Faculty of Mathematics, Physics and Natural Sciences
Master’s Degree in Mathematics
Master’s Thesis
Multiple Yield Curve
Modelling
Candidate:
Yinglin Zhang
Supervisor:
Prof. Maurizio Pratelli
Contents
1 Classical Theory of Bond Market 7
1.1 General market model and First Fundamental Theory . . . 7
1.2 The pre-crisis bond and interest rate market . . . 8
1.2.1 Bonds and interest rates . . . 8
1.2.2 Interest rate derivatives and their pricing formula . . . 12
1.2.3 Some pre-crisis bond market models . . . 20
1.2.4 Black’s formula . . . 24
2 Post-crisis Problem: Multiple Yield Curve Modelling 27 2.1 The post-crisis interest rate market . . . 27
2.1.1 Risk-free discounting yield curve . . . 29
2.2 Some modelling approaches . . . 31
2.3 How to use a model . . . 32
2.3.1 Bootstrapping technique . . . 33
2.3.2 Calibration . . . 37
2.3.3 Monte Carlo simulation . . . 38
2.3.4 Discretization of associated PDEs . . . 41
3 Short Rate Models 45 3.1 Dynamics under risk-neutral measure Q . . . . 46
3.2 Pricing formula . . . 50
3.2.1 FRA . . . 50
3.3 Calibration . . . 55
3.4 Comments . . . 55
4 HJM Models: Libor Dynamics Approach 57 4.1 Dynamics under T -forward measure QT and risk-neutral measure Q . . . 59
4.1.1 Volatility assumption for Markovian setting . . . 60
4.1.2 Generalization of a shifted d-factor Hull-White model . . 65
4.2 Pricing formulas . . . 65
4.2.1 Swap rates dynamics . . . 65
4.2.2 Basis swap spreads dynamics . . . 67
4.2.3 Caps and swaptions . . . 67 3
4.3 Calibration . . . 69 4.4 Comments . . . 69
5 A Numerical example 71
5.1 Assumptions . . . 71 5.2 Bootstrapping initial yield curves . . . 72 5.3 Calibration . . . 75
A A Stochastic Fubini-Tonelli Theorem 79
Introduction
The last crisis of 2007 has affected massively the financial market with the raising of the credit crunch. Credit and liquidity risk even among the biggest fi-nancial institutions manifests in significant spreads which were negligible before the crisis, like Eonia-Euribor spread, OIS-IRS spread, basis swap spread etc. The classical interest rate theory, which was sufficient enough to explain mar-ket dynamics pre-crisis, now becomes problematic, since the usual no-arbitrage relationships began to be violated in a macroscopic way.
Many new models are being developed concerning this problem. The main idea is to model risky interbank rates of different tenors as separated assets, as in general risk increases with tenor length. Such approach is called multi-curve approach, in contrast with the classical single-curve approach where all market issues are described by a single yield curve.
In this thesis we give an introduction and some simple examples of multiple yield curve modelling methods, since the argument is still in continuous devel-opment.
The thesis is composed by five chapters.
In the first Chapter we recall shortly the classical interest rate theory. We be-gin with the First Fundamental Theory which is the basis of every market model. After that, we give the basic definition of bonds and interest rates as mathe-matical issues and general pricing formulas of the main interest rate derivatives (both linear and non linear), like IRS, OIS, FRA, basis swap, caps/floors, swap-tions etc. We give also a synthetic view of principal pre-crisis bond market models which will be extended in the multi-curve context.
The second Chapter presents the problem formulation of multiple yield curve modelling.
Overnight rates (e.g. Eonia, OIS etc.) which have reduced credit and liq-uidity exposure were used to construct the so called discounting curve. This reference yield curve is used as the discount factor (i.e. numeraire for martin-gale measure) to value future cash flows, it completes the risk-free picture where the classical theory continues to hold.
Interbank rates, considered risky, now must be modelled separately. Their 5
yield curves of different tenors are called forwarding curves. Our aim is to construct a coherent market model consisting of both discounting curve and forwarding curves.
In this Chapter we will also show how to use an interest rate model in practice.
1. Bootstrapping technique gives a method of constructing initial yield curves, both discounting and forwarding.
2. Calibration is the most important step in which the market parameters are adapted to market data.
3. Calibrated model is then used to price future cash flows of market instru-ments. If tractable formula is not available, numerical methods must be used. The most used method is Monte Carlo simulation which is however computationally slow and expensive; a more sophisticate method requires the discretization of associated PDEs when original SDEs have a Marko-vian setting, such as Fokker-Planck equations whose strong solution gives the correspond SDE solution’s density function.
A first simple multi-curve model is shown in Chapter three. It is an extended short rate model which uses a Vasiček-type factor structure for modelling both risk-free short rate and short rate spreads between the risk-free one and the risky one. This modelling approach leads to a generalization of affine term structure, and the calculation of an adjustment factor between pre-crisis and post-crisis FRA prices. However, this model has several problems, like the assumption of negative values with non zero probability and the difficulty to apply calibration procedure.
Chapter four gives a more elaborated modelling method. It is inspired both by the classial HJM framework and Libor model, i.e. it models both risk-free instantaneous forward rate and risky forward Libor rates under T -forward mea-sure. Particular volatility assumption ensures that SDE solutions are Markovian processes; moreover, the entire model dynamics can be driven by a finite family of Markov processes, thus the original model becomes a generalization of shifted multi-factor Hull-White model. Unlike the previous model, this model leads to a real Black’s formula for caps and an approximated one for swaptions, which simplified enormously the calibration problem.
Finally, in the last Chapter a numerical example is given for a simplified case of HJM-type model presented in Chapter four. Initial yield curve graphs and calibrated model parameters are obtained by concrete market data.
Chapter 1
Classical Theory of Bond
Market
1.1
General market model and First
Fundamen-tal Theory
Let be there a filtered probability space (Ω, F , (Ft)t>0, P) and a general market
model on the time interval [0, T ], with asset prices {Si
t : i ∈ I, 0 6 t 6 T },
where I is a general set and every Si is an adapted stochastic process.
We recall the First Fundamental Theorem which gives a sufficient (and nec-essary if I is a finite set) condition for the absence of arbitrage in every market model. The theorem gives also a pricing formula for the par value of a T -claim X (assumed to be a random variable) at time t such that 0 6 t 6 T , denoted by Π(t; X), that is the price value with which the expanded market including the new asset process Π(t; X) continues to be free of arbitrage.
Theorem 1.1.1 (First Fundamental Theorem). If S0 is strictly positive, then 1. If I is a finite set, i.e. I = {1, 2, ..., n} with n ∈ N, n < ∞, then the market is free of arbitrage if and only if there exists a martingale measure or martingale probability Q0
∼ P such that every discounted process Si
t
S0 t
for i = 1, ..., n
is a (local) martingale under Q0; the discount asset S0is called numeraire.
2. If I is a general set, then the existence of a martingale measure is just a sufficient condition for absence of arbitrage.
3. The par value of a T -claim X is given by Π(t; X) = St0E0 X S0 T Ft , (1.1.1) 7
where E0
denotes expectation under Q0.
The following theorem ensures that the no-arbitrage condition is independent of the numeraire choice, and it gives a transformation formula to pass from one numeraire to an other.
Theorem 1.1.2 (Numeraire change formula). Let be there a price system {Si:
i ∈ I}. We assume the existence of a martingale measure Q0 related to the
numeraire S0.
• If Si
(i ∈ I) is a strictly positive asset, then Qi is a martingale measure
with numeraire Si if and only if
Qi =S i TS00 Si 0ST0 Q, i.e. the ration
STiS00
Si 0ST0
is the Radon-Nikodym derivative for the measure change Q0
→ Qi. • In particular dQi dQ0 Ft = S i tS00 Si 0S0t .
1.2
The pre-crisis bond and interest rate market
In this section, we recall shortly the basic assets in a bond market and the definition of different interest rates, as well as some pricing formulas for typical interest rate derivatives (both linear or non-linear) and finally, some main models of the pre-crisis bond and interest rate market, like short rate models, HJM type models, Libor models and swap models. However, the classical pricing formulas don’t correspond to the real situation of the post-crisis bond market as we will see in the next chapter.
1.2.1
Bonds and interest rates
Zero coupon bonds
The basic bonds traded in the bond market are zero coupon bonds, that is bonds without coupons.
Definition 1.2.1. A zero coupon bond (ZCB) with maturity date T > 0, or a T -bond, is a contract which guarantees the holder 1 dollar (or euro) to be paid on the date T . We denote its price at time t such that 06 t 6 T by B(t, T ).
As a mathematical entity, B(t, T ) is assumed to have the following proper-ties:
1.2. THE PRE-CRISIS BOND AND INTEREST RATE MARKET 9 Assumption 1.2.1.
• B(t, T ) is traded for every T > 0; in other word, we assume the existence of a frictionless bond market for every maturity.
• 0 < B(t, T ) < 1 for 0 < t < T and B(T, T ) = 1 for every T > 0.
• For each fixed t, B(t, T ) is decreasing and differentiable w.r.t. time of maturity T .
• For each fixed T , B(t, T ) is a scalar adapted stochastic process.
Based on zero coupon bond, we can define some abstract rates not existing in real life only for modelling reason.
Definition 1.2.2.
1. The instantaneous forward rate at t with maturity T > 0 (06 t 6 T ) is defined by
ft(T ) := −
∂ log B(t, T )
∂T .
2. The instantaneous short rate at time t> 0 is defined by rt:= lim
T ↓tft(T ) = ft(t).
Observation 1.2.2.
1. These definitions yield the following relationship between zero coupon bond and instantaneous forward rate for 06 t 6 T :
log B(t, T ) = − Z T t ft(u)du, B(t, T ) = exp − Z T t ft(u)du . (1.2.1) Thus for every fixed t> 0, the zero coupon bond curve {B(t, T ), T > t} and the forward rate curve {ft(T ), T > t} give exactly the same
informa-tion from a mathematical point of view.
2. We notice that knowing only the dynamic of the short rate (rt)t>0 can
not help to determine the entire zero coupon bond curve or forward rate curve.
We introduce the basic numeraire in the bond market:
Definition 1.2.3 (Money market account). We call money market account the quantity Bt:= exp Z t 0 rudu . for every t> 0.
Definition 1.2.4.
1. In a bond market we call risk neutral probability or martingale probability without mentioning the numeraire, if the numeraire is money market ac-count. It is denoted by Q (EQthe correspondent expectation). Under this
probability, every zero coupon bond discounted by money market account is a (local) martingale, i.e. for every 06 t 6 T ,
B(t, T ) Bt
is a (local) martingale.
2. We call T-forward measure or T-forward probability, denoted by QT
(ET
the correspondent expectation), the probability defined as above with a T-bond as numeraire, instead of money market account.
Observation 1.2.3. Because of the term structure of assets in the bond mar-ket, in many cases using a T-bond as numeraire, so modelling with T-forward measure may be very useful.
Proposition 1.2.4. Assuming integrability conditions, for every fixed T > 0 we have the following relations for 06 t 6 S < T
B(t, T ) = EQ exp − Z T t rudu Ft , (1.2.2) B(t, S) B(t, T ) = E T 1 B(S, T ) Ft , (1.2.3) ft(T ) = ET[rt|Ft]. (1.2.4)
Proof. (1.2.2) and (1.2.3) are direct consequence of Definition 1.2.4 of the mar-tingale probability and the T -forward measure, as
B(t, T ) Bt = E Q B(T, T ) BT Ft = EQ 1 BT Ft and B(t, S) B(t, T ) = E T B(S, S) B(S, T ) Ft = ET 1 B(S, T ) Ft .
For (1.2.4), by using the general pricing formula (1.1.1) with S0(t) = Bt and
X = rt, and the change of numeraire (Theorem 1.1.2)
Π(t; rt) = EQ rt − Z T t rudu Ft = B(t, T )ETrt|Ft,
1.2. THE PRE-CRISIS BOND AND INTEREST RATE MARKET 11 we get ET[rt|Ft] = 1 B(t, T )E Q rtexp − Z T t rudu Ft = − 1 B(t, T )E Q ∂ ∂T exp − Z T t rudu Ft = − 1 B(t, T ) ∂ ∂TE Q exp − Z T t rudu Ft = −BT(t, T ) B(t, T ) = − ∂ log B(t, T ) ∂T = ft(T ). Libor rate
One of the most common benchmark interest rate indexes for the bond market is Libor /Euribor rate.
Definition 1.2.5 (Libor/Euribor). The London Interbank Offered Rate (Libor) (resp. Euro Interbank Offered Rate (Euribor))is the daily published average interbank linear interest rate at which a selection of banks on the London (resp. Euro) money market are prepared to lend to one another for a certain time frame. The Libor (or Euribor) value at t for [t, T ] is denoted by L(t, T ). Remark 1.2.5. We will use the same notations both for London Interbank Offered Rate and European Interbank Offered Rate, and we will call them Libor rate indifferently. The numerical values will usually refers to Euribor rate as we will always consider the Euro area.
As a mathematical entity, L(t, T ) (with t6 T ) is assumed to be • defined for every T > 0 and 0 6 t 6 T ;
• strictly positive for every (t, T ) with 0 < t 6 T ; • an adapted stochastic process for every fixed T > 0.
In an ideal arbitrage-free world, the following relations between ZCB and Libor should be hold for 06 t < T
1
B(t, T )= 1 + (T − t)L(t, T ), i.e.
L(t, T ) = 1 − B(t, T )
B(t, T )(T − t). (1.2.5)
In real life, there can be a small spread (i.e. difference) between the Libor rate and the actual rate applied. However, in the pre-crisis context, these spreads are ignorable, so the arbitrage-free relations can be considered true.
1.2.2
Interest rate derivatives and their pricing formula
In this section we are going to present typical interest rate derivatives and their abstract pricing formula in the classical theory.
There are two types of interest rate products: linear products and non-linear products. For the first ones we have a direct and handleable price expression in terms of the Libor rate or zero-coupon bonds in the classical context, and the pricing formula is independent of the model choice. The second ones have valuation formulas which are not always tractable, unless market model has particular structure. They are typically used as calibration instruments to de-termine model parameters. However, as we will see later, in the multi-curve framework, linear products will be used as calibration instruments too, since the simple classical no-arbitrage relations don’t hold any longer.
We will begin with linear products. Interest rate swap (IRS)
The simplest of all interest rate derivatives is interest rate swap.
Definition 1.2.6. An interest rate swap or IRS provides a payment exchanging between a payment stream at a fixed interest rate K (fixed leg ) called swap rate, and a payment stream at a floating rate (typically Libor/Euribor rate) (floating leg ).
1. A swap can be made by swaplet settled in arrears, i.e. we can have a number of payment dates T1, ...Tnwith 0 < T1< ... < Tnand Ti+1−Ti = δ
for every i = 1, ..., n − 1. The swap is initiated at time T0 ∈ [0, T1) with
T1− T0= δ. At each payment date Ti, i = 1, ..., n, the value of a swaplet
for the receiver of the floating rate with notional N is ΠIRS(Ti; Ti−1, Ti, K, N ) = N δ(L(Ti−1, Ti) − K).
Using the pricing formula (1.1.1) by setting (Ti)-claim X as
N δ(L(Ti−1, Ti) − K)
the value of the Ti-swaplet at time t6 T0 is then
N B(t, Ti)δETi[L(Ti−1, Ti) − K|Ft].
The price of an IRS is the sum of all swaplets, i.e. ΠIRS(t; T1, Tn, K, N ) = N
n
X
i=1
B(t, Ti)δETi[L(Ti−1, Ti) − K|Ft].
We call par swap rate KIRS(T
1, Tn, δ) the value for K such that the
con-tract has zero value at inception, namely KtIRS(T1, Tn, δ) =
Pn
i=1B(t, Ti)δETi[L(Ti−1, Ti)|Ft]
δPn
i=1B(t, Ti)
1.2. THE PRE-CRISIS BOND AND INTEREST RATE MARKET 13 The no-arbitrage relation between Libor rate and zero coupon bond (1.2.5) leads to an simple expression for every Ti-swaplet
N B(t, Ti)δETi[L(Ti−1, Ti) − K|Ft] =N B(t, Ti)δETi 1 − B(Ti−1, Ti) δB(Ti−1, Ti) − K Ft =N B(t, Ti)ETi 1 B(Ti−1, Ti) Ft − (1 + δK)B(t, Ti) =N B(t, Ti−1) − (1 + δK)B(t, Ti)
where we have used also (1.2.3). Then the par value of an IRS is given by
ΠIRS(t; T1, Tn, K, N ) = N n X i=1 B(t, Ti)δETi[L(Ti−1, Ti) − K|Ft] = N n X i=1 B(t, Ti−1) − (1 + δK)B(t, Ti) = N B(t, T0) − B(t, Tn) − Kδ n X i=1 B(t, Ti) .
and the par swap rate is exactly
KtIRS(T1, Tn, δ) =
B(t, T0) − B(t, Tn)
δPn
i=1B(t, Ti)
.
2. The previous valuation formula can be extended to the general case of swaps with a δ tenor floating leg and a ¯δ tenor fixed leg paying at times T = {T1, ..., Tn} and ¯T = { ¯T1, ..., ¯Tm} respectively. We will use the notations
τi= Ti+1− Ti for i = 1, ..., n − 1, and ¯τj = ¯Tj+1− ¯Tj for j = 1, ..., m − 1.
At time t6 min(T1− δ, ¯T1− δ) the value of such an IRS is
ΠIRS(t; T , ¯T , K, N ) =N n X i=1 τiB(t, Ti)ETi[L(Ti− δ, Ti)|Ft] − K m X j=1 ¯ τjB(t, ¯Tj) .
The par swap rate KIRS(T , ¯T , δ) is then given by KtIRS(T , ¯T , δ) = Pn i=1τiB(t, Ti)ETi[L(Ti− δ, Ti)|Ft] Pm j=1τ¯jB(t, ¯Tj) . (1.2.6)
coupon bond (1.2.5), the value of IRS becomes ΠIRS(t; T1, Tn, K, N ) =N n X i=1 τi δB(t, Ti)E Ti 1 B(Ti− δ, Ti) Ft −τi δB(t, Ti) − K m X j=1 ¯ τjB(t, ¯Tj) =N n X i=1 τi δ B(t, Ti− δ) − B(t, Ti) − K m X j=1 ¯ τjB(t, ¯Tj) ; thus the par swap rate is
KtIRS(T , ¯T , δ) = Pn i=1 τi δ B(t, Ti− δ) − B(t, Ti) Pm j=1τ¯jB(t, ¯Tj) .
Definition 1.2.7. We call swap measure, denoted by QIRS
(EIRS the
corre-spond expectation), the martingale measure for the numeraire
m
X
j=1
¯
τjB(t, ¯Tj).
Observation 1.2.6. Clearly, the swap rate KtIRS(T , ¯T , δ)
is a (local) martingale under the swap measure QIRS. Overnight indexed swap (OIS)
Definition 1.2.8 (EONIA). EONIA (Euro Overnight Index Average) rate is an effective rate calculated from the weighted average of all overnight unsecured lending transactions undertaken in the Euro interbank market.
Observation 1.2.7. While Euribor rates usually refer to middle-long tenor (for example: 1 month, 3 months, 6 months, 12 months etc.), Eonia rates refer only to very short tenor (namely one day).
Definition 1.2.9 (OIS). An OIS (overnight indexed swap) contract is a par-ticular swap contract where the floating rate is Eonia rates; we call OIS rates the market quotes for the fixed leg.
As before, the value of an OIS can be expressed as (we assume for simplicity payment dates T1, ...Tn with 0 < T1 < ... < Tn and Ti+1 − Ti = δ for all
i = 1, ..., n − 1) ΠOIS(t; T1, Tn, K, N ) = N B(t, T0) − B(t, Tn) − Kδ n X i=1 B(t, Ti)
and the OIS rate KOIS is given by KtOIS(T1, Tn, δ) =
B(t, T0) − B(t, Tn)
δPn
i=1B(t, Ti)
1.2. THE PRE-CRISIS BOND AND INTEREST RATE MARKET 15 Observation 1.2.8. Before the financial crisis of 2007, the spread between Overnight Indexed Swaps (OIS) and standard Interest Rate Swaps (IRS) (with Euribor rate as floating leg) was quite low and constant, indicating that the classical valuation formulae fitted almost perfectly the market quotes. We’ll see further that with the beginning of the crisis, this situation changed drastically and OIS rate will play an important role in the post-crisis context.
Forward rate agreement (FRA)
Definition 1.2.10. A forward rate agreement or FRA is an OTC contract with the floating Libor rate L(T, T + δ) (with tenor δ, fixed at time T , and spanning the time interval [T, T + δ]) as floating leg, versus a fixed rate (fixed leg) K spanning the same time interval. The FRA contract with notional N pays at maturity T + δ the amount
N δ(L(T, T + δ) − K).
We call FRA rate or forward Libor rate the market quote at t for the fixed rate K with tenor δ and we denote it by Lt(T, T + δ).
We notice that the FRA rate or forward Libor rate is actually the par swap rate K at t for a swaplet over [T, T + δ] paying fixed rate at T in exchange for L(T, T + δ). So the value of a FRA contract at time t 6 T and the par FRA rate (or forward Libor rate), denoted by Lt(T, T + δ), are respectively
ΠF RA(t; T, T + δ, K, N ) = N B(t, T + δ)δET +δ[L(T, T + δ) − K|Ft] (1.2.7)
and
Lt(T, T + δ) = ET +δ[L(T, T + δ)|Ft]. (1.2.8)
Observation 1.2.9. We notice that
LT(T, T + δ) = L(T, T + δ)
since L(T, T + δ) is FT-measurable.
Observation 1.2.10. The formula (1.2.8), which can be considered a definition of FRA or forward Libor rates, allows to express every interest rate swap as a FRA rate derivative. Indeed, if we consider the general case of swaps with a δ tenor floating leg and a ¯δ fixed leg paying at times T = {T1, ..., Tn} and
¯
T = { ¯T1, ..., ¯Tm}, the value of an IRS is then
ΠIRS(t; T , ¯T , K, 1) = N n X i=1 τiB(t, Ti)Lt(Ti− δ, Ti) − K m X j=1 ¯ τjB(t, ¯Tj) , (1.2.9) and the valuation formula of par swap rate (1.2.6) becomes
KtIRS(T , ¯T , δ) = Pn i=1τiB(t, Ti)Lt(Ti− δ, Ti) Pm j=1τ¯jB(t, ¯Tj) . (1.2.10)
Assuming the classical no-arbitrage relation (1.2.5) we have ΠF RA(t; T, T + δ, K, N ) = N B(t, T ) − B(t, T + δ)(1 + δK) and Lt(T, T + δ) = 1 δ B(t, T ) B(t, T + δ)− 1 .
Observation 1.2.11. In particular the following relationship holds among zero-coupon bonds and FRA rates for every 06 t 6 S < T :
Lt(S, T ) = − B(t, T ) − B(t, S) (T − S)B(t, T ) (1.2.11) i.e. 1 B(t, S)(1 + (T − S)Lt(S, T )) = 1 B(t, T ). Basis swap
Definition 1.2.11. A basis swap is another special type of interest rate swap where two cash flows related to Libor rates associated to different tenors are exchanged between two counterparties. It is equivalent to a long/short position on two different interest rate swaps which share the same fixed leg.
Let be there three date set T1 = {T1
0, ...Tn11}, T
2 = {T2
0, ...Tn22} and T
3 =
{T3
0, ...Tn33}, with the same initial time and final time: T
1 0 = T02 = T03, Tn11 = T2 n2 = T 3 n3; such that T 1 ⊂ T2, n
1 < n2. The first two tenor structures are
associated to the two floating legs with corresponding tenor lengths δ1 > δ2,
and the third one (without any restrictions) is associated to a δ3tenor-fixed leg.
We assume δ1= T1
i+1− Ti1for every i = 0, ..., n1− 1, δ2= Tj+12 − Tj2 for every
j = 0, ..., n2− 1.
The value at time t6 T1 0 is ΠBSW(t; T1, T2, T3, N ) =N n1 X i=1 δ1B(t, Ti1)ETi1L T1 i−1(T 1 i−1, T 1 i) Ft − n2 X j=1 δ2B(t, Tj2)ETj2L T2 j−1(T 2 j−1, T 2 j) Ft − K n3 X l=1 δ3B(t, Tl3) .
The par value of basis swap spread KBSW such that the value of the contract
at initiation is zero is then KtBSW(T1, T2, T3, δ1, δ2, δ3) = Pn1 i=1δ 1B(t, T1 i)ET 1 iL(T1 i−1, Ti1) Ft Pn3 l=1δ3B(t, Tl3) − Pn2 j=1δ 2B(t, T2 j)E T2 jL(T2 j−1, Tj2) Ft Pn3 l=1δ3B(t, Tl3)
1.2. THE PRE-CRISIS BOND AND INTEREST RATE MARKET 17 Observation 1.2.12. In particular, the basis swap spread is the difference of two IRS rates with the same fix leg:
KtBSW(T1, T2, T3, δ1, δ2, δ3) = KtIRS(T1, T3, δ1) − KtIRS(T2, T3, δ2) Observation 1.2.13. Recalling Observation 1.2.10, the par basis swap spread can be considered as a FRA rate derivative
KtBSW(T1, T2, T3, δ1, δ2, δ3) = Pn1 i=1δ 1B(t, T1 i)Lt(Ti−11 , T 1 i) − Pn2 j=1δ 2B(t, T2 j)Lt(Tj−12 , T 2 j) Pn3 l=1δ3B(t, T 3 l) . As usual, the classical no-arbitrage relation (1.2.5) leads to
ΠBSW(t; T1, T2, T3, N ) =N n1 X i=1 B(t, Ti−11 ) − B(t, T 1 i) − n2 X j=1 B(t, Tj−12 ) − B(t, T 2 j) − K n3 X l=1 δ3B(t, Tl3) , KtBSW(T1, T2, T3, δ1, δ2, δ3) = Pn1 i=1 B(t, Ti−11 ) − B(t, Ti1) − P n2 j=1 B(t, Tj−12 ) − B(t, Tj2) Pn3 l=1δ3B(t, Tl3) ! =0.
It’s important to notice that the classical no-arbitrage valuation of basis swap spread gives a value exactly equal to zero. Before the financial crisis this was a quite good estimation as the real value of quoted basis swap spread was indeed very near to zero.
Now we are going to consider some most traded non-linear interest rate derivatives, such as caps and swaptions. These products don’t have simple valuation formula neither in the classical context, unless there are any particular model structure assumptions.
Caps and floors
Definition 1.2.12. An interest rate cap (resp. floor ) is a contract against the high (resp. low) value of a floating rate (typically Libor rate), i.e. it protects from having to pay more (resp. less) than a fixed rate, called cap rate (resp. floor rate) K. It is composed by caplets (resp. floorlets) just as an IRS. Every caplet with notional N , strike price K and maturity T , settled in arrears at T + δ gives at time T + δ
its valuation formula is hence ΠCP LT(t; T, T + δ, K, N ) = N Btδ E 1 BT +δ (L(T, T + δ) − K)+ Ft .
Every floorlet with notional N , strike price K and maturity T , settled in arrears at T + δ gives at time T + δ
ΠF LLT(T + δ; T, T + δ, K, N ) = N δ (K − L(T, T + δ))+; with valuation formula
ΠF LLT(t; T, T + δ, K, N ) = N Btδ E 1 BT +δ (K − L(T, T + δ))+ Ft .
We notice that, formally speaking, the caplet value at t can be seen as a call option on the underlying L(T, T + δ) with strike price K.
Definition 1.2.13 (Black’s Formula for Caplets). We call Black’s formula or Black-76 formula for the caplet with notional N = 1
δ (L(T, T + δ) − K)+ the valuation formula
ΠCP LT(t; T, T + δ, K, N ) ≈ δB(t, T + δ) Lt(T, T + δ)N [d1] − KN [d2]
where N is the cumulative distribution function for the standard normal variable N (0, 1) and d1= 1 σ√T − t ln Lt(T, T + δ) K +1 2σ 2(T − t) , d2= d1− σ √ T − t.
The quantity σ is called implied Black volatility.
Usual market practices is to compute cap prices by this Black’s formula for caplets and to quote prices in terms of the implied Black volatilities, but the formula has sense only under particular modelling assumption, as we will see in Section 1.2.4.
Swaptions
Let be there an interest rate swap starting at T0 = T with notional N , a δ
tenor floating leg and a ¯δ tenor fixed leg paying at times T = {T1, ..., Tn} and
¯
T = { ¯T1, ..., ¯Tm} respectively, where = Ti+1− Ti = τi for i = 1, ..., n − 1, and
¯
1.2. THE PRE-CRISIS BOND AND INTEREST RATE MARKET 19 Definition 1.2.14. A swaption or swap option with swaption strike price K is a contract which at the exercise date T gives the holder the right but not the obligation to enter into a swap with the fixed swap rate K. So its’s a T -claim defined by
ΠSW P T N(T ; T , ¯T , K, N ) = ΠIRS(T ; T , ¯T , K, N )+ . Its par value at t6 min(T1− δ, ¯T1− δ) is thus
ΠSW P T N(t; T , ¯T , K, N ) = BtE 1 BT ΠIRS(T ; T , ¯T , K, N )+ Ft ,
which can be expressed in term of the FRA rate using (1.2.9)
ΠSW P T N(t; T1, Tn, K, N ) = N E Bt BT n X i=1 τiB(t, Ti)Lt(Ti−δ, Ti)−K m X j=1 ¯ τjB(t, ¯Tj) + Ft .
On the other hand, if τi = ¯τj = δ for every i = 1, ..., n − 1 and for every
j = 1, ..., m − 1, the classical relationship (1.2.5) between Libor rate and zero coupon will lead to
ΠSW P T N(t; T1, Tn, K, N ) = BtE 1 BT B(t, T0)−B(t, Tn)−Kδ n X i=1 B(t, Ti) + Ft . We recall that ΠIRS(t; T , ¯T , K, 1) = KIRS t (T , ¯T , δ) − K m X j=1 ¯ τjB(t, ¯Tj) ,
then a T -swaption is formally a call option on KIRS
T (T , ¯T , δ) with strike price
K ΠSW P T N(T ; T , ¯T , K, 1) = m X j=1 ¯ τjB(T, ¯Tj) KTIRS(T , ¯T , δ) − K + ,
whose par value is given by ΠSW P T N(t; T1, Tn, K, N ) =E Bt BT ΠIRS(T ; T1, Tn, K, N ) + Ft =E exp − Z T t rudu m X j=1 ¯ τjB(T, ¯Tj) KTIRS(T , ¯T , δ) − K+ Ft . (1.2.12)
Definition 1.2.15 (Black’s Formula for Swaptions). The Black formula for a swaption with strike K is defined as
ΠSW P T N(t; T , ¯T , K, 1) ≈ m X j=1 ¯ τjB(t, ¯Tj) KtIRS(T , ¯T , δ)N [d1] − KN [d2] where d1= 1 σ√T − t ln K IRS t (T , ¯T , δ) K +1 2σ 2(T − t) , d2= d1− σ √ T − t.
As for Black’s formula for caps, Black’s formula for swaptions is very used in quoting prices, even though it needs particular model structure to have the-oretical interpretation (see Section 1.2.4.).
1.2.3
Some pre-crisis bond market models
We can classify bond market models in base of the objects in: 1. short rate models (Vasiček, C.I.R., Ho-Lee, Hull-White etc.); 2. forward rate models (HJM framework etc.);
3. Libor rate models; 4. swap rate models.
We can also classify models by the probability used, so we have 1. P models (modelling under P);
2. risk neutral models (modelling under Q); 3. T -forward models (modelling under QT)
We are going to briefly call back these models without going into the details, as we will give generalized versions later.
Short rate models
In short rate models, the short rate of interest is the only object given a priori, whose dynamic under the martingale probability Q (assumed to exist) is the solution of a SDE of the form
drt= µ(t, rt)dt + σ(t, rt)dWt
where W is a Q-Wiener process, µ (drift) and σ (volatility) are continuous functions (or at least measurable functions), “good” enough to ensure the strong existence and uniqueness of SDE solution.
1.2. THE PRE-CRISIS BOND AND INTEREST RATE MARKET 21 Observation 1.2.14. A short rate with the above dynamics is always a Markov process, just as all strong solutions of stochastic differential equations of the form
dXt= µ(t, Xt)dt + σ(t, Xt)dWt.
The Markov property is a consequence of the flow property of a SDE solution (or diffusion), namely
Xs0,x= Xt,Xtx
s P − a.s. where we denote by (Xt,x
s ; s > t) the solution of SDE starting from x at time x.
The money market account B has dynamics dBt= rtBtdt.
We can interpret it as the dynamics of the value of a bank account with stochas-tic short rate of interest r.
Once the short rate dynamics is known, a T -zero coupon bond price with T > 0 is given by the risk neutral valuation (1.2.2):
B(t, T ) = EQ exp − Z T t rudu Ft .
Here we list some well-known models. If a parameter isn’t explicitly time dependent, then it is constant.
• Vasiček drt= (b − art)dt + σdWt, (a > 0); • Cox-Ingeroll-Ross (CIR) drt= a(b − rt)dt + σ √ rtdWt; • Ho-Lee drt= Θ(t)rtdt + σdWt;
• Hull-White (extended Vasiček)
drt= (Θ(t) − a(t)rt)dt + σ(t)dWt, (a(t) > 0);
• Hull-White (extended CIR)
drt= (Θ(t) − a(t)rt)dt + σ(t)
√
rtdWt, (a(t) > 0);
Apart form the Markovian dynamics as mentioned in Observation 1.2.14, all enumerated models have affine term structure (defined below). These two are very useful properties from an analytical and computational point of view, since they facilitate the yield curve inversion, that is the procedure to adapt the model parameters to the observable bond data.
Definition 1.2.16. If the zero coupon bond B(t, T ) has the form B(t, T ) = exp(A(t, T ) − C(t, T )rt)
where A and C are deterministic functions, then the model is said to possess an affine term structure.
The following proposition shows that an affine choice of µ and σ2is a
suffi-cient condition for the existence of an affine term structure. Proposition 1.2.15. If µ and σ are of the form
(
µ(t, r) = α(t) + β(t)r, σ(t, r) =pγ(t) + δ(t)r,
then the model admits an affine term structure where A and C satisfy the system (
Ct(t, T ) + β(t)C(t, T ) − 12δ(t)C
2(t, T ) + 1 = 0, C(T, T ) = 0,
At(t, T ) = α(t)B(t, T ) −12γ(t)B2(t, T ), A(T, T ) = 0.
Furthermore, if µ and σ2are time independent, then the affine choice is also
a necessary condition for the existence of an affine term structure.
Proposition 1.2.16. If the model admits an affine term structure and µ and σ are time independent, i.e. µ(t, r) = µ(r) and σ(t, r) = σ(r), then
(
µ(t, r) = α + βr, σ(t, r) =√γ + δr, and A, C satisfy the system
(
Ct(t, T ) + βC(t, T ) −δ2C2(t, T ) + 1 = 0, C(T, T ) = 0,
At(t, T ) = αB(t, T ) −γ2B2(t, T ), A(T, T ) = 0.
HJM framework
HJM framework is based on instantaneous forward rate under the martingale probability Q (assumed to exist). It isn’t a model but a framework as it doesn’t propose a specific model.
It assumes that, for every fixed maturity T > 0, the dynamic of forward rate under Q is describe by
dft(T ) = αt(T )dt + σt(T ) · dWt,
f0(T ) = f0∗(T ),
here {f0∗(T ), T > 0} gives the observed initial condition curve, W is a
(d-dimensional) Q-Wiener process,
α : (Ω × R+) × R+, P ⊗ B(R+) −→ R, B(R) (t, T, ω) 7→ αt(T )(ω)
1.2. THE PRE-CRISIS BOND AND INTEREST RATE MARKET 23 σ : (Ω × R+) × R+, P ⊗ B(R+) −→ Rd, B(Rd)
(t, T, ω) 7→ σt(T )(ω)
are two measurable processes (P denotes the σ-algebra generated by all progres-sively measurable processes) which will be always assumed to be continuous a.s. in both t and T , and to satisfy a.s. for all t, T ∈ R+ the following conditions
Z t 0 Z T 0 |αs(u)|duds < ∞, (1.2.13) Z t 0 Z T 0 ||σs(u)||2duds < ∞, (1.2.14)
Observation 1.2.17. The coefficient conditions (1.2.13) and (1.2.14) implies that for every fixed T > 0
Z T 0 |αs(u)|ds < ∞, Z T 0 ||σs(u)||2ds < ∞,
that is, the solution ft(T ) is an Itô process for every fixed T > 0;
further-more, with conditions (1.2.13) and (1.2.14) it is possible to use both classic and stochastic Fubini-Tonelli Theorem (see Appendix A) respectively for the drift process αt(T ) and the volatility process σt(T ).
Observation 1.2.18. It is important to notice that without further assump-tions, the above general dynamics of instantaneous foward rate do not ensure the Markovianity of the solution.
The zero coupon bond B(t, T ) can be obtained by the relation (1.2.1)
B(t, T ) = exp − Z T t ft(u)du .
However, in order for the formulas (1.2.1) and (1.2.2) to hold simultaneously, i.e. both B(t, T ) = exp − Z T t ft(u)du and B(t, T ) = EQ exp − Z T t rudu Ft
we must have the following consistency relation between α and σ, which is also a necessary and sufficient condition for the existence of the martingale measure Q.
Theorem 1.2.19 (HJM drift condition). For very fixed maturity T > 0, B(t, T )
Bt
is a (local) martingale if and only if the processes α and σ satisfy the following relation for every t such that 06 t 6 T
αt(T ) = σt(T )
Z T
t
σt(s)ds. (1.2.15)
In other word, the dynamic of forward rate under Q is completely determined by its volatility structure:
dft(T ) = σt(T ) · Z T t σt(s)ds dt + σt(T ) · dWt. (1.2.16) Recalling (1.2.4) ft(T ) = ET[rt|Ft],
we notice that for every fixed T > 0, ft(T ) is a martingale under T -forward
measure. Hence we can just model instantaneous forward rate under T -forward measure.
Proposition 1.2.20. The dynamics of ft(T ) under QT as given by
dft(T ) = σt(T ) · d ˜Wt
where ˜W is a d-dimensional QT-Wiener process.
Proof. Since ft(T ) is a QT-martingale, the drift term is equal to zero.
Further-more, the volatility term σt(T ) doesn’t change under a Girsanov transformation.
Hence comparing with (1.2.16) we obtain the Girsanov kernel for the transfor-mation Q → QT
Z T
t
σt(s)ds.
This quantity satisfy the Novikov condition since it is even bounded for condition (1.2.14) (using Jensen disequality), thus ˜W defined by
d ˜Wt= dWt+ Z T t σt(s)ds dt is a QT-Wiener process.
1.2.4
Black’s formula
1.2. THE PRE-CRISIS BOND AND INTEREST RATE MARKET 25 Definition 1.2.17. We call Black’s formula for a T -claim S with strike price K and implied Black volatility Γ at current time t the following expression
Bl(K, S, Γ) := SN [d1] − KN [d2] (1.2.17)
where N is the cumulative distribution function for the standard Brownian motion and d1= 1 Γ√T − t ln S K +1 2Γ 2(T − t) , d2= d1− Γ √ T − t.
If the underlying price process St has a log-normal dynamics under some
martingale measure ¯Q, namely
dSt= Stσ(t)dWt
where W is a (d-dimensional) ¯Q-Wiener process, σ is a (d-dimensional) mea-surable deterministic function such that
Z T
0
||σ(s)||2ds < ∞,
then the classical Black-Samuelson theory ensures that the par value of a call option (ST − K)+ at time t < T is given by StN [d1] − KN [d2] where d1= 1 q RT t ||σ(s)|| 2ds ln St K +1 2 Z T t ||σ(s)||2ds , d2= d1− s Z T t ||σ(s)||2ds.
In particular, the implied Black volatility is such that Γ2(T − t) = Z T t ||σ(s)||2ds. Recalling (1.2.8) Lt(T, T + δ) = ET +δ[L(T, T + δ)|Ft]
we notice that the forward Libor rate is a (local) martingale under T -forward measure QT. If L
t(T, T + δ) have dynamics under QT described by
where W is a QT-Wiener and σ is a good enough deterministic function, then
we will get a log-normale dynamics under QT and Black’s valuation formula for
caplets will hold theoretically.
These models are called Libor market model. In Appendix B we will give a proof of the existence of Libor market model with finite maturities.
For swaptions, we know that a par swap rate is a (local) martingale under swap measure (Observation 1.2.6). Hence if it has dynamics under swap measure of the form
dKtIRS(T , ¯T , δ) = KIRS
t (T , ¯T , δ)σ(t)dWt
then Black’s formula will hold.
Chapter 2
Post-crisis Problem: Multiple
Yield Curve Modelling
2.1
The post-crisis interest rate market
After the last financial crisis (2007), there is an average credit quality deteriora-tion even among the biggest banks which were considered undefaultable before crisis. Interbank rates (e.g. Libor) can not be considered risk-free (i.e. free of credit or liquidity risk) any more, since they are affected by credit and liquidity risk of the panel of contribution banks, especially if these rates have long tenor. The most important consequence of this situation is the raise of spreads (which in the pre-crisis environment were negligible) between interbank rates and overnight rates, as well as between interbank rates associated to different tenor lengths. In particular, we have the following significant spread values.
• EONIA-Euribor spreads:
Figure 2.1: FRA 3 × 6 vs forward EONIA. 27
In Figure 2.1 spread between FRA 3 × 6 and forward rate obtained by two consecutive EONIA rates 3M and 6M. Figure obtained form Bloomberg platform.
Figure 2.2: FRA 6 × 12 vs forward EONIA.
In Figure 2.2 spread between FRA 6×12 and forward rate obtained by two consecutive EONIA rates 6M and 12M. Figure obtained form Bloomberg platform.
• OIS rate-IRS rate spreads: in Figure 2.3 spread between EONIA swap rate with maturity one year (orange line) and EURIBOR swap-rate (white line) on the same maturity. Figure obtained form Bloomberg platform.
Figure 2.3: OIS vs IRS.
• spreads between FRA rates of different tenors: in Figure 2.4 term structure of additive spreads between FRA rates and OIS forward rates, at time T0=
2.1. THE POST-CRISIS INTEREST RATE MARKET 29
Figure 2.4: Spread FRA of different tenors.
• Basis swap spreads: in Figure 2.5 basis-swap spread for six-months vs. three-months EURIBOR rates on a swap with maturity one year. Figure obtained form Bloomberg platform.
Figure 2.5: Basis swap spreads.
Clearly, the classical bond market theory is not suitable to the new financial situation, as its usual no-arbitrage relationships are violated in a macroscopic way, being thus unable to describe the market dynamics nor explain the presence of all these remarkable spreads. Therefore, we need to create new models which should be able to handle all these problems in an arbitrage-free context.
2.1.1
Risk-free discounting yield curve
Firstly, as interbank rates are now risky (that is affected by credit and liquidity risk), we need a proxy for risk-free rates as reference.
We notice that overnight deposits have very reduced credit/liquidity expo-sure, while the longer is the tenor, the greater will be the risk. This leads us to make a basis classification.
Definition 2.1.1.
1. We classify as risk-free, all the overnight rates (e.g. EONIA rates) and the rates that have overnight rates as underlying (e.g. OIS rates). 2. We call risky the interbank rates (e.g. Libor/Euribor rates), which have
typically longer tenor than one day, and all their derivatives (e.g. FRA, swaps, caps/floors, swaptions etc.).
OIS rate has overnight rate EONIA as its underlying; moreover, unlike EO-NIA, OIS rate has a plurality of available maturities (even up to thirty years). For these reasons, OIS rate is usually considered the best available proxy for Euro market risk-free rates.
Using bootstrapping techniques (see Section 2.3.1) we can obtain from OIS rates the risk-free OIS zero coupon bond curve T → B(t, T ) where we use the notation B(t, T ) as in Chapter 1.
OIS zero coupon bonds then lead to other definitions as in Definition 1.2.2. Definition 2.1.2. For all 06 t 6 T we define
• OIS instantaneous forward rate ft(T ) := −∂Tlog B(t, T ),
• OIS short rate rt:= ft(t),
• OIS money market account Bt:= exp(
Rt
0rsds).
Definition 2.1.3. We call simply compounded (risk-free) OIS forward rates at t for the time interval [T, T + δ]
Et(T, T + δ) := 1 δ B(t, T ) B(t, T + δ)− 1 .
We notice that Et(T, T + δ) corresponds to the pre-crisis risk-free par value
of FRA rate (or forward Libor rate) at time t for the interval [T, T + δ] as in (1.2.8).
In this way, OIS rates provide a complete picture of the risk-free yield curve. Definition 2.1.4. We call the yield curve bootstrapped by OIS rates discount-ing yield curve.
Remark 2.1.1.
• Risk-free OIS yield curve is called discounting because, as we will see later, it is used as the discount factor in order to value future cash flows.
2.2. SOME MODELLING APPROACHES 31 • From now on, the notation Q will be used exclusively for the martingale measure for the risk-free OIS money market account Bt as numeraire,
under which every risk-free tradable asset instantaneously increases its value at the risk-free OIS short rate rt; QT will be used as the T -forward
measure for the risk-free OIS T-bond B(t, T ) as numeraire.
• We notice that the choice of the discount curve is not unique, and it doesn’t influence modelling structure.
Observation 2.1.2. The actually quoted FRA rates, still denoted by Lt(T, T +
δ), which have Libor rates (risky) for some tenor δ > 1/365 (i.e. for some tenor longer than one day, as the common assumption is setting unit time = 1 year) since underlying quantities are now typically higher than their theoretical values:
Lt(T, T + δ) > Et(T, T + δ).
Observation 2.1.3. Similarly, all other typical fixed income products, such as swaps, caps/floors and swaptions, which have Libor rates as underlying quanti-ies, must be revalued.
2.2
Some modelling approaches
There are many possibilities to handle the problem. The basic idea is to consider Libor rates of different tenors as different assets. If in the classical bond market theory it was enough to use one yield curve to describe the entire bond market dynamics, now we should use more yield curves for different tenor lengths.
The common scheme adopted in all models is
1. Describe the risk-free discounting yield curve with the classical theory. 2. Use some new method to describe dynamics of risky yield curves for
dif-ferent tenors, called forwarding yield curve, so that the market consisting of all these assets (both risk-free and risky) is coherently free of arbitrage. 3. *Consider the correlation between different tenor lengths in the stochastic
evolution modelling.
The last point which we have signed with * , is due to the fact that spreads are usually positive and increasing with respect to the tenor’s length δ, as we can see in Figure 2.4.
So an optimal model should be able to ordered the spreads according to the tenor length.
In this thesis we will discuss two different modelling approaches, respectively in Chapter 3 and Chapter 4. In Chapter 5 we will give a numerical example of a particular case of the second modelling approach.
We notice that Markovianity is a very useful property which allows us to value prices using only current information at time t instead of all information of the entire path taken from date 0 to date t. In particular, it allows to pass to the associated PDEs of a given SDE, this fact can generate a useful numerical method as we will see later. For these reasons, we will always try to create models with Markovian dynamics.
Short rate models (Chapter 3)
In Chapter 3 we present an extension of the classical short rate models. We keep the classical relationship
L(T, T + δ) = 1 δ
1
Bδ(T, T + δ)− 1
between the forward Libor rate and the zero coupon bond, introducing abstract zero coupon bond Bδ(t, T ) for every tenor δ. Then we define short rate spread sδ
t defined as
sδt := rtδ− rt
where rδ
t is the abstract “short rate” associated to the theoretical bond Bδ(t, T )
for given tenor δ and rt is the risk-free OIS short rate. We model both OIS
short rate rt and the short rate spread sδt with a Vasiček type affine factor
model, which can be extended to a Hull-White version.
This method gives a generalized affine structure. However, this way may yield several problems, like over-parametrization issues that affect the calibra-tion procedure.
HJM models: Libor Dynamics Approach (Chapter 4)
In Chapter 4 we presents a second method based on both classical HJM frame-work and Libor models. It models observable quantities and gives a direct stochastic evolution of L(t, T, T +x) for every tenor δ and every maturity T > 0. A good choice of volatility structure allows to get Markovian setting and dy-namics driven by a finite set of Markovian processes crossing all maturities. This modelling approach supports Black’s formula for caps and “approximated” Black’s formula for swaptions. However, such an approach can’t fully capture the characteristic properties of observed spreads with different tenors, such as the positivity and monotonicity with respect to the tenor.
2.3
How to use a model
Before we study single models, we explain synthetically how to use a bond market model according to multi-curve approach.
Market models are mathematical instruments aiming to describe market development and to (possibly) predict future prices of market assets. For multi-curve bond market models, we follow these steps:
2.3. HOW TO USE A MODEL 33 1. Construction of initial yield curves for both discounting curve and
for-warding curves of different tenors, using current market data.
2. Adaptation of model parameters to market data using already calculated initial yield curves and inverting handleable pricing formulas (e.g. Black’s formula for caps or swaptions if available) for currently quoted instru-ments.
3. Pricing future prices of interest rate derivatives using model dynamics. In the first step, the most used method is the so called bootstrapping technique which allows to fit exactly N point of the initial yield curve to current market data. This step is common for all multi-curve models and it is independent of particular model choices.
The second step is called calibration. It is the most crucial and problematic step, since there isn’t a universal method but it is fully dependent on model choice. A good model should be able to allow both simple and accurate calibra-tion.
In the final step, there are several methods. The easiest case is when we already have an explicit formula for target instrument. If this isn’t the case, the only information we have is that the par value of our T -claim X is given by
˜ E St X ST Ft
where Stis a certain numeraire and ˜E is the associated expectation. The
clas-sical method used to handle this problem is Monte Carlo simulation. However, this method is particularly heavy from a numerical and computational point of view. An alternative method is to pass to the associated PDEs of the SDE which describes underlying dynamics according to the model setting.
Here we give description of every step enumerated above.
2.3.1
Bootstrapping technique
Bootstrapping is a procedure used to construct yield curve (term structure) from market data, usually the initial yield curve.
The basic idea is the following:
1. Not every interest rate or bond of any maturities are always quoted in real life. Indeed, many interest rates are only mathematical objects (e.g. short rate or instantaneous forward rates); even for real rates (e.g. Libor rates) there are only a few quoted indexations/maturities (e.g. for Libor rates the most quoted indexations are 1m, 3m, 6m and 12m). If we want to construct entire yield curve, we select a finite set (N elements) of the most liquid interest rate instruments having the given interest rate as underlying, traded in real time on the market with increasing maturities.
2. If the function system linking these instrument prices to our initial interest rates or bonds is invertible, then we can calculate the exact value of the given interest rate (or bond) using the market quotes of these derivatives, in this way we get N points (pillars) of yield curve, called bootstrapping grid.
3. The rest of the yield curve is obtained by some interpolation methods, which should be chosen carefully, in order to get a smooth enough curve. This construction scheme is called bootstrapping.
The above scheme is a very simplified one, since the second step needs a very strong requirement. Usually instrument-price functions them-selves depend on not observable interest rate (or bond) data. A common method adapted in these cases is to anticipate the interpolation step in order to get not available interest rate (or bond) values. In other word, interpolation is already partially used to obtain pillars. However, this way requires a very careful use of interpolation method and a careless use may lead to meaningless results.
Single-curve context
In the classical single yield curve scenery, the interbank curve was usually boot-strapped using a selection from the following market instruments:
1. Euribor fixing from one day up to one year; 2. FRA rates from one month up to two years; 3. Swaps from one to thirty years;
4. etc.
These instruments are not homogeneous, since their underlying interest rates have mixed tenors.
At time 0, let us call {P (0, t) : 0 6 t 6 T } (T > 0) the wanted yield curve (e.g. the initial instantaneous forward rate curve), and {Si : i = 1, ..., N } the
set of quoted instruments, where for every i = 1, ..., N Si= Si(P (0, t1), ...., P (0, tN)),
where t1 < t2 < .... < tN. That is, every product depends on a set of yield
curve points, but it is not necessary for every instrument to depend on all {P (0, ti) : i = 1, ..., N }. If the instrument set choice is such that the system
S1= S1(P (0, t1), ...., P (0, tN)),
.. .
SN = SN(P (0, t1), ...., P (0, tN)),
admits solutions, then it is possible to obtain the bootstrapping grid {P (0, ti) :
2.3. HOW TO USE A MODEL 35 Multi-curve context
Multi-curve approach separates the discounting curve from the forwarding curves, as well as forwarding curves of different tenors.
Bootstrapping the discounting curve needs the following overnight instru-ments:
• Discounting: EONIA fixing, OIS from one to thirty years etc. Here we continue to use the usual single-curve scheme.
For the forwarding curves, the most quoted Libor rate tenors are given by {1m, 3m, 6m, 12m}, we consider the following market quotes:
• Indexation over 1m: Euribor one-month fixing, swaps from one to thirty years paying an annual fix rate in exchange for the Euribor 1m rate; • Indexation over 3m: Euribor three-months fixing, FRA rates up to one
year, swaps form one to thirty years paying an annual fix rate in exchange for the Euribor 3m raet;
• Indexation over 6m: Euribor six-months fixing, FRA rates up to one year and a half, swaps form one to thirty years paying an annual fix rate in exchange for the Euribor 6m rate;
• Indexation over 12m: Euribor twelve-months fixing, FRA rates up to two years, six-vs-twelve months basis-swaps form two to thirty years.
If we denote with {P (0, t) : 06 t 6 T } the already obtained discounting curve, {Pδ
(0, t) : 0 6 t 6 T } the δ-tenor forwarding yield curve (e.g. the initial δ-tenor Libor rate curve) which should be calculated, and {Si : i = 1, ..., N } the set of
quoted instruments, then we get a system of the following form S1= S1(P (0, t1), ...., P (0, tN), Pδ(0, t1), ...., Pδ(0, tN)),
.. .
SN = SN(P (0, t1), ...., P (0, tN), Pδ(0, t1), ...., Pδ(0, tN)).
Using information of the discounting yield curve, if we are able to resolve the above system, then we get the bootstrapping grid {Pδ(0, t
i) : i = 1, ..., N }.
A more sophisticate method is the following one.
Once the discounting curve is known, in order to obtain smooth curves for both forwarding curves (which we assume to be forward Libor rate curve) and basis spreads, we bootstrap the spread term structures between forward Libor rates of different tenor, along with their spread with respect to the discounting curve.
In particular, we start form the six-months tenor, which corresponds in the Euro area to the family of most liquid instruments. We bootstrap the curve of the rate spread between risk-free rate and forward Libor rate of six-months tenor, and then get the curve of the six-months Libor rate adding the spread curve to the discounting curve.
Once we know the six-months curve, we proceed in a similar way to obtain the curves corresponding to the other tenors. Considering the liquidity of the underlying instruments, we select to bootstrap the following rate spreads:
• We obtain the three-months curve using as staring point the six-months curve, since the market quotes the three-vs-six-months basis swaps; • We obtain the one-months curve using as staring point the three-months
curve, since the market quotes the one-vs-three-months basis swaps; • We obtain the twelve-months curve using as staring point the six-months
curve, since the market quotes the six-vs-twelve-months basis swaps. Interpolation
A good interpolation method is the one based on cubic splines which link piece-wisely and smoothly the interpolation nodes with cubic polynomials.
If f is the target function
f : I −→ R
where I = [t0, tn] ⊆ R, and {fi : i = 0, ..., n} is the set of interpolation nodes
where fi= f (ti) with ti∈ I for every i = 0, ..., n, then the cubic spline s
s : [t0, tn] −→ R
is piecewise-defined by
s(t) = si(t) for t ∈ [ti, ti+1], i = 1, ..., n
where every si is a cubic polynomial such that
• for i = 0, ..., n − 1
si(ti) = fi, si(ti+1) = fi+1;
• for i = 1, ..., n − 1
s0i−1(ti) = s0i(ti), s00i−1(ti) = s00i(ti);
• if the bound derivatives f0(t
0) and f0(tn) are unknown, we set
s000(ti) = s00n(tn) = 0.
Cubic spline method is numerical stable and ensures good convergence in infinite norm || · ||∞ up to the third derivative for every regular function f .
2.3. HOW TO USE A MODEL 37
2.3.2
Calibration
As we have anticipated, calibration is the most problematic step which heavily depends on model choice. However, it is possible to give the essential scheme of how it works in some particular cases. If model parameters are constants, the basic idea is to use Least Square Principle to find correct parameters which best fit to market data.
In the simplest case of Linear Statistical Model, the fitting parameters given by Least Square Principle are optimal estimate among all correct linear esti-mates.
Theorem 2.3.1 (Linear Statistical Model). Let be there a linear statistical model on Rn, B(Rn), {Pθ,σ2 : θ ∈ Rm, σ ∈ R} with n > m
Y = AΘ + σZ where
Y = (Y1, ..., Yn)
is the vector of observed data,
A ∈ Rn×m
is a maximum ranked deterministic matrix, Z = (Z1, ..., Zn)
is a stochastic vector with zero mean and covariance matrix equal to identity. Then
min
θ ||Y − Aθ|| 2
is the optimal estimate among all correct linear estimates of θ.
Let {αi : i = 1, ..., m} be the (finite) set of our model free parameters and
{Xj : j = 1, ..., n} be a set of quoted vanilla instruments, i.e. instruments
with explicit and handleable pricing formula. This means that these quoted instruments or derivatives are “simple” functions of underlying interest rates or bonds whose dynamics are described by our particular modelling choice. Using information of the already constructed initial yield curves and particular model structure we can get that for every j = 1, ..., n the instrument Xj is a “simple
function” of free model parameters
Xj= Xj(α1, ..., αm) If we denote α := (α1, ....αm), X := (X1, ..., Xn) and ¯ X = ( ¯X1, ..., ¯Xn)
the current market prices of these instruments, then Least Square Principle ensures that, for n> m, the parameter vector α which minimized the quantity
|| ¯X − X(α)||2=
m
X
i=1
( ¯Xi− X(α))2
gives the best estimate of correct model parameters.
2.3.3
Monte Carlo simulation
Monte Carlo is a numerical method used to approximate solutions to quantita-tive problems. It simulate the underlying process with a big set of input and then calculate the average result of the process.
In finance this method is typically used to price exotic products which do not have a handleable pricing formula.
Let be there a filtrated stochastic space (Ω, F , (Ft)06t6T, P), the Monte
Carlo simulation is based on the well-known Law of Large Numbers and Central Limit Theorem:
Theorem 2.3.2 (Law of Large Numbers). Let X1, ..., Xn be a real-valued
ran-dom sample of size n, i.e. a set of n independent and identically distributed random variables, with finite mean value m. If we denote the sample average with Sn:= Pn i=1Xi n , then for n → ∞ Sn a.s. −→ m.
Theorem 2.3.3 (Central Limit Theorem). Let X1, ..., Xn be a real-valued
ran-dom sample of size n, with mean value m and finite non-zero variance σ2. If
we denote the sample average with Sn, then the following convergence in
distri-bution holds for n → ∞ √ nSn− m σ d −→ N (0, 1). Basic case
Let be there a random variable X and a measurable deterministic function g, we want to estimate the quantity
E[g(X)] = Z
g(x)f (x)dx =: m
where f is the probability density function of X w.r.t. Lebesgue measure, i.e. Z
2.3. HOW TO USE A MODEL 39 We pick a (big) set of sample values {xi: i = 1, ..., n} with distribution described
by the probability density f . We consider the sample mean
Sn:= 1 n n X i=1 g(xi),
and the sample variance
S2:= 1 (n − 1) n X i=1 (g(xi) − Sn)2 of {g(x1), ..., g(xn)}.
For Law of Large Number 2.3.2, the sample mean is a good estimate of the integral value m. Further, it is also a correct estimate. But since the law of X is arbitrary, we don’t have a-priori a statistical structure which can ensure us the confidence limits with given sample size n.
We notice that, as a consequence of Law of Large Numbers 2.3.3 and Lemma of Slutsky we have
S2 a.s.−→ σ2 (2.3.1)
where σ2is the variance of g(X).
The convergence (2.3.1) together with Central Limit Theorem 2.3.3 yields √ nSn− m S = Sn− m S √ n d −→ N (0, 1)
for n → ∞. In other word, for very big value of n, the distribution of Sn− m
S √ n
can be regarded as normal.
The basic idea of Monte Carlo method is thus to pick a very large sample set (of size n), calculate the sample mean Sn, and use the usual Student Test
for Gaussian sample with hypothesis
H0) : m = Sn against H1) : m 6= Sn.
In such a way, the confidence interval, called Monte Carlo confidence interval, of level α is hence D = Sn− t(1−α 2,n−1) S √ n, Sn+ t(1−α2,n−1) S √ n
where t(·,n−1) is the (n − 1)-grade Student quantile.
For example, a 99% Monte Carlo confidence interval (i.e. α = 0, 01) for very big sample set is approximately
D = Sn− 2.576 S √ n, Sn+ 2.576 S √ n .
More general cases
In finance, derivative pricing problem usually requires to estimate quantities of the form
E[g(XT)]
where Xtis the underlying price process which is the (strong) solution of a given
SDE describing its price dynamics
dXt(ω) = µ(ω, t, Xt)dt + σ(ω, t, Xt)dWt(ω)
where W is a dimensional) P-Wiener process, µ (1-dimensional) and σ (d-dimensional) are two coefficient process (or deterministic functions if they are independent of ω) “good” enough to ensure the strong existence and uniqueness of the SDE solution.
Unlike the previous case, here we do not even know the probability law of X. Thus we need to make further approximations.
The most usual method is consecutive Euler approximation which discretizes the original SDE in
Xt+δt− Xt= µ(t, Xt)δt + σ(t, Xt) Wt+δt− Wt
where δ is a small time increment.
If 0 is the current time and ∆n = {0 = t (n) 0 < t (n) 1 < ... < t (n) kn = T } is
a partition of the interval [0, T ], this method requires that the Riemman-type sum Rn := kn−1 X i=0 µ t(n)i , Xt(n) i t(n)i+1− t(n)i + kn−1 X i=0 σ t(n)i , Xt(n) i Wt(n) i+1 − Wt(n) i
converges to the real solution
Xt− X0
at least in distribution for n → ∞ and |∆n| → 0, where the initial condition
X0 is observable from current market data (in the case of interest rate model,
we need a initial yield curve which can be obtained by bootstrapping method as we have already seen).
Since in this thesis we consider only Itô processes, i.e. µ(t, Xt) and σ(t, Xt)
are progressively measurable processes and a.s. Z
|µ(s, Xs)|ds < ∞,
Z
||σ(s, Xs)||2ds < ∞,
we have actually convergence in probability for n → ∞ Rn−→ XP t− X0
2.3. HOW TO USE A MODEL 41 which is stronger than convergence in distribution
Rn d
−→ Xt− X0.
In conclusion, in order to estimate E[g(Xt)], we should
1. choose a partition ∆ of the time interval [0, t] such that |∆| 1;
2. generate a big set (size = n) of paths using Wiener distribution and per-forming consecutive Euler approximation;
3. estimate the expectation value with the mean of obtained sample {g(x1), ..., g(xn)}
and calculate the Monte Carlo confidence interval.
The previous method can be easily generalized to the case when derivative price depends on more than one instant and/or underlying, namely
E[g(Xt1, ..., Xtm)]
where X is a k-dimensional process.
2.3.4
Discretization of associated PDEs
Monte Carlo method, and in particular consecutive Euler approximation, ex-plained in the previous section is computationally slow and complicated. An alternative approach is to pass to the associated PDEs of original SDEs de-scribed by the model, if these SDEs admits markovian solutions.
In general, let be there a filtrated probability space (Ω, F , (Ft)t>0, P), and a
d-dimensional SDE on the time interval [0, T ] of the form dXt= b(t, Xt)dt + σ(t, Xt)dWt
where W is a k-dimensional P-Wiener process, b (d-dimensional) and σ (d × k-dimensional) are two measurable deterministic functions, “good” enough to ensure the strong existence and uniqueness of solution, which is in particular a markovian process under these conditions.
To this SDE we can associate three types of PDEs, namely • Kolmogorov equation ∂u ∂t + 1 2 X i,j aij ∂2u ∂xi∂xj +X i bi ∂u ∂xi = 0 u|t=T = ϕ where aij:=Pkσikσjk;
• Fokker-Planck equation ∂p ∂t − 1 2 ∂2 ∂xi∂xj X i,j (aijp) + div(bp) = 0 p|t=0= p0 where div(v) =P i ∂vi ∂xi; • Dirichlet equation 1 2 X i,j aij(x) ∂2u ∂xi∂xj +X i bi(x) ∂u ∂xi = 0 u|∂D= g
where D is a certain domain of Rd.
Under suitable conditions, these PDEs are all linked to the original SDE with particular relations which we do not explain here in details.
Concerning us, as we have already mentioned in the previous section for Monte Carlo simulation, our main problem is to determine the distribution of the SDE solution with dynamics given by models, in order to generate different paths according to this distribution and then calculate the mean value, which should give us the estimated derivative price.
The starting point is the following theorem.
Theorem 2.3.4. If b and σ are continuous functions, then the law of SDE so-lution µt:= L(Xt) coincides with the weak solution of Fokker-Planck equation,
namely for all ψ ∈ Cc∞([0, T ]) s 7→
Z
ψ(x)µs(dx)
is measurable, and the following equation holds Z ψ(x)µt(dx) = Z ψ(x)µ0(dx) + Z t 0 Z Lψ(x)µs(dx)ds
where the operator L is defined by (Lu)(t, x) = ∂u ∂t + 1 2 X ij aij ∂2u ∂xi∂xj + b · Ou.
We notice the following important fact.
Observation 2.3.5. If Fokker-Planck equation admits classical strong solution pt, it will give the density function of SDE solution Xt w.r.t. the Lebesgue
2.3. HOW TO USE A MODEL 43 Our problem thus reduces to approximate a PDE solution. A classical nu-merical method is finite difference method, which we synthesised as following
1. determine bound conditions and a discretization grid of input variables; 2. choice a discretization scheme for all derivatives;
3. write the matrix form of the original equation and calculate its solution; 4. verify the stability and convergence of the method;
Chapter 3
Short Rate Models
Here we give a two-curve setup made of OIS risk-free curve (discounting curve) and Libor risky curve for a given tenor (forwarding curve for generating future cash flows) extending classical short rate models.
In analogy to the classical single curve arbitrage-free relationship between Libor rate and zero coupon bond (1.2.5), to every risky Libor rate L(T, T + δ) for a given tenor δ > 0 we can associate an artificial risky bond price Bδ(t, T )
(which we will define further) such that
L(T, T + δ) = 1 δ 1 Bδ(T, T + δ)− 1 (3.0.1)
for all maturities T > 0. We assume the same assumptions for these risky bonds as in Assumption 1.2.1.
We remember that rt is the OIS short rate as in Definition 2.1.2, then the
following relation holds just as in the single curve risk-free setting (1.2.2)
B(t, T ) = EQ exp − Z T t rudu Ft , If we call rδ
t = rt+ stthe short rate associated to the risky bond Bδ(t, T ),
where st denotes the short rate spread (an instantaneous credit spread), then
by knowing the risky short rate dynamics we can define Bδ(t, T ) as
Bδ(t, T ) := EQ exp − Z T t (ru+ su)du Ft , (3.0.2)
Remark 3.0.6. We notice that Bδ(t, T ) are not actually traded bond prices,
and they are supposed to be affected by the same factors as the Libor rates. In conclusion, in order to describe the bond market, it is enough to give the dynamics of the OIS short rate rs and the short rate spread st.
