0, Tj₋₁, Tj) represents floating rate. Those forward rates, respectively the curve of those forward rates Tj₋₁→ F3M(0, Tj₋₁, T j), are the objects we are interested in.

As we illustrated early, the net value of the floating leg and the fixed leg should be zero. we conclude with the following set of equations the forwards should satisfy to match market prices [9, p.7]: s3M_i ni

∑

j=1 (Tj− Tj₋₁)D(0, Tj) = ni

∑

j=1 D(0, Tj)F3M(0, Tj₋₁, Tj)(Tj− Tj₋₁), (3.20)

for 1_{≤ i ≤ n, where n}icorresponds to the number of payment dates for the i−th swap and s3Mi is corresponding par rate. Note that I use the same assumption again that floating and fixed leg have equal payment dates for simplicity. Here, this does not make a difference at all. As the previous section mentioned about forward rates, here we can also express forward rate in terms of discount factor [9, p.7]:

F3M(0, Tj−1, Tj) = 1 T_{j− Tj−1} _P3M_{(0, T} j₋₁) P3M_{(0, T} j) − 1  . (3.21) This transforms (3.20) to [9, p.8]: s3M_i ni

∑

j=1 (Tj− Tj−1)POIS(0, Tj) = ni

∑

j=1 POIS(0, Tj) _P3M_{(0, T} j₋₁) P3M_{(0, T} j) − 1  . (3.22)

Note that here, s3M_i is quoted fixed rate for the i_{−th swap ,P}3M(0, Tj) represents discount factor for Euribor curve which is different with OIS discount factor3 POIS(0, Tj) (here we replace D(0, Tj)). Since the two discount curves are different, that is why we call this boot-strapping as dual curve stripping. In our case, to strip a 3 months-Euribor curve we have to mention the frequency of the floating payments is every 3 months. For instance, the tenor "EUR 1Y IRS 3M" what represents 1 year maturity IRS with 3 months as tenor tells us that within one year there are 4 times floating payments or ni equals to 4. They are 2020-07-16, 2020-10-16, 2021-01-18 and 2021-04-16. Discount factor P3Mat 2020-07-16 and 2020-10-16 can be obtained via deposit and FRA. Discount factor P3M at 2021-04-16 is what we want. Obviously we can use linear interpolation to get unknown discount factor P3M at 2021-01-18.

The right side of (3.22) can be rewritten as:

ni

∑

j=1 POIS(0, Tj)  P3M(0, Tj−1) P3M_{(0, T} j) − 1  = ni

∑

j=1  P3M(0, Tj−1) P3M_{(0, T} j) POIS(0, Tj)− POIS(0, Tj)  = ni

∑

j=1 P3M(0, Tj₋₁) P3M_{(0, T} j) POIS(0, Tj)− ni

∑

j=1 POIS(0, Tj) = ni−1

∑

j=1 P3M(0, Tj−1) P3M_{(0, T} j) POIS(0, Tj)− ni

∑

j=1 POIS(0, Tj) +P 3M_{(0, T} j₋₁) P3M_{(0, T} j) POIS(0, Tj). (3.23)

According to (3.22) we finally obtain discount factors:

P3M(0, Tj) = POIS(0, Tj)· P3M(0, Tj₋₁) POIS_{(0, T} j) + s3M_i · ∑n_j=1i ∆jPOIS(0, Tj)− ∑n_j=1i−1  P3M_(0,T j−1) P3M_(0,T j) − 1  POIS_{(0, T} j) , (3.24) where ∆j= Tj− Tj₋₁. And the zero rate is calculated as:

Zi(0, Tj) =−100 · lnP3M(0, Tj)· 365 di_{(0, T}

j)

. (3.25)

Here I continue use Python to bootstrap 3-months Euribor curve and the detail is included in AppendixB. A new table of 3-months Euribor after bootstrapping is expressed in table3.4. The bootstrapped discount curve and yield curve are illustrated in Figure 3.3and Figure3.4, respectively.

Now let’s look at 6-months Euribor rates, see table3.5. For example, for FRAs, 1x7 means that the forward contract starts 1 month from now (2020-04-14) and will end up 7 months from now (2020-04-14). And the forward contract will last 6 months long. Furthermore, such as aforementioned here we also have 2020-04-14 as measure date. The algorithm of bootstrapping is same as for 3-months Euribor. The detail will show in AppendixC. All the bootstrapped results are showed in table3.6. And the bootstrapped discount curve and yield curve are illustrated in Figure3.5and Figure3.6, respectively.

3.2

The Smith–Wilson model

The smoothness possible curve needs an acceptable term structure of interest rates. Moreover, risk-free yield curves are the basic building blocks for the valuation of future financial claims and long-term risk management work. However, measuring and estimating suitable risk-free

Name Quote From date To date Days Discount Forward Spot EONIA O/N -0.451 2020-04-14 2020-04-15 1 1.00001252793472-0.450999999999585-0.457266753162583 EUR 1W OIS -0.4583 2020-04-14 2020-04-23 9 1.00012711604027-0.515580936367857-0.515493400291399 EUR 1M OIS -0.4606 2020-04-14 2020-05-18 34 1.00044769541751-0.461427724142001-0.480506645922278 EUR 2M OIS -0.4645 2020-04-14 2020-06-16 63 1.0008258696548-0.469070282961994-0.478282565742515 EUR 3M OIS -0.4696 2020-04-14 2020-07-16 93 1.00122661312711-0.480303020784519-0.48111766658352 EUR 6M OIS -0.4841 2020-04-14 2020-10-16 185 1.00250434353045-0.498732464755508-0.493482542749158 EUR 9M OIS -0.4978 2020-04-14 2021-01-18 279 1.00387971576616-0.52470260606523-0.506579323897994 EUR 1Y OIS -0.5077 2020-04-14 2021-04-16 367 1.00520412748153-0.538999769137221-0.516234598122799 EUR 15M OIS -0.5169 2020-04-14 2021-07-16 458 1.00661600467764-0.554873777331133-0.525521498946875 EUR 18M OIS -0.5217 2020-04-14 2021-10-18 552 1.00805059208214-0.545028649458078-0.530199459381391 EUR 21M OIS -0.5242 2020-04-14 2022-01-17 643 1.00942662282304-0.539280218371716-0.532597411889658 EUR 2Y OIS -0.5232 2020-04-14 2022-04-18 734 1.01074540674105-0.516170255472784-0.531491971317548 EUR 27M OIS -0.5261 2020-04-14 2022-07-18 825 1.0121509384075-0.54935927469716-0.534347097481486 EUR 30M OIS -0.5245 2020-04-14 2022-10-17 916 1.01345767973635-0.51008801250395-0.532674046186954 EUR 2Y 9M OIS -0.5205 2020-04-14 2023-01-16 1007 1.01469016224648-0.480516631238024-0.528590572308455 EUR 3Y OIS -0.5149 2020-04-14 2023-04-17 1098 1.01585441051151-0.453393445413074-0.522902129681962 EUR 3Y 3M OIS -0.5095 2020-04-14 2023-07-17 1189 1.01699804005016-0.444863072114309-0.517421724564593 EUR 3Y 6M OIS -0.5037 2020-04-14 2023-10-16 1280 1.01810094488266-0.428556718139346-0.511543884644207 EUR 3Y 9M OIS -0.4973 2020-04-14 2024-01-16 1372 1.0191663387116-0.409053185504009-0.505066837835035 EUR 4Y OIS -0.4902 2020-04-14 2024-04-16 1463 1.02015692291336-0.384136455512101-0.49788847789851 EUR 4Y 3M OIS -0.4823 2020-04-14 2024-07-16 1554 1.02107708684452-0.356506771656568-0.489908863706539 EUR 4Y 6M OIS -0.4736 2020-04-14 2024-10-16 1646 1.02193394798843-0.328097027948742-0.481127201322867 EUR 4Y 9M OIS -0.4645 2020-04-14 2025-01-16 1738 1.02272673203556-0.303326230567583-0.471944730672365 EUR 5Y OIS -0.4549 2020-04-14 2025-04-16 1828 1.02342110401107-0.27139247873125-0.462260891980387 EUR 5Y 3M OIS -0.4453 2020-04-14 2025-07-16 1919 1.02407979838881-0.254455162191572-0.452578158870841 EUR 5Y 6M OIS -0.4353 2020-04-14 2025-10-16 2011 1.02467918453885-0.228893501572751-0.442493491309696 EUR 5Y 9M OIS -0.4251 2020-04-14 2026-01-16 2103 1.02521491033727-0.204475990398409-0.432207542589984 EUR 6Y OIS -0.4147 2020-04-14 2026-04-16 2193 1.02566162717677-0.174216068015335-0.42172053863055 EUR 6Y 3M OIS -0.4042 2020-04-14 2026-07-16 2284 1.02606053035214-0.153799746629003-0.411132251765344

Figure 3.3: The bootstrapped discount curve for 3-months Euribor at 2020-04-14 (Table 3.4)

Name Quote Measured date From date To date EURIBOR 6M -0.195 2020-04-14 2020-04-16 2020-10-16 EUR 1X7 FRA -0.223 2020-04-14 2020-04-16 2020-11-16 EUR 2X8 FRA -0.25 2020-04-14 2020-04-16 2020-12-16 EUR 3X9 FRA -0.269 2020-04-14 2020-04-16 2021-01-18 EUR 4X10 FRA -0.288 2020-04-14 2020-04-16 2021-02-16 EUR 5X11 FRA -0.307 2020-04-14 2020-04-16 2021-03-16 EUR 6X12 FRA -0.316 2020-04-14 2020-04-16 2021-04-16 EUR 7X13 FRA -0.326 2020-04-14 2020-04-16 2021-05-17 EUR 8X14 FRA -0.335 2020-04-14 2020-04-16 2021-06-16 EUR 9X15 FRA -0.337 2020-04-14 2020-04-16 2021-07-16 EUR 10X16 FRA -0.338 2020-04-14 2020-04-16 2021-08-16 EUR 11X17 FRA -0.339 2020-04-14 2020-04-16 2021-09-16 EUR 12X18 FRA -0.336 2020-04-14 2020-04-16 2021-10-18 EUR 2Y IRS -0.294 2020-04-14 2020-04-16 2022-04-18 EUR 3Y IRS -0.287 2020-04-14 2020-04-16 2023-04-17 EUR 4Y IRS -0.262 2020-04-14 2020-04-16 2024-04-16 EUR 5Y IRS -0.227 2020-04-14 2020-04-16 2025-04-16 EUR 6Y IRS -0.191 2020-04-14 2020-04-16 2026-04-16 EUR 7Y IRS -0.154 2020-04-14 2020-04-16 2027-04-16 EUR 8Y IRS -0.114 2020-04-14 2020-04-16 2028-04-17 EUR 9Y IRS -0.075 2020-04-14 2020-04-16 2029-04-16 EUR 10Y IRS -0.036 2020-04-14 2020-04-16 2030-04-16 EUR 11Y IRS 0.003 2020-04-14 2020-04-16 2031-04-16 EUR 12Y IRS 0.04 2020-04-14 2020-04-16 2032-04-16 EUR 13Y IRS 0.075 2020-04-14 2020-04-16 2033-04-18 EUR 14Y IRS 0.105 2020-04-14 2020-04-16 2034-04-17 EUR 15Y IRS 0.13 2020-04-14 2020-04-16 2035-04-16 EUR 16Y IRS 0.15 2020-04-14 2020-04-16 2036-04-16 EUR 17Y IRS 0.164 2020-04-14 2020-04-16 2037-04-16 EUR 18Y IRS 0.174 2020-04-14 2020-04-16 2038-04-16 EUR 19Y IRS 0.18 2020-04-14 2020-04-16 2039-04-18 EUR 20Y IRS 0.182 2020-04-14 2020-04-16 2040-04-16 EUR 21Y IRS 0.181 2020-04-14 2020-04-16 2041-04-16 EUR 22Y IRS 0.178 2020-04-14 2020-04-16 2042-04-16 EUR 23Y IRS 0.172 2020-04-14 2020-04-16 2043-04-16 EUR 24Y IRS 0.165 2020-04-14 2020-04-16 2044-04-18 EUR 25Y IRS 0.156 2020-04-14 2020-04-16 2045-04-17 EUR 26Y IRS 0.146 2020-04-14 2020-04-16 2046-04-16 EUR 27Y IRS 0.135 2020-04-14 2020-04-16 2047-04-16 EUR 28Y IRS 0.124 2020-04-14 2020-04-16 2048-04-16 EUR 29Y IRS 0.113 2020-04-14 2020-04-16 2049-04-16 EUR 30Y IRS 0.102 2020-04-14 2020-04-16 2050-04-18 EUR 40Y IRS 0.02 2020-04-14 2020-04-16 2060-04-16