Multi-curve models and a national bonds market model

Università di Pisa

Facoltà di Scienze Matematiche, Fisiche e Naturali

Tesi di Laurea Magistrale

Multi-curve models and a national bonds

market model

27 Ottobre 2017

Relatore: Prof. Maurizio Pratelli

Controrelatore: Prof. Marco Romito

Contents

1 Introduction 3

1.1 Preliminaries and notations . . . 5

1.2 Main denitions . . . 6

1.3 The classical interest rate models . . . 8

1.4 The change of numeraire . . . 9

1.5 Main derivatives . . . 11

1.6 The reasons for a new model . . . 15

1.7 Bootstrapping techniques . . . 17

2 Short rate approach 20 3 HJM approach 28 3.1 Parsimonious approach . . . 28

3.2 Complete approach . . . 36

3.2.1 Choice of the volatility . . . 40

3.2.2 Pricing swaptions . . . 41

3.2.3 Bootstrapping and calibration . . . 43

4 National bonds model 44 4.1 Introduction . . . 44

4.2 Hull-White spread . . . 46

4.3 Pricing derivatives with Fourier transform . . . 48

4.4 Application to the model . . . 50

4.5 Calibration . . . 53

4.6 CIR++ spread . . . 53

4.7 Adding covariance . . . 57

Chapter 1

Introduction

In the classical interest rate models the important no-arbitrage relationships that relate the prices of the zero-coupon bonds and the LIBOR rates have always been assumed to hold. Inside a model these equalities allow to hedge perfectly forward-rate agreements in terms of zero-coupon bonds. As a consequence forward rates of dierent tenors are related to each other by sharp constraints that however might not hold in practice. For long time the gap between the models and the real market has been considered negligible, but starting from summer 2007, with the beginning of the credit crunch, the market quotes of zero-coupon bonds and of the LIBOR rates started to violate the classical constraints in a macroscopic way. This because the counterparties began to consider the default of their creditors a realistic possibility, and lending money became a risky investment. In Figure 1.1 the historical data of the EURIBOR rates from 2004 to the end of 2017 are plotted. The graphic shows that, the nancial crisis caused a marked increase of the spreads between the rates, a symptom of the increasing didence between the counterparties. Other evidences can be seen for example in the historical data of the spread between the OIS swap rate with maturity one year and the EURIBOR swap rate with the same maturity.While before the crisis it was nearly zero, consistently with the usual interest rate models, after the summer 2007 the situation drastically changed, and the OIS-IRS spread started growing, making the nancial scenario totally inconsistent with the traditional models. This brought to the need to model separately the forward rates of the various tenors quoted in the market, since they couldn't be obtained from the classical one-curve models any more. The result is a multi-curve model where each forward rate dynamic of a tenor quoted in the market is described by a stochastic equation. The aim of the work is to present the main approaches proposed in literature for the construction of a multi-curve framework.

The rst chapter is an introduction of the main denitions, the classical no-arbitrage for-mulas and the main interest rate derivatives, with a focus on the dierences of the forfor-mulas in the case of the classical models and in the multi-curve models. Also, the main techniques needed to bootstrap the forward rates curve are described.

Figure 1.1: The gure shows the daily quotations of the EURIBOR rates of tenors 1 month, 6 months, 6 months and one year, from the 1st _{January 2004 to the 3}rd _{August 2017. Data}

obtained from http://www.bundesbank.de .

The second chapter is about a multi-curve model adopting a short rate approach. It is assumed that a risk-free short rate and a risky short rate exist, and dynamics for the risk-free rate and for the spread between the two rates are chosen. Moreover, a new formula for FRA contracts is obtained and it can be used in the calibration.

In the third chapter two models adopting an HeathJarrowMorton approach are described. For each tenor quoted in the market it is assumed that an instantaneous forward rate exists and an HJM-dynamic is chosen. In both models formulas for pricing swaptions are deduced and they can be used in the calibration.

In the fourth chapter a model for the national bonds market is presented. The shapes of the yield curves of the European states, while are similar for short maturities, present signicant spreads for long maturities. This reects the dierent reliability the investors has in the states, and the dierent liquidity risk attributed to the states by the market. This suggests, as in multi-curve frameworks, to produce a model, where each national bond dynamic is described by a dierent stochastic equation. Hence, as in the multi-curve short rate model, for each state it is assumed that a short rate exists, modelled as sum of a risk-free rate and a spread. The result is a multi-curve model for the bonds market. The last part also presents a possible pricing methodology for call and put options with underlying a national bond.

1.1 Preliminaries and notations

EONIA

The EONIA rate (Euro OverNight Index Average) is a rate published every business day at 19.00 and calculated by the European Banking Federation (EBF). It's a weighted average of the rates of all the loans between the most important European banks, happened during the business day, of a tenor of approximately one day. The weights used are the amounts of the transactions. Since the EONIA rate refers to transactions with so brief tenor, it can be approximately considered a risk-free rate.

EURIBOR

The EURIBOR rates (Euro OverNight Inter Bank Oered Rate) are rates published every business day and calculated by the European Banking Federation (EBF). Each of them is a weighted average of all the rates of loans between the most important European banks, happened during the business day, of a xed tenor. The weights used are the amounts of the transactions. The LIBOR rates are available for the tenors of 1 month, 3 months, 6 months and one year. With the nancial crisis the LIBOR rates began to violate the classical no-arbitrage relationships, and for this we considered these rates risky rates.

LIBOR

The LIBOR rates (London Inter Bank Oered Rate) are rates published every business day and calculated by Thomson Reuters. Each of them is a weighted average of all the rates of loans of a xed tenor, between the most important international banks, happened during the business day. The weights used are the amounts of the transactions. The LIBOR rates are available for seven tenors ranging from overnight to one year.

We will refer indierently with LIBOR rates both to LIBOR and to EURIBOR rates. We indicate with L(T − ∆, T ) the LIBOR rates published in T − ∆ with tenor ∆. We now recall the rst fundamental theorem, and the denition of the martingale measure.

Theorem 1 (First fundamental theorem). Suppose (Ω, F, P) a probability space, and FW

t a

ltration generated by a d-dimensional Brownian motion Wt, and suppose there exists a set of

positive Ito's processes {Bt, St1, . . . , Stn}, adapted to the ltration Ft where

dBt= rtdt

Suppose also there exist a probability Q equivalent to P such that each proces Sit

Bt is a

The equivalent probability Q will be called indierently martingale probability or risk-neutral measure. In the models presented in the work we will always suppose the existence of the martingale measure to guarantee the no-arbitrage hypothesis.

Corollary 1 (Pricing formula). If X is a positive T -measurable random variable, then its price πX

t at time t, under the Q measure, is

π_tX = BtE  X BT     Ft 

We will use the notation Et[ ] instead of Et[ |Ft] when it will be clear from the context

respect to what ltration we are evaluating the conditional expectation.

1.2 Main denitions

Denition 1. A T -maturity zero-coupon bond is a contract that guarantees its holder 1 unit of currency at time T . The contract price at time t < T is denoted by Pt(T ). Clearly, PT(T ) = 1

for all T .

We suppose the existence of the maximum future bond maturities T∗ _{and, at each time}

instant t, we suppose the existence of all the zero-coupon bond Pt(T ), for all 0 ≤ T ≤ T∗.

At each time instant t the function T −→ Pt(T ) is quite regular while, xed T , the function

t −→ Pt(T ) turn up to have marked oscillations, hence we model t −→ Pt(T ) as a stochastic

process.

Now we present the principal rates and the so called no arbitrage relationships between the interest rates and the zero-coupon bond. We begin with the following observation. At time t 1 unit of currency can be used for buying a zero-coupon bond fraction 1

Pt(T ), since 1 = Pt(T )

1 Pt(T ).

This gives the right at time T to the amount 1

Pt(T ). The continuously-compounded interest rate

between t and T of this investment is called Yield. It must satises 1

Pt(T )

= eYt(T )(T −t)

Hence we have the following denition

Denition 2 (Yield). The continuously-compounded interest rate at time t for the maturity T is the constant rate at which an investment of 1 unit of currency at time t accrues continuously to yield the amount of currency 1

Pt(T ) at maturity T , namely

Y (t, T ) = −log Pt(T ) T − t

Similarly we give the denition of the linear interest rate L(t, T ), that is the rate such that such that

1 Pt(T )

= 1 + L(t, T )(T − t)

Denition 3 (LIBOR). The linear interest rate at time t for the maturity T is the constant rate at which an investment of 1 unit of currency at time t accrues to yield the amount of currency 1

Pt(T ) at maturity T , namely

L(t, T ) = 1 − Pt(T ) Pt(T )(T − t)

(1.1) We will refer to the important formula (1.1) as the no-arbitrage relationship for the LIBOR rates.

Let t < S < T three time instants and Pt(S), Pt(T ) the prices of the zero coupon bonds

with maturity S and T . An investment of 1 at time t gives the amount 1

Pt(S) at time S and

1

Pt(T ) at time T . Now we can dene the forward rate Et(S, T ) as the rate that satisfy

1 Pt(S)

= eEt(S,T )(T −S)₌ 1

Pt(T )

Denition 4 (Forward rate). The continuously-compounded forward rate at time t from S to T is

Et(S, T ) =

log Pt(S) − log Pt(T )

T − S

Similarly we give the denition of the forward LIBOR rate, that is the rate such that 1

Pt(T )

= 1 + L(t, T )(T − t)

Denition 5 (Forward LIBOR rates). The linear interest rate at time t from S to T is Ft(S, T ) =

Pt(S) − Pt(T )

Pt(T )(T − S) (1.2)

We will refer to the important formula (1.1) as the no-arbitrage relationship for the forward LIBOR rates. We eventually dene the dierential form of the rates introduced, since they are often the starting point of interest rates models.

Denition 6 (Instantaneous forward rate). We dene the instantaneous forward rate as ft(T ) = lim

∆T →0Et(T, T + ∆T ) = −

∂

∂T log Pt(T ) Denition 7 (Instantaneous short rate). We dene the short rate as

r(t) = ft(t) = lim

∆t→0Y (t, t + ∆t)

From the denition of the forward rate it's easy to see that Pt(T ) = e− RT

t ft(u)du. In all the

models we will present we will use as discounting factor the bank account Bt = e Rt

1.3 The classical interest rate models

The traditional models can be classied in two families • short rate models

• HJM models

We recall briey for both the main features.

Short rate models

The short rate models consist in choose a dynamic for the short rate rt, directly under the

risk-neutral measure Q, that is choose µt and σt such that

drt= µtdt + σtdWt

We recall some of important models for the short rate • Vasicek: drt = (b − art)dt + σdWt • C.I.R: drt= a(b − rt)dt + σ √ rtdWt • Ho-Lee: drt= θ(t)dt + σdWt • Hull-White: drt= (θ(t) − art)dt + σdWt

Once chosen a dynamic for the short, can be dened the bank account Bt = e Rt

0rudu and the

risk-free bond Pt(T ) = Et[e− RT

t rudu]. Since the model is directly under the risk-neutral measure,

the processes Pt(T )

Bt are all martingales. Hence if we set

Pt(T ) = Et[e− RT

t rudu] = eA(t,T )−rtB(t,T ) A(T, T ) = 0 B(T, T ) = 0 (1.3)

we can derive conditions for the functions A(t, T ) and B(t, T ) that guarantee that the processes

Pt(T )

Bt are martingales. The following theorem clarify when can be derived conditions for the

processes A(t, T ) and B(t, T ).

Theorem 2 (Due-Kan). Suppose the short rate rt satisfy, under the risk-neutral measure Q,

the stochastic equation drt= µ(r, t)dt + σ(r, t)dWt, where the functions µ and σ are in the form

µ(t, r) = α(t)r + β(t) σ(t, r) =pγ(t)r + δ(t)

with α, β, γ, δ are continuous functions, thus exist functions A(t, T ) and B(t, T ) such that (1.3) holds and such that

(

Bt(t, T ) + α(t)B(t, T ) − 1₂γ(t)B2(t, T ) = −1 B(T, T ) = 0

At(t, T ) = β(t)B(t, T ) − 1₂δ(t)B2(t, T ) A(T, T ) = 0

If there exist functions that solve the system (1.4) we say that the model has an ane term structure.

If the system (1.4) admits an explicit solution we can obtain explicit expressions for the zero-coupon bonds Pt(T ), and for the LIBOR rates, from the no-arbitrage relationships.

HJM models

The Heath Jarrow Morton (HJM) models consist in choose a dynamic for the instantaneous forward rate ft(T ), directly under the risk-neutral measure Q, that is choose processes αt(T )

and σt(T ) such that

ft(T ) = αt(T )dt + σt(T )dWt

where Wt is a Brownian motion under the risk-neutral measure. We can dene directly the

zero-coupon bond as Pt(T ), the short rate rt and th bank account Bt with the usual formula.

Under the risk-neutral measure the processes Pt(T )

Bt must be martingales for all T . The following

important theorem shows the conditions on the processes αt(T )and σt(T ), that guarantee that

the processes Pt(T )

Bt are martingales.

Theorem 3 (Heath Jarrow Morton). The processes Pt(T )

Bt are martingales if and only if

αt(T ) = σt(T )

Z T t

σt(u)du (1.5)

Once chosen an expression for the volatility σt(T ) we can set

ft(T ) = f0∗(T ) + Z t 0 αs(T )ds + Z t 0 σs(T )dWs where f∗

0(T )are the instantaneous forward rates values observed in the market, hence the model

is automatically calibrated to the bond prices.

1.4 The change of numeraire

In this section we recall the denition of the numeraire and the important change of numeraire theorem.

Denition 8. A positive-valued stochastic process Dt adapted to the ltration is called

Theorem 4. Suppose the existence of a market {S1

t, . . . , Stk} and suppose that the processes Si

Bt

and Dt

Bt are martingales under a probability Q. We consider

dQD dQ = DT BTD0 The processes Si t

Dt are martingales under Q

D_.

Often will be useful, in the discussion of the following chapters, taking as numeraire the process Pt(T ). If Pt_B(T )_t is a martingale under Q, a property that will always be respected in

the models will be presented, Pt(T ) can be taken as numeraire. The equivalent probability

obtained by this change of numeraire will be indicated QT _{and called T -forward measure, and}

the relative expectation indicated with ET. Now we can present the important equality that

relates the instantaneous forward rate with short rate. Proposition 1. We have that

ft(T ) = EtT[rT] Proof. E_tT[rT] = 1 Pt(T ) Et[rTe− RT t rudu_] = − 1 Pt(T ) Et[ ∂ ∂Te −RT t rudu] = − 1 Pt(T ) ∂ ∂TEt[e −RT t rudu] = − ∂ ∂TPt(T ) Pt(T ) = ft(T )

Last proposition in particular shows that the forward rate is a martingale under the T -forward measure. We recall that, from the HJM theorem, we have that under the martingale measure the forward rate must follow an equation of the form

dft(T ) = σt(T )

Z T

t

σt(u)dudt + σt(T )dWt

where Wt is a Brownian motion under the martingale measure. If we take the probability QT

obtained taking the numeraire Pt(T ), from the Girsanov's theorem there exist a process Ht

such that

is a Brownian motion under QT_{. Hence the dynamic of f} t(T ) is dft(T ) = σt(T ) Z T t σt(u)dudt + σt(T )Htdt + (. . . )dWtT

hence, since ft(T ) is QT-martingale we have that

Ht= Z T t σt(u)du and hence dW_tT = dWt+ Z T t σt(u)dudt

is a Brownian motion under the T -forward measure.

1.5 Main derivatives

Forward rate Agreement (FRA)

Denition 9. A Forward Rate Agreement (FRA) is an agreement between two counterparties, where they exchange a payment at the oating rate with a payment at a xed rate.

A FRA is a contract involving three time instants, namely the current time t and two future instants S < T . The contract allows the holder to exchange a oating rate payment with a xed rate payment, for the period between S and T . Formally at time T the holder pays

(T − S)(L(S, T ) − K)

where L(S, T ) is the oating rate observed in S for the period [S, T ] and K is the xed rate. The par value of the payo at time t is

π_tFRA(S, T, K) = (T − S)Et  Bt BT (L(S, T ) − K)  = (T − S)Pt(T )EtT [L(S, T ) − K]

We dene the KFRA

t the value of K that makes the contract fair at the time t, namely the value

of K such that the price of the FRA contract is 0 at time t. We obtain

K_tFRA(S, T ) = E_tT[L(S, T )] (1.6) where ET _{is the expectation in the probability obtained with the change of numeraire P}

t(T ).

If we suppose that the classical no-arbitrage formula (1.1) holds we obtain K_tFRA(S, T ) = 1 T − S  Pt(S) Pt(T ) − 1  = Ft(S, T )

This shows that in the case of the classical one-curve model, the fair rate of a FRA contract is equal to the forward rate Ft(S, T ). We can generalize the denition of the forward rate to the

case that the usual no arbitrage formulas don't hold. Let's suppose that we have 1 unit at time tand loan it to a counterparty for the period between S and T at a oating rate. Let's suppose that the counterparty wants to exchange the oating rate with a xed rate. The rate that we oer is obviously the fair rate of the FRA contract. In other words we oered at time t a xed interest rate KFRA

t (S, T ) for a loan for the period between S and T . After this observation we

can generalize the denition of the forward rate.

Denition 10 (Simply-compounded forward interest rate). We dene the simply-compounded forward rate at time t between the future dates S < T as

Ft(S, T ) = EtT[L(S, T )] (1.7)

We observe that the forward rate at time t, under the measure PT_{, is a martingale and so}

the models that we will introduce must respect this property.

Interest Rate Swap (IRS)

Denition 11. An Interest Rate Swap (IRS) is an agreement between two counterparties, where they exchange a ow of payments at the oating rate with a ow of payments at a xed rate.

Let's consider some future dates T = {T0, . . . , Tn} and let's suppose that we are exposed

to a oating payment at each Ti for i = 1, . . . , n, at the LIBOR rate, observed in Ti−1, for the

period from Ti−1 to Ti. At each Ti for i = 1, . . . , n we pay

τiL(Ti−1, Ti)

where τi = Ti− Ti−1. If we exchange the oating rate with a xed rate K, we have to pay to

the counterparty at time Ti the quantity

τi(L(Ti−1, Ti) − K)

The price of this exchange at time t is obviously Et  Bt BT τi(L(Ti−1, Ti) − K)  = τiPt(Ti)EtTi[L(Ti−1, Ti) − K]

hence the price at time t of the IRS is πIRS_t (T , K) =

n

X

i=1

We dene the swap rate the value of K that makes the contract fair at time t, namely the value of K that makes the price equal to 0 in t, that is

K_tIRS(T ) = Pn i=1τiPt(Ti)E Ti t [L(Ti−1, Ti)] Pm i=1τiPt(Ti) (1.8) This denition can be extended to the case that the dates of the oating leg are dierent from those of the xed leg. If we call with T = {T0, . . . , Tn}the oating leg payment dates and with

T0 _{= {T}0

0, . . . , T 0

m} the xed leg payment dates, we can obtain in the same way the price of the

IRS at time t πIRS_t (T , T0, K) = n X i=1 τiPt(Ti)EtTi[L(Ti−1, Ti)] − K m X j=1 τ_j0Pt(Tj0) (1.9)

and the value of the swap rate at time t K_tIRS(T , T0) = Pn i=1τiPt(Ti)E Ti t [L(Ti−1, Ti)] Pm j=1τ 0 jPt(Tj0) (1.10) We can also deduce the formula in the case the usual no-arbitrage formulas hold. We obtain

K_tIRS(T , T0) = PPt(Tm0) − Pt(Tn) j=1τ

0 jPt(Tj0)

(1.11) Denition 12. An IRS contract where the oating rate is the EONIA rate is called overnight indexed swap (OIS)

The formulas for the OIS price and the OIS rate can be simply deduced from (1.9) and (??).

Swaptions

Denition 13. A swaption is a contract that gives the holder the right to sell (or to buy) an IRS at a future date.

Let's consider the case of the sale of an IRS at a future date T with oating leg future dates T = {T0, . . . , Tn}, xed leg future dates T0 = {T00, . . . , Tn0} and xed rate K. Obviously we

suppose T ≤ min{T0, T=0}. The holder of the swaption at time T will sell the IRS only if its

price is positive, thus the price of the swaption at time T is (πIRS_T (T , T0, K))+ Hence the par value of the swaption at time t is

π_tSWPT(T , K) = Et  Bt BT (π_TIRS(T , T0, K))+  (1.12)

Now we dene the swap numeraire as At(T0) = m X j=1 τ_j0Pt(Tj0)

It's easy to see that

πIRS_T (T , T0, K) = (K_TIRS(T , T0) − K)AT(T0)

hence from (1.12) we obtain

π_tSWPT(T , K) = Et  Bt BT AT(T0)(KTIRS(T , T 0_{) − K)}+  Now we can observe that the process At(T0)

Bt is a martingale under the risk-neutral measure and

so At(T0) can be used as numeraire. We call the probability obtained the swap measure, we

indicate it with QT0

and the expectation respect to that probability with ET0

. So, using the theorem of the change of numeraire we have that

πSWPT_t (T , T0, K) = At(T0)ET

0

t (KTIRS(T , T 0

) − K)+ (1.13) Last formula shows how the price of a swaption can be calculated as the price of a PUT with underlying the swap rate and the xed rate as strike. Moreover KIRS

t is a martingale under the

swap measure and so the price of the swaption can be calculated as

π_tSWPT = At(T0)BlPUT(KtIRS, K, Σ2) = At(T0)(Kφ(−d2) − KtIRSφ(−d1))

d1,2 = log(KtIRS K ) ± 1 2Σ 2_{(t, T )} pΣ2_{(t, T )} Σ 2_{(t, T ) =} Z T t σ(s)2ds The volatility σ(t) can be calculated in two ways

• Implicit volatility: we consider some market swaptions price, we invert the Black's formula and we nd numerically a funciton σ(t) that ts all the market prices. Then the volatility determined can be used to calculate other swaptions prices. The function σ can be chosen also dependent from the strike K;

• Inside a model: the process K_tIRS is a positive martingale under the forward measure, so inside a model its dynamic will be

dK_tIRS = σtKtIRSdW τ0

t

where Wτ0

t is a Brownian motion under the forward measure and σtis a stochastic process.

Basis Swap (BS)

Denition 14. A basis swap is a contract between two counterparties where they exchange a ow of oating payments with a ow of xed payments and an other ow of oating payments. Let's consider three sets of dates T = {T0, . . . , Tn}, T0 = {T00, . . . , Tm0 }, T00 = {T000, . . . , Tp00}

with T0 = T00 = T 00

0, Tn= Tm0 = T 00

p, and three ows of payments: two payments at the oating

rate related to the dates T and T0 _{where, for the rst ow, the rate is observed in T}

i−1 and the

payment occurs in Ti for i = 1, . . . , n, while for the second ow the rate is observed in Tj−10 and

the payment occurs in T0

j for j = 1, . . . , m; one payment at a xed rate K related to the set of

dates T00_{. If we dene as usual τ}

i = Ti − Ti−1, τj0 = T 0 j − T 0 j−1, τ 00 h = T 00 h − T 00 h−1, we obtain the

par value of the contract at time t π_tBS(T , T0, T00, K) = n X i=1 τiPt(Ti)EtTi[L(Ti−1, Ti)] − m X j=1 τ_j0Pt(Tj0)E Tj0 t [L(T 0 j−1, T 0 j)] − K p X h=1 τ_h00Pt(Th00) (1.14)

We call basis swap spread the value of K that makes the contract fair at time t, obtainable by simply setting formula (1.14) equal to zero

K_tBS(T , T0, T00) = Pn i=1τiPt(Ti)EtTi[L(Ti−1, Ti)] Pp h=1τ 00 hPt(Th00) − Pm j=1τ 0 jPt(Tj0)E T_j0 t [L(T 0 j−1, T 0 j)] Pp h=1τ 00 hPt(Th00) (1.15) Observation 1. It's easy to see that

K_tBS(T , T0, T00) = K_tIRS(T , T00) − K_tIRS(T0, T00)

so the basis swap spread is the dierence between two swap rates. If we use the no-arbitrage relationship we have that

K_tBS(T , T0, T00) = PPt(Tp 0) − Pt(Tn) j=1τ 00 jPt(Tj00) −Pt(T 0 0) − Pt(Tn0) Pp j=1τ 00 jPt(Tj00) = 0

This observation shows that, if the classical no-arbitrage formulas remains true, the value of the basis swap spread of an IRS must be equal to zero. This has been true until August 2007 when with the beginning of the crisis some of basis swap spreads began to growth. This was an evidence of the fact that the classical non-arbitrage relationships where no more valid.

1.6 The reasons for a new model

Consider a counterparty A, exposed to a a ow of future payments, with dates {0 = T0, T1, . . . , Tn},

Figure 1.2: The gure shows the daily quotations of the 3 months EURIBOR and of the OIS rate with the same maturity, from the 1st _{January 2006 to the 30}th _{June 2014. Data obtained}

from http://www.bundesbank.de.

the ow of payments with only one payment in Tn. If we suppose the classical no arbitrage

relationships for the LIBOR rate hold, the par oered rate, given by formula (1.10), becomes K₀OIS= ∆1d Pn i=1P0(Ti)E Ti 0 [L(Ti−1, Ti)] ∆_3mP0(Tn) = 1 ∆_3m  1 P0(Tn) − 1 

Last expression, in the classical single curve model, is the xed LIBOR rate at time 0, with tenor ∆. Hence, the par rate of the swap is equal to the LIBOR rate with tenor 3. The rst gure shows the daily quotations of the three months EURIBOR rate and of the OIS swap rate with the same maturity, while the second is the daily spread between the two quotes. The gures shows clearly that the indexes were almost equal before 2009, with a spread near to 0, but after the nancial crisis the spread couldn't be considered negligible any more. Let's try to understand the reasons that brought to this situation. Since the oating leg pays a oating interest rate with tenor 1 day, that can be considered a riskless rate, it can't be inclined to accept a EURIBOR rate as xed rate, because it would pay a risk-rate and sell a risk-free payment. So the contract wouldn't be at par. So the oered rate can't be the EURIBOR, but a risk-free rate for the maturity of three months. This explains the beginning of the raise of the observed spread. This is an example of the inconsistency of the classical no-arbitrage relationships, but many others could be shown.

Since the LIBOR rates couldn't be any more obtained form the classical no-arbitrage re-lationships, they can be modelled as dierent processes and choose a dynamic for each one of

Figure 1.3: The gure shows the daily dierence between the daily quotations of the 3 months EURIBOR and of the OIS rate with the same maturity, from the 1st _{January 2006 to the 30}th

June 2014. Data obtained from http://www.bundesbank.de.

them. The so done frameworks are the multi-curve models. So, at time 0, for each LIBOR rates Ft(T − ∆, T ), must be available the quantities F0(T − ∆, T ) for each maturity T , that

is, at time 0 must be built the family of the curves T −→ F0(T − ∆, T ), one for each tenor ∆.

This explains the name multi-curve model. The next section will describe in details the main procedures for the bootstrapping of the curves.

1.7 Bootstrapping techniques

Typically the curves that we will need to bootstrapped in the model will be presented are T −→ P0(T ) T −→ F0(T − ∆, T )

The rst is called zero-coupon curve and the second is a family of curves called forward curves, one for each tenor ∆ that appears in the model. We now explain for each curve the needed instruments, and the procedure for the bootstrapping.

The zero-coupon curve

The procedure we describe for the bootstrap of the zero-coupon curve need the xed EONIA and a set of OIS prices with T = T0 _{and ∆}

rate we can suppose the usual no-arbitrage relationship between the LIBOR rates and the bond prices, since we have supposed the risk in the EONIA rate is negligible. This means that

EOt(T − ∆1d, T ) = 1 ∆_1d  Pt(T − ∆1d) Pt(T ) − 1  = E_tT [L(T − ∆_1d, T )] (1.16) thus we can obtain the rst value of the curve P0(∆1d) from EONIA xing as

P0(∆1d) =

1

EO0(0, ∆1d)∆1d+ 1

Other values are determined with the set of OIS prices. The OIS fair rate at time 0 is K₀OIS(T ) = Pn i=1P0(Ti)E Ti 0 [L(Ti−1, Ti)] Pn i=1P0(Ti)

Using again equality (1.16), we can obtain a formula for P0(Tn) depending from the previous

dates P0(Tn) = P0(T0) − K0OIS(T )∆1d Pn−1 i=1 P0(Ti) 1 + ∆_1dK₀OIS(T )

Once computated a set of P0(Ti) for enough i we can simply interpolate them with the usual

numerical tecniques and obtain the zero-coupon curve.

The forward curves

Suppose we have already bootstrapped the zero-coupon curve and see we how to bootstrap the others. The procedure that will be illustrated need the following instruments for the bootstrap: • for ∆_1m: EURIBOR 1m xing, FRA rates up to one year, swap rates of contract paying an annual x rate for the EURIBOR 1m rate (some of them can be substituted with FRA rates);

• for ∆_3m: EURIBOR 3m xing, FRA rates up to one year, swap rates of contract paying an annual x rate for the EURIBOR 3m rate (some of them can be substituted with FRA rates);

• for ∆_6m: EURIBOR 6m xing, swap rates of contract paying an annual x rate for the EURIBOR 6m rate (some of them can be substituted with FRA rates);

• for ∆_1y: EURIBOR 1y xing, six-vs-one-year basis-swaps rates (some of them can be substituted with FRA rates).

Let's see how to use swap rates and FRA rates in the bootstrapping of the curve of tenor ∆. We suppose the swaps to be referred to dates such that τi = ∆ for all i and τj0 = ∆

0 _{for all j.}

Formula (1.10) becomes at time t = 0 K₀IRS(T , T0) = ∆ Pn i=1P0(Ti)F0(Ti− ∆, Ti) ∆0Pm j=1P0(T 0 j)

This brings to a formula for the forward rate F0(Tn− ∆, Tn)

F0(Tn− ∆, Tn) = K₀IRS∆0Pm j=1P0(Tj0) − ∆ Pn−1 i=1 P0(Ti)F0(Ti− ∆, Ti) P0(Tn) (1.17) This shows that the forward rate F0(Tn− ∆, Tn) can be calculated from the previous rates. So

we can use FRA rates for the rst values and then the swap rates for the last values using the formula (1.17). Once determined a set of P0(Ti) we can bootstrap the forward curve with the

usual numerical techniques. Once bootstrapped the 6 months forward curve, we can bootstrap the 1 year forward curve with the basis swaps. For the basis-swap considered we have that T0 = T00 and τi = ∆6m for all i and τj0 = ∆1y. Formula (1.15) reduces to

K₀BS(T , T0, T0) = ∆6m Pn i=1P0(Ti)F0(Ti− ∆6m, Ti) ∆_1yPm j=1P0(T 0 j) − Pm j=1P0(T 0 j)F0(Tj0− ∆1y, Tj0) Pm j=1P0(T 0 j)

and we can obtain again that F0(Tm− ∆1y, Tm) = (K₀IRS(T , T0) − K₀BS(T , T0, T0))Pm j=1P0(T 0 j) P0(Tm0 ) − Pm−1 j=1 P0(T 0 j)F0(Tj0− ∆1y, T 0 j) P0(Tm0 ) (1.18)

We point out that, for the considered basis swaps, m is equal to 1, thus formula (1.18) give explicitly an expression for F0(Tm − ∆1y, Tm) only depending from the 6 months curve and

the zero-coupon curve. Again, ones obtained a set of values of the curve, we proceed with the interpolation.

Chapter 2

Short rate approach

In this section we will present a short rate model to price FRA contracts of a xed tenor ∆, that typically is 1 month, 2 months, 6 months or 1 year. We will model

• the short rate rt

• a spread st, related to the tenor ∆, between a risky short rate rt+ st and the short rate

rt

We point out that the rate rt+ st is ctitious and not observable in the market. The model

can be generalized to a set of tenors, and in that case we will model a family of spreads s∆ t ,

one for each ∆. Starting from the two rates we can dene their respective bonds: the risk-free bond and a risky bond

Pt(T ) = Et[e− RT

t rudu] P¯_t(T ) = E_t[e−

RT

t ru+sudu]

In the classical one curve models the LIBOR rates can be dened by the risk-free bond as in (1.1), but in the multi-curve framework they will be modelled as risky rates. Hence we dene the LIBOR rates substituting in (1.1) the risk-free bond with the risky bond. Thus we have the following denition of the LIBOR rates

L(T, T + ∆) = 1 ∆  1 ¯ Pt(T + ∆) − 1  (2.1)

The model

We suppose the existence of three independent processes ψ1

t, ψt2, ψt3 and we dene the short rate

rt and the spread st as

rt= ψt2− ψ 1 t st = kψt1+ ψ 3 t

where k is a positive constant. We suppose the following dynamics for the processes introduced under the risk-neutral measure

dψ1_t = (a1− b1_ψ1 t)dt + σ1dWt1 dψ2_t = (a2− b2_ψ2 t)dt + σ 2p ψ2 tdW 2 t dψ3_t = (a3− b3_ψ2 t)dt + σ 3p ψ3 tdWt3

where ai_{, b}i_{, σ}i _{are positive constants with a}2 _{≥ (σ}2₎2_{/2, a}3 _{≥ (σ}3₎2_{/2, and W}i

t are independent

Brownian motions. We recall that the rst stochastic equation is a Vasicek-type equation and can be explicitly solved.

Pricing FRA

We now present a formula to price FRA, that can be used in the calibration of the model. All over the section we will consider a FRA at time t, with future dates T and T + ∆ and strike K. We have seen in Section 1.5 that its price is given by

π_tFRA(T, T + ∆, K) = ∆Pt(T + ∆)EtT +∆[L(T, T + ∆) − K]

By the LIBOR denition (2.1)

π_tFRA(T, T + ∆, K) = Pt(T + ∆)EtT +∆  1 ¯ PT(T + ∆) − (1 + ∆K)  thus we have only to compute the quantity

¯ νt,T = EtT +∆  1 ¯ PT(T + ∆) 

We dedicate the next part of the section to the computation of ¯νt,T. The xed rate that makes

the FRA contract fair at time t becomes

K_tFRA(T, T + ∆) = 1

∆(¯νt,T − 1)

Moreover we dene the respective quantities in the case of a simple one-curve model νt,T = EtT +∆  1 PT(T + ∆)  = Pt(T ) Pt(T + ∆) Kt = 1 ∆(νt,T − 1) = 1 ∆  Pt(T ) Pt(T + ∆) − 1 

We now nd two expressions for the risk-free bond Pt(T ) and the risky bond ¯Pt(T ), that

will be useful in the proof of the main equality for the factor ¯νt,T. For the risk-free bond we

have Pt(T ) = Et  exp  − Z T t rudu  = Et  exp Z T t (ψ_u1− ψ2_u)du  = exp(A(t, T ) − B1(t, T )ψ_t1− B2(t, T )ψ_t2) (2.2)

where the coecients satisfy      B1 t − b1B1− 1 = 0 B1(T, T ) = 0 B2 t − b2B2− (σ2₎2 2 (B 2₎2_{− 1 = 0 B}2_{(T, T ) = 0} At= a1B1− (σ1₎2 2 (B 2₎2_{+ a}2_B2 _{A(T, T ) = 0}

A simple computation brings to

B1(t, T ) = 1 b1



e−b1(T −t)− 1 (2.3)

We can do the same for the risky bond ¯Pt(T ). We dene rst a money account associated to

the risky rate

¯ Bt = e Rt 0ru+sudu and we force P¯t(T ) ¯

Bt to be a martingale under the risk-neutral measure. If we set

¯ Pt(T ) = Et  exp  − Z T t ru+ sudu  = Et  exp  − Z T t ((k − 1)ψ_u1 + ψ2_u+ ψ3_udu)  = exp( ¯A(t, T ) − ¯B1(t, T )ψ_t1− ¯B2(t, T )ψ_t2− ¯B3(t, T )ψ_t3 we can nd the following constraints

           ¯ B_t1− b1_B¯1_{+ (k − 1) = 0 B¯}1_{(T, T ) = 0} ¯ B2 t − b2B¯2− (σ2₎2 2 ( ¯B 2₎2_{+ 1 = 0 B}¯2_{(T, T ) = 0} ¯ B3 t − b3B¯3− (σ3₎2 2 ( ¯B 3₎2_{+ 1 = 0 B¯}3_{(T, T ) = 0} ¯ At= a1B¯1−(σ 1₎2 2 ( ¯B 1₎2_{+ a}2_B¯2 _{+ a}3_B¯3 _{A(T, T ) = 0¯} leading to ¯ B1(t, T ) = 1 − k b1  e−b1(T −t)− 1= (1 − k)B1(t, T )

From the system we can obtain that                  ¯ B1(t, T ) = (1 − k)B1(t, T ) ¯ B2(t, T ) = B2(t, T ) ¯ A(t, T ) = A(t, T ) − a1k Z T t B1(u, T )du − (σ 1₎2 2 k 2 Z T t (B1(t, T ))2du + (σ1)2k Z T t B1(u, T )du + a3 Z T t ¯ B3(u, T )du If we set for simplicity

˜

A(t, T ) = ¯A(t, T ) − A(t, T ), B˜1 = B1(T, T + ∆) we obtain that ¯ Pt(T ) = exp ¯A(t, T ) − B1(t, T )ψt1− B 2 (t, T )ψ_t2− ¯B3(t, T )ψ_t3+ kB1(t, T )ψ1_t
= Pt(T ) exp ˜A(t, T ) + kB1(t, T )ψt1− ¯B3(t, T )ψt3  and hence PT(T + ∆) ¯ PT(T + ∆) = exp− ˜A(T, T + ∆) − k ˜B1ψ_T1 + ¯B3(T, T + ∆)ψ_T3 (2.4) Denition 15. We call adjustment factor the process

AdT,∆_t = Et

 PT(T + ∆)

¯

PT(T + ∆)



The next proposition is the main result obtained for the model. Proposition 2. We have that

¯ νt,T = νt,TAdT ,∆t exp  k (σ 1₎2 2(b1₎3  1 − e−b1∆ 1 − e−b1(T −t) 2 (2.5) and AdT ,∆_t = eA(T,T +∆)˜ Et h e−k ˜B1ψT1+ ¯B3(T ,T +∆)ψT3 i (2.6) Proof. First of all we perform a change of numeraire from the T + ∆-forward measure to the risk-neutral measure. The density process is

Lt =

Pt(T + ∆)

where Bt is the bank account. We can write then ¯ νt,T = EtT +∆  1 ¯ PT(T + ∆)  = L−1_t Et  LT +∆ ¯ PT(T + ∆)  = 1 Pt(T + ∆) Et  exp  − Z T t rudu  PT(T + ∆) ¯ PT(T + ∆)  Recalling the expression (2.4), we can write

¯ νt,T = 1 Pt(T + ∆) Et h e−RtTruduexp  − ˜A(T, T + ∆) − k ˜B1ψ_T1 + ¯B3(T, T + ∆)ψ_T3i = 1 Pt(T + ∆)

exp(− ˜A(T, T + ∆))Etexp ¯B3(T, T + ∆)ψ3T

 · Et  exp Z T t ψ2_u− ψ1 udu  exp−k ˜B1ψ_T1  Now we dene the process

Nt = Et  exp Z T t ψ_u2 − ψ1 udu  exp(−k ˜B1ψ_T1) 

Due to the independence and the ane dynamics of the processes ψi_{, i = 1, 2 we have}

Nt= Et  exp Z T t ψ_u1du  exp  −k ˜B1ψ_T1  Et  exp  − Z T t −ψ_u2du  = exp(α1(t, T ) − β1(t, T )ψ_t1) exp(α2(t, T ) − β2(t, T )ψ_t2)

where the coecients satisfy            β_t1− b1_β1_{− 1 = 0 β}1_{(T, T ) = k ˜B}1 β2 t − b2β2− (σ2)2 2 (β 2₎2_{+ 1 = 0 β}2_{(T, T ) = 0} α1 t = − (σ1₎2 2 (β 1₎2_{+ α}1_β1 _α1_{(T, T ) = 0} α2 t = a2β2 α2(T, T ) = 0 (2.7)

(2.3) for B1_{(t, T )}                                  β1(t, T ) = _b11  (b1k ˜B1+ 1)e−b1(T −t)− 1= B1(t, T ) + k ˜B1e−b1(T −t) β2(t, T ) = B2(t, T ) α1(t, T ) = (σ 1₎2 2 Z T t (β1(u, T ))2du − a1 Z T t β1(u, T )du = (σ 1₎2 2 Z T t (B1(u, T ))2du − a1 Z T t B1(u, T )du + (σ 1₎2 2 (k ˜B 1 )2 Z T t e−2b1(T −u)du + k ˜B1(σ1)2 Z T t B1(u, T )e−b1(T −u)du − a1k ˜B1 Z T t e−b1(T −u)du α2(t, T ) = −a2R_tT B2(u, T )du

Hence for the process Nt we have that

Nt = exp  (σ1₎2 2 Z T t (B1(u, T ))2du − α1 Z T t β1(u, T )du − a2 Z T t B2(u, T )du − B1(t, T )ψ1_t − B2(t, T )ψ2_t  · exp (σ 1₎2 2 (k ˜B 1 )2 Z T t e−2b1(T −u)du − a1k ˜B1 Z T t e−b1(T −u)du − k ˜B1e−b1(T −t)ψ_t1  · exp  k ˜B1(σ1)2 Z T t B1(u, T )e−b1(T −u)du  = Pt(T ) exp  (σ1₎2 2 (k ˜B 1₎2 Z T t e−2b1(T −u)du − a1k ˜B1 Z T t e−b1(T −u)du − k ˜B1e−b1(T −t)ψ1_t  · exp  k ˜B1(σ1)2 Z T t B1(u, T )e−b1(T −u)du  Recalling the formula (2.4), we obtain

Et

 PT(T + ∆)

¯

PT(T + ∆)



= exp(− ˜A(T, T + ∆))Etexp( ¯B3(T, T + ∆)ψ3T) Et

h

exp(−k ˜B1ψ1_T)i Using again the ane dynamics of ψi

t we can write that

Et

h

exp(−k ˜B1ψ1_T) i

= exp( ¯α(t, T ) − ¯β(t, T )ψ1_t) where the processes ¯α and ¯β satisfy

( ¯_β t− b1β = 0¯ β(T, T ) = 0¯ ¯ αt= a1β −¯ (σ 1₎2 2 ( ¯β) 2 _{α(T, T ) = 0¯}

So we can simply obtain that

( ¯_{β(t, T ) = k ˜B}1_e−b1_{(T −t)}

¯

α(t, T ) = −a1k ˜B1R_tT e−b1(T −u)du +(σ₂1)2(k ˜B1)2R_tT e−2b1(T −u)du and so that Et h exp(−k ˜B1ψ1_T) i = exp  −k ˜B1e−b1(T −t)ψ_t1  · exp  k(σ1)2B˜1 Z T t B1(u, T )e2b1(T −u)ψT3  So we have that ¯ νt,T = 1 Pt(T + ∆)

exp(− ˜A(T, T + ∆))Etexp( ¯B3(T, T + ∆)ψ3T) Nt

= Pt(T ) Pt(T + ∆) Et  PT(T + ∆) ¯ PT(T + ∆)  exp  k(σ1)2B˜1 Z T t B1(u, T )e−b1(T −u)du  The result follows noticing that

˜ B1 Z T t B1(u, T )e−b1(T −u)du = 1 2(b1₎3  1 − e−b1∆   1 − e−b1(T −t) 2

Corollary 2. We have that ¯ K_tFRA(T, T + ∆) =  Kt+ 1 ∆  AdT ,∆_t exp  k(σ 1₎2 2(b1₎3  1 − e−b1∆ 1 − e−b1(T −t) 2 − 1 ∆ For the calibration on the FRA we lack an explicit formula for the conditional expectation that appears in formula (2.6). We nd the solution in the same way we have done many times in the previous proof.

Et

h

eB¯3(T,T +∆)ψT3

i

= exp( ˜α(t, T ) − ˜β(t, T )ψ_t3) where the coecients satisfy

( − ˜βt+ ˜βb3 + (σ3_β)˜2 2 = 0 β(T, T ) = ¯B 3_{(T, T )} ˜ αt− ˜βa3 = 0 α(T, T ) = 0

The rst equation is a Bernoulli equation and it has an explicit solution ˜

β(t, T ) = e−b3(T −t)( ¯B3(T, T + ∆) − 1 2b3(σ

3₎2_(e−b3_{(T −t)}

− 1))−1 The function α(t, T ) can be easily obtained integrating β(t, T )

Bootstrapping and calibration

In this section a procedure to calibrate the model on a set of FRA prices observed in the market will be present. The unknown parameters are k, ai_{, b}i_{, σ}i _{and the initial values ψ}i

0 and s0 that

are not observable in the market and that also must respect the constraints r0 = ψ02− ψ

1

0, s0 = kψ01+ ψ 3 0

We refer to the price of a FRA observed in the market, with strike K and referring to a loan between the future dates T and T + ∆ as

πFRA∗₀ (T, T + ∆, K)

We call U the set of the sets {T, T +∆, K} of all the FRA observed in the market. The distance between the prices observed and the prices obtained from the model is

φ(k, a, b, σ, ψ0) = X (T ,T +∆,K)∈U (πFRA∗₀ (T, T + ∆, K) − π₀FRA(T, T + ∆, K))2 where obviously a = (a1_{, a}2_{, a}3₎, b = (b1_{, b}2_{, b}3₎, σ = (σ1_{, σ}2_{, σ}3₎, ψ 0 = (ψ01, ψ02, ψ03). This

func-tion depends also from the curve P0(T ) that is the only one that have to be bootstrapped. The

other quantities that appear in the formula, that is ˜A(T, T +∆), ¯α(0, T ), ¯β(0, T ), ˜α(0, T ), ˜β(0, T ), can be explicitly computed as solutions of simple dierential equations as we have seen previ-ously. So the optimal parameters are found computing

argmin

k,a,b,σ,φi 0

Chapter 3

HJM approach

In this section we present two models that use a HJM approach. In the rst, presented below, the existence of the instantaneous forward rate ft(T ) and of the forward rates Ft(T − ∆, T )

are supposed and dynamics are chosen under the T -forward measure. This approach is called parsimonious approach. In the second a family of instantaneous forward rates f∆

t (T ) is dened,

one for each tenor ∆ quoted in the market, and then the family of the standard forward rates Ft(T −∆, T )is dened. We will call this approach complete approach. In both models swaptions

pricing formulas are found, and they can be used in the calibration.

3.1 Parsimonious approach

In this section we suppose the existence

• of a risk-free instantaneous forward rate ft(T )

• of the LIBOR rates L(T − ∆, T ) and of the associated risky forward rates Ft(T − ∆, T ),

one for each tenor ∆, where typically ∆ ∈ {1 month, 3 months, 6 months, 1 year}

The construction consists in the following dynamics of ft(T )and Ft(T −∆, T ), modelled directly

under the T -forward measure

dft(T ) = σt∗(T )dW T t d(k(T, ∆) + Ft(T − ∆, T )) k(T, ∆) + Ft(T − ∆, T ) = Σ∗_t(T, ∆)dW_tT σt(T ) = σt(T, T, 0) Σt(T, ∆) = Z T T −∆ σt(u, T, ∆)du

where σt(u, t, ∆) is a vector stochastic process, a WtT is a d-dimensional T -forward Brownian

motion and k(T, ∆) are deterministic shift functions. We will use shift functions such that k(T, ∆) ≈ 1/∆ for short tenors ∆. We point out that, our assumptions, make Ft(T − ∆, T )

a martingale under the T -forward measure. Since WT

t is a Brownian motion under the T

-forward measure, it can be obtained from a Brownian motion under the risk-neutral measure. Specically holds that

dW_tT = dWt+

Z T

t

σt(u, u, 0)du

where Wt is a d-Brownian motion under the risk-neutral measure. Now we can rewrite the

processes dynamics under the risk-neutral measure d(k(T, ∆) + Ft(T − ∆, T )) k(T, ∆) + Ft(T − ∆, T ) = Σ∗_t(T, ∆) Z T t σt(u, T, ∆)dudt + dWt  (3.1) dft(T ) = σ∗t(T ) Z T t σt(u, T, ∆)dudt + dWt  (3.2) Integrating equation (3.1) we have that

log k(T, ∆) + Ft(T − ∆, T ) k(T, ∆) + F0(T, ∆)  = Z t 0 Σ∗_s(T, ∆)  dWs− 1 2Σs(T, ∆)ds + Z T s σs(u, u, 0)duds  (3.3)

Volatility constraints

We will discuss the case where the volatility respects these constraints σt(u, T, ∆) = htq(u, T, ∆)g(t, u) g(t, u) = exp  − Z u t λ(u)du  q(u, u, 0) = 1

where h is a d × d matrix adapted process, q is a d × d deterministic diagonal matrix and λ is a d vector of deterministic functions. We call qi the element of place (i, i) of the diagonal of q

and λi the function of place i of λ, for i = 1, . . . , d.

Dynamics of the forward rate

The volatility hypothesis introduced, allows to derive simple SDE for the forward rate Ft(T −

∆, ∆)under the risk-neutral measure, that can be useful to price derivatives and in Monte Carlo simulations. We rst introduce the denition of the element-wise product between vectors, that will be useful to make formulas compact.

Denition 16. Let be u, v ∈ Rn _{with u}

i, vi their elements and a matrix M ∈ Rn×n with

elements Mij. We dene the vector u. ∗ v and the matrix M. ∗ u as

(u. ∗ v)i = uivi

(M. ∗ u)ij = Mijui

Proposition 3. If u, v, w are vectors in Rn and M ∈ Rn×n we have

u∗(v. ∗ w) = (u∗. ∗ v∗)w

This simple property will be important in the passages of the following proposition. Proposition 4. We have that

log k(T, ∆) + Ft(T − ∆, T ) k(T, ∆) + F0(T − ∆, ∆)  = G∗(t, T − ∆, T, T, ∆)(Xt+ Yt(G0(t, t, T ) − 1 2G(t, T − ∆, T, T, ∆)) where Xt = Z t 0 g(s, t). ∗  h∗_sdWs+ h∗shs Z t s g(s, y)dyds  Yt= Z t 0 g(s, t). ∗ (h∗_shs). ∗ g∗(s, t)ds and G0(t, T0, T1) = Z T1 T0 g(t, y)dy G(t, T0, T1, T, ∆) = Z T1 T0 q(y, t, ∆)g(t, y)dy Proof. Substituting in equation (3.3) the expression of the volatility, we obtain

Z t 0 Z T T −∆ g∗(s, u)q∗(u, T, ∆)h∗_sdu  dWs− 1 2 Z T T −∆ hsq(u, T, ∆)g(s, u)du + Z T s hsg(s, u)duds  We now consider the vector g∗_{(s, u)q}∗_{(u, T, ∆)h}∗

s. We can write the element i

(g∗(s, u)q∗(u, T, ∆)h∗_s)i = d

X

j=1

gj(s, t)gj(t, u)qj(u, T, ∆)h∗ji,s

=

d

X

j=1

So we have that

g∗(s, u)q∗(u, T, ∆)h∗_s = (q(u, T, ∆)g(t, u))∗(g(s, t). ∗ h∗_s) Using this fact we have that

= Z t

0

Z T

T −∆

(q(u, T, ∆)g(t, u))∗(g(s, t). ∗ h∗_s)du  dWs− 1 2 Z T T −∆ hsq(u, T, ∆)g(s, u)du + Z T s hsg(s, u)duds  Splitting the integral RT

s (...)du in R t s(...)du + RT t (...)du, we obtain = G∗(t, T − ∆, T, T, ∆) Z t 0 g(s, t). ∗ h∗_s  dWs− 1 2 Z T T −∆ hsq(u, T, ∆)g(s, u)du + Z t s hsg(s, u)duds + Z T t hsg(s, u)duds  and using the second property of the element-wise product we deduce

= G∗(t, T − ∆, T, T, ∆) Z t 0 g(s, t). ∗  h∗_sdWs− 1 2h ∗ shs Z T T −∆ q(u, T, ∆)g(s, u)du +h∗_shs Z t s g(s, u)duds + h∗_shs Z T t g(s, u)duds  Moreover since q is diagonal we have

q(u, T, ∆)g(s, u) = g(s, t). ∗ (q(u, T, ∆)g(t, u)) so the expression becomes

= G∗(t, T − ∆, T, T, ∆)  Xt+ Z t 0 g(s, t). ∗  − 1 2h ∗ shs  g(s, t). ∗ Z T T −∆ q(u, T, ∆)g(t, u)du  +h∗_shs  g(s, t). ∗ Z T t g(t, u)duds  and the thesis follows.

The formula just introduced could seem complicated but functions G0 and G, once the

choice of the volatility parameters is done, are simply computable, and the processes Xt and

Proposition 5. The processes Xt and Yt follow the dynamics dX_ti = d X k=1 (Y_tik− λi(t)Xti)dt + d X k=1 h∗_ik,tdW_tk dY_tik =h∗ tht− (λi(t) + λk(t))Ytik dt

Proof. We begin deriving the dynamic of Xt. From the formula of Xt we can deduce that

X_ti = d X k=1 Z t 0 gi(s, t)  h∗_ik,sdW_sk+ (h∗_shs)ik Z t s gk(s, y)dyds  hence dX_ti = d X k=1 gi(t, t)h∗ik,tdW k t − λi(t) d X k=1 Z t 0 gi(s, t)h∗ik,sdWk,s ! + d X k=1 gi(t, t)(h∗tht)ik Z t t gk(s, u)du ! dt − λi(t) d X k=1 Z t 0 gi(s, t)(h∗shs)ik Z t s gk(s, y)dyds ! dt + d X k=1 Z t 0 gi(s, t)(h∗shs)ikgk(s, t)ds ! dt = d X k=1 Y_tik− λi(t)Xti ! dt + d X k=1 h∗_ik,tdW_tk

Now we calculate the dynamic of Yt. From the formula of Yt we have that

Y_tik = Z t

0

gi(s, t)(h∗shs)ikgk(s, t)ds

Deriving last expression we obtain

dY_tik = gi(t, t)(h∗tht)ikgk(t, t)dt − (λi(t) + λk(t))Ytik Z t 0 gi(s, t)(h∗shs)ikgk(s, t)ds  dt =(h∗_tht− (λi(t) + λk(t))Ytik 

We now derive the dynamic of the instantaneous forward rate ft(T ), using the same

Proposition 6. The dynamic of ft(T ) under the risk-neutral measure is

ft(T ) − f0(T ) = g∗(t, T )(Xt+ YtG0(t, t, T ))

Proof. Integrating equation (3.2) and substituting the expression of the volatility we obtain that ft(T ) − f0(T ) = Z t 0 g∗(s, T )h∗_s Z T s hsg(s, u)duds + dWs  = Z t 0 (g∗(t, T ). ∗ g∗(s, t))  h∗_shs Z T s g(s, u)duds + h∗_sdWs  = g∗(t, T ) Z t 0 g(s, t). ∗  h∗_shs Z T s g(s, u)duds + h∗_sdWs  Splitting the integral RT

s (...)du in R t s(...)du + RT t (...)du, we obtain = g∗(t, T )  Xt+ Z t 0 g(s, t). ∗ (h∗_shs) Z T t g(s, u)duds 

Using again the property of the point-wise product in Proposition (3) we have that = g∗(t, T )(Xt+ YtG0(t, t, T ))

From the expression of ft(T )it's easy to deduce an expression for the short rate rtby simply

setting T = t. We obtain rt = f0(t) + d X i=1 X_ti

Weighted Gaussian model

In this section we will present a specic example of the model introduced, giving explicit expression to the terms of the volatility. We begin setting

h∗_t = (t)hR∗

where h is a diagonal constant matrix, whose elements of place (i, i) are hi, and R is a lower

triangular matrix representing the square root of a correlation matrix ρ, that is R∗R = ρ

Also we set

(t) = 1 + (β0− 1 + β1t)e−β2t

where β0, β1, β2 are positive constants. We also suppose that

qi(u, T, ∆) = e−∆ηi

With these assumptions we can write the dynamics of the processes Xt, Yt

dX_ti = n X j=1 Y_tij − λi(t)Xti ! dt + (t)hid ˆWti dY_tij = 2(t)hihjρij + (λi(t) + λj(t))Y ij t
dt d[ ˆW_tiWˆ_tj]t= ρijdt

Pricing swaption

The framework introduced gives the possibility to nd an approximated formula for swaptions. We will use an approximation technique called freezing, that simply consists in approximating a future stochastic value of a process with its present value. We have seen, in Section 1.5, that the price of a swaption can be computed by Black's formula, with a swap rate as underlying. Hence, let's consider a swap, related to dates of the oating leg T = {T0, . . . , Tn} and of the

xed leg T0 _{= {T}0

0, . . . , Tm0 }. We recall the formula for the par value of the xed rate at time t

K_tIRS= Pn i=1τiPt(Ti)Ft(Ti−1, Ti) Pm j=1τ 0 jPt(Tj0)

We introduce the weights

wi(t) = τiPt(Ti) Pm j=1τ 0 jPt(Tj0)

The process KIRS

t is a martingale under the swap measure Q T0

. We can easly derive its dynamic dK_tIRS = n X i=1 Ft(Ti−1, Ti)dwi(t) + i X i=1 wi(t)dFt(Ti−1, Ti)

We can now approximate the processes wi(t) with their present values wi(0). This

approxima-tion is the freezing technique. This leads to dK_tIRS ≈ b X i=1 wi(t)(k(Ti, ∆) + Ft(Ti−1,Ti))Σ ∗ t(Ti, ∆)dWT 0 t

Using again the same approximation we have dK_tIRS ≈ K_tIRS+ ψ
n X i=1 δiΣt(Ti, ∆)dWτ 0 t

where we have set ψ = Pn i=1τiP0(Ti)k(Ti, ∆) Pm j=1τ 0 jP0(Tj0) , δi = Pn i=1τiP0(Ti)(k(Ti, ∆) + F0(Ti−1, Ti)) Pm j=1τ 0 jP0(Tj0) (3.4) So we have seen that the process KIRS

t + ψ, under the swap measure, solve an approximated

Black and Scholes equation, so we can explicitly calculate the price of a swaption of maturity T with Black's formula

π_tSWPT = AtET 0 t (KTIRS− K) + = AtET 0 t (KTIRS− ψ − (K + ψ)) + = AtBlPUT(KtIRS+ ψ, K + ψ, Γ) where Γ = v u u t Z T t n X i=1 δiΣs(Ti, ∆) !2 ds (3.5)

Substituting in (3.5) the expression of Σs(Ti, ∆), we obtain that

Γ =pc(∆)∗_{Q(t, T}

0)c(∆) (3.6)

where c(∆) is a deterministic vector c(∆) = he−η∆ n X i=1 δi(∆) e−λTi−1 − e−λTi λ and Q(t, T0)is a deterministic matrix

Q(t, T0) =

Z T

t

(eλu)∗ρeλu2(u)du

Bootstrapping and calibration

In this section we give the instructions to calibrate the weighted Gaussian model introduced in Section 3.1 and we show what curves need to be bootstrapped. We present a procedure to calibrate the model with a set of market swaptions prices with strike K and payments date

T = T0_{. The oating rate usually is the 3 months LIBOR or the 6 months LIBOR. The formula}

for the swaptions, that can be obtained from the weighted Gaussian model, is

π₀SWPT= A0BlPUT(K0IRS+ ψ, K + ψ, Γ) (3.7)

where ψ and Γ are dened in (3.6) and (3.4). The pricing formula depends on the zero-coupon curve P0(T ) and from the forward curve F0(T − ∆, T ), that must be bootstrapped with the

bootstrapping techniques illustrated in Section 1.7. Once the curves are built, we can proceed with the calibration. We dene the dierence between the market swaptions prices and the swaption prices calculated from the model with formula (3.7), depending from the parameters ηi, λi, ρij, β0, β0, β1, β2 for i, j = 1, . . . , d.

Φ(η, λ, ρ, β) = X

(T ,K)∈M

πSWPT∗₀ (T , K) − π₀SWPT(T , K)
2

where πswpt∗0 (T , K)is the price of the swaption observed in the market, with dates T and strike

K, and M is the set of the couples of dates and strikes (T , K) of all the swaptions observed in the market. The calibrated values can be obtained calculating

argmin

η,λ,ρ,β

Φ(η, λ, ρ, β)

3.2 Complete approach

We introduce a instantaneous forward rate ft(T ) and a family of instantaneous forward rates

f∆

t (T ), one for each tenor ∆, and we suppose the following dynamics under the risk-neutral

measure df_t∆(T ) = σ_t∆(T ) Z T t σ_t∆(u)du  dt + σ_t∆(T )dW_t∆ (3.8) dft(T ) = σt(T ) Z T t σt(u)du  dt + σt(T )dWt (3.9) where σ∆

t (u) and σt(u) are volatility processes and Wt∆ and Wt are Brownian motions such

that

ρ∆= ∂

∂t[W, W

∆_] t

We can dene the short rates

rt= ft(t) r∆t (t) = f ∆ t (t)

and the bank account

Bt= exp

Z t 0

rudu

Notice that the dynamics of the forward rates respect the HJM conditions. We also dene the relatives bonds P_t∆(T ) = exp  − Z T t f_t∆(u)du  Pt(T ) = exp  − Z T t ft(u)du  The quantities r∆

t and Pt∆(T ) are ctitious quantities, not observable in the market, but they

will be useful for the denition of the LIBOR rates. We can derive the dynamic of the ∆-zero-coupon bond under the risk-neutral measure, and obtain that

dP_t∆(T ) P∆ t (T ) = r∆_t dt − Z T t σ∆_t (u)du  dW_t∆

We use the denition of the ∆-zero-coupon bond to dene a new family of forward rates ˜ Ft(T − ∆, T ) = 1 ∆  P∆ t (T − ∆) P∆ t (T ) − 1  (3.10) Starting from this denition we can dene the LIBOR rates as

L(T − ∆, T ) = ˜FT −∆(T − ∆, T ) = 1 ∆  1 P_{T −∆}∆ (T ) − 1  Now we can dene also the standard forward rates as

Ft(T − ∆, T ) = EtT[L(T − ∆, T )]

where ET

t is the conditional expectation obtained taking Pt(T ) as numeraire. We recall that,

under the T -forward measure, by Girsanov's theorem, the process dW_tT = dWt+

Z T

t

σt(u)dudt

is a Brownian motion under the T -forward measure.

Proposition 7. The dynamic of ˜Ft(T − ∆, T )under the T -forward measure is

d ˜Ft(T − ∆, T ) ˜ Ft(T − ∆, T ) + ∆−1 = Z T T −∆ σ_t∆(u)du   θ∆_t (T )dt + dW_t∆,T  where θ∆(t) = Z T t σ∆_t (u) − ρ∆σt(u)du

Proof. From (3.10) we can only derive the process P∆ t (T −∆)

P∆

t (T ) . Under the risk-neutral measure

we have dP∆ t (T ) P∆ t (T ) = r∆_t dt − Z T t σ∆_t (u)du  dW_t∆ We can easly deduce the dynamic of 1

P∆ t (T ) d  1 P∆ t (T )  = 1 P∆ t (T ) −r∆ t + Z T t σ_t∆(u)du 2! dt + Z T t σ_t∆(u)dW_t∆ !

Using Ito's formula we obtain d P ∆ t (T − ∆) P∆ t (T )  = P ∆ t (T − ∆) P∆ t (T ) Z T t σ∆_t (u)du 2 − Z T −∆ t σ_t∆(u)du Z T t σ_t∆(u)du ! dt + Z T T −∆ σ_t∆(u)du  dW_t∆ and thus d P ∆ t (T − ∆) P∆ t (T )  = P ∆ t (T − ∆) P∆ t (T ) Z T T −∆ σ∆_t (u)du  Z T t σ∆_t (u)dudt + dW_t∆  (3.11) Now we dene ˜ Wt = 1 p1 − (ρ∆₎2(W ∆ t − ρ ∆_W t)

We can easily prove that ˜Wtis a Brownian motion under the risk-neutral measure and

indepen-dent of Wt, but in addiction we can prove that it's also a Brownian motion under the T -forward

measure. In fact by Girsanov's theorem there exists ˜Ht such that the process

d ˜Wt T

= d ˜Wt− ˜Htdt

is a Brownian motion under the T -forward measure. Moreover _W˜_tBt

Pt(T ) is a martingale under the

T-forward measure. By Ito's formula, directly under the T -forward measure, we obtain that d ˜ WtBt Pt(T ) ! = Bt(d ˜Wt T + ˜Htdt) Pt(T ) + Bt ˜ Wt Pt(T ) Z T t σt(u)dudWtT hence ˜Ht = 0, so ˜Wt = ˜Wt T

is a Brownian motion under the T -forward measure. Once again we have from Girsanov's theorem that there exists H∆

t such that the process

dW∆,T = dW_t∆− H∆ t dt

is a Brownian motion under the T -forward measure. We have that dW_t∆,T = dW_t∆− H_t∆dt =p1 − (ρ∆₎2_{d ˜W} t+ ρ∆dWt− Ht∆dt =p1 − (ρ∆₎2_{d ˜W} t+ ρ∆(dWtT − Z T t σt(u)dudt) − Ht∆dt Since p1 − (ρ∆₎2_{d ˜W}

t+ ρ∆dWtT is a Brownian motion under the T - forward measure, we have

that

dW_t∆ = ρ∆ Z T

t

σt(u)dudt + dWt∆,T

Substituting last equation in (3.11) we have the thesis.

The next proposition shows the relationship between the forward rates ˜F and F . It will be useful for nding the dynamic of the forward rate Ft.

Proposition 8. We have that

Ft(T − ∆, T ) = ˜Ft(T − ∆, T ) 1 + 1 + ∆ ˜Ft(T − ∆, T ) ∆ ˜Ft(T − ∆, T ) (Θ∆_t (T ) − 1) ! (3.12) where Θ∆_t (T ) = exp Z T −∆ t Z T T −∆ σ_u∆(v)θ∆_u(T )dvdu  Proof. We begin setting

Mt= ˜Ft(T − ∆, T ) 1 + 1 + ∆ ˜Ft(T − ∆, T ) ∆ ˜Ft(T − ∆, T ) (Θ∆_t (T ) − 1) ! We observe that MT −∆= ˜FT −∆(T − ∆, T ) = L(T − ∆, T )

Since Ft(T − ∆, T ) is a martingale under the T -forward measure, it's sucient to prove that

the process Mt is a martingale under the T -forward measure. We omit for simplicity the

dependences from T and so we set ˜Ft(T − ∆, T ) = ˜Ft and Θ∆t (T ) = Θ∆t . It's easy to see that

Mt = ∆−1(Θ∆t − 1) + ˜FtΘ∆t If we set Xt= Z T −∆ t Z T T −∆ σ_u∆(v)θ_u∆(T )dvdu

we have that dXt = −θt∆(T ) Z T T −∆ σ∆_t (v)dvdt and so dΘ∆_t = d(eXt_{) = Θ}∆ t dXt

Hence, deriving by Ito's formula

dMt= ∆−1dΘ∆t + ˜FtdΘ∆t + Θ ∆ t d ˜Ft = (∆−1+ ˜Ft)dΘ∆t + Θ ∆ t d ˜Ft

and using the dynamic of ˜Ft from the previous proposition, we have

dMt = (∆−1+ ˜Ft)Θ∆t  dXt+ θ∆t (T ) Z T T −∆ σ_t∆(u)dudt  + Θ∆_t ( ˜Ft+ ∆−1) Z T T −∆ σ_t∆(u)dudW_t∆,T = Θ∆_t ( ˜Ft+ ∆−1) Z T T −∆ σ_t∆(u)dudW_t∆,T So Mt is a martingale under the T -forward measure.

The dynamic of the forward rate Ft(T − ∆, T )can be deduced simply retracing the passages

of last proposition.

Corollary 3. The dynamic of the forward rate Ft(T − ∆, T ) under the T -forward measure is

dFt(T − ∆, T ) = (Ft(T − ∆, T ) + ∆−1)

Z T

T −∆

σ_t∆(u)dudW_t∆,T

3.2.1 Choice of the volatility

For our purpose we need only to choose a volatility σ∆

t (T ) for a xed tenor ∆. We suppose

the existence of a d- dimensional Brownian motion ¯Wt and its generated ltration ¯Ft. We take

a correlation matrix ρ with elements ρkh for k, h = 1, . . . , d, a lower triangular matrix R such

that R∗_{R = ρ and we dene the process W}

t = R∗W¯t. We have that each Wk,t is a Brownian

motion in the ltration ¯Ft and that

ρkh =

∂

∂t[Wk, Wh]t

We now consider a random variable I independent of the Brownian motion ¯Wt, taking values

in {1, . . . , m}, with probabilities ωi = Q(I = i) > 0 and where

P

each k = 1, . . . , d, we consider positive constants ak(I), positive functions σk(t, I) and integers

q(I). We call vI

t(T ) the vector of elements vk,tI (T ) given by

( vI

k,t(T ) = σk(t, I)e−ak(I)(T −t) k = 1, . . . , q(I)

vI k,t(T ) = 0 k = q(I) + 1, . . . , j We set σ∆_t (T ) = v_tI(T )∗ρv_tI(T ) (3.13) and W_t∆ = d X i=1 Z t 0 vI k,s(T ) kRvI s(T )k Wk,s (3.14)

It's easy to see that W∆

t is a Brownian motion under the risk-neutral measure. With (3.13)

and (3.14) we have that

σ_t∆(T )dW_t∆ = kRV_tI(T )kdW_t∆=

d

X

k=1

σk(t, I)e−ak(I)(T −t)dWk,t

With these denitions we can obtain a simple dynamic formula for the short rate. From the denition of the forward rate (3.8) we have that

r∆_t = f0(t) + Z t 0 σ∆_s (t) Z t s σ∆_s (u)duds + Z t 0 σ∆_s (t)dW_s∆ We set for simplicity

ϕ∆(t) = f0(t) + Z t 0 σ∆_s (t) Z t s σ_s∆(u)duds, xk(t, I) = Z t 0

σk(s, I)e−ak(I)(t−s)dWk,s

and hence r∆_t = ϕ∆(t) + q(I) X k=1 xk(t, I)

By Ito's formula we can easly derive that

dxk(t, I) = −ak(I)xk(t, I)dt + σk(t, I)dWk,t

3.2.2 Pricing swaptions

Now we illustrate a procedure that can be used for pricing swapions, as alternative approach to the method of the implied volatility. We will nd a formula for a swaption with oating leg payment dates T = {T0, . . . , Tn}, xed leg payment dates T0 = {T00, . . . , Tm0 }, strike K and

maturity T . The price of a swaption at time t, under the risk-neutral measure, can be written as

π_tSWPT= Et[X]

where X is a random variable. We dene the probability restricted, to the event I = j Qj = Q|I=j

so the price of the swaption can be written as π_tSWPT= m X j=1 ωjE j t[X] = m X j=1 ωjπSWPT(j)t

where Ej _{is the expectation respect to Q}j_{, and π}SWPT(j)

t the price of the swaption under the

probability Qj. So we can just nd a formula for the price of the swaption in the case that

I = j. Moreover in this section we will suppose σ_k∆(t, T ) to be constant respect to t. So we have that σ_t∆(T )dW_t∆ = q(j) X k=1 σk(j)e−ak(j)(T −t)dWk,t

From Corollary 3 we can nd the dynamic Ft= Ft(T − ∆, T ) under the T -forward measure

dFt= (Ft+ ∆−1) Z T T −∆ σ_t∆(u)du  dW_t∆,T we call Σt(T, ∆) = Z T T −∆ σ∆_t (u)du

and using the freezing technique, as in the section of the parsimonious approach, we obtain an approximated formula for the swap rate under the swap measure Qτ0

dK_tIRS ≈ K_tIRS+ ψ
n X i=1 δiΣt(Ti, ∆)dWτ 0 t where ψ = Pn i=1τiP0(Ti)k(Ti, ∆) Pm j=1τ 0 jP0(Tj0) , δi = Pn i=1τiP0(Ti)(k(Ti, ∆) + F0(Ti−1, Ti)) Pm j=1τ 0 jP0(Tj0) and where WT0 t is a QT 0

-Brownian motion. By Black's formula we can calculate the price of the swaption in scenario j

where Γj = v u u t Z T t n X i=1 Σs(Ti, ∆) !

Using the expression of the volatility given in the previous section we obtain an explicit formula for Γj Γj = v u u t e2(T −t) 2 n X i=1 δiM (Ti, ∆) !2 M (Ti, ∆) = X k,h ρkhσk(j)σh(j) e−A(I)khTi A(I)kh (1 − eAkh(I)∆₎

Akh(I) = ak(I) + ah(I)

3.2.3 Bootstrapping and calibration

For the calibration we can proceed as in Section 3.1 and choose a set of swaptions with strike K and payments dates T = T0 with τi = ∆, where ∆ is usually 3 months or 6 months. We call

with πSWPT∗

0 (T , K) the market price of the swaption with dates T and strike K and with M

the set of the couples of dates and strike (T , K) of all the swaptions chosen for the calibration. We also call πSWPT

0 (T , K) the price of the swaption obtainable from the model with formula

(3.15). Also we set

Φ(a, σ, ρ) = X

(T ,K)∈M

π₀SWPT∗(T , K) − πSWPT₀ (T , K)
2

where we have dened for simplicity the matrices a, σ such that akj = ak(j), σkj = σk(j) and

where are supposed bootstrapped the curves T 7−→ P0(T ) and T 7−→ F0(T − ∆, T ). The last

step for the calibration is the evaluation of argmin

a,σ,ρ