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Interest rates stochastic models
Ioane Muni Toke
Ecole Centrale ParisOption Mathematiques AppliqueesMajeure Mathematiques Financieres
December 2010 - January 2011
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 1 / 61
Course outline
Lecture 1 Basic concepts and short rate models
Lecture 2 From short rate models to the HJM framework
Lecture 3 Libor Market Models
Lecture 4 Practical aspects of market models - I(E.Durand, Societe Generale)
Lecture 5 Practical aspects of market models - II(E.Durand, Societe Generale)
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 2 / 61
Useful bibliography
This short course uses material from :
Brigo D. and Mercurio F. (2006). Interest rates models - Theory and
Practice, 2nd edition, Springer.
Martellini L. and Priaulet P. (2000). Produits de taux d’interet,Economica.
Shreve S. (2004). Stochastic Calculus for Finance II:
Continuous-Time Models, Springer.
Original research papers (references below).
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 3 / 61
Part I
Basic concepts
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 4 / 61
Spot interest rates
r(t) : Instantaneous (interbank) rate, or short rate.
P(t,T ) : Price at time t of a T -maturity zero-coupon bond
R(t,T ) : Continuously-compounded spot interest rate
R(t,T ) = − lnP(t,T )
T − ti.e. P(t,T ) = e−R(t,T )(T−t) (1)
L(t,T ) : Simply-compounded spot interest rate
L(t,T ) =1− P(t,T )
P(t,T )(T − t)i.e. P(t,T ) =
1
1 + L(t,T )(T − t)(2)
Y (t,T ) : Annually-compounded spot interest rate
Y (t,T ) =1
P(t,T )1/(T−t)− 1 i.e. P(t,T ) =
1
(1 + Y (t,T ))(T−t)
(3)
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 5 / 61
Term structure of interest rates
Reproduced from “Danish Government Borrowing and Debt 1998”, Danmarks National Bank, 1999.
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 6 / 61
Forward interest rates
Forward-rate agreement : exchange of a fixed-rate payment and afloating-rate payment
L(t,T ,S) : Simply-compounded forward interest rate
L(t,T ,S) =1
S − T
(
P(t,T )
P(t,S)− 1
)
i.e. 1 + (S − T )L(t,T ,S) =P(t,T )
P(t,S)
(4)
f (t,T ) : Instantaneous forward interest rate
f (t,T ) = −∂ lnP(t,T )
∂Ti.e. P(t,T ) = exp
(
−∫ T
tf (t, u)du
)
(5)
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 7 / 61
Swap rates
Exchange of fixed-rate cash flows and floating-rate cash flows
Exchanges at dates Tα+1, . . . ,Tβ, with τi = ti − Ti−1
Value at time t of a receiver swap :
ΠRS(t, α, β,N,K ) = −N(P(t,Tα)− P(t,Tβ) + N
β∑
i=α+1
τiKP(t,Ti)
(6)
Swap rate
Sα,β(t) =P(t,Tα)− P(t,Tβ)∑β
i=α+1 τiP(t,Ti)(7)
Link with simple forward rates
Sα,β(t) =1−∏β
j=α+11
1+τjL(t,Tj−1,Tj)∑β
i=α+1 τi∏i
j=α+11
1+τjL(t,Tj−1,Tj )
(8)
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 8 / 61
Caps, floors and swaptions
Cap : Payer swap in which only positive cash flows are exchanged
Floor : Receiver swap in which only positive cash flows are exchanged
Caplet (floorlet) : One-date cap (floor), i.e. contract with payoff attime Ti
Nτi [L(Ti−1,Ti )− K ]+ . (9)
Swaption : A European payer swaption is an option giving the rightto enter a payer swap (α, β) at maturity T , i.e. contract with payoffat time T if T = Tα
N
(
β∑
i=α+1
τiP(Tα,Ti ) [L(Tα,Ti−1,Ti )− K ]
)+
. (10)
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 9 / 61
Part II
Short-rate models
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Table of contents
1 The Vasicek model
2 The CIR model
3 The Hull-White (extended Vasicek) model
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The Vasicek model
Table of contents
1 The Vasicek model
2 The CIR model
3 The Hull-White (extended Vasicek) model
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 12 / 61
The Vasicek model
Model definition
Original paper
Vasicek, O. (1977). “An equilibrium characterization of the termstructure”, Journal of Financial Economics, 5 (2), 177–188.
Dynamics of the short rate
Short rate r(t) follows an Ornstein-Uhlenbeck process
drt = κ[θ − rt ]dt + σdWt (11)
with κ, θ, σ positive constants.
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 13 / 61
The Vasicek model
Dynamics of the short rate
Proposition
In the Vasicek model, the SDE defining the short rate dynamics can beintegrated to obtain
r(t) = r(s)e−κ(t−s) + θ(1− e−κ(t−s)) + σ
∫ t
se−κ(t−u)dWu (12)
Short rate r(t) is normally distributed conditionally on Fs .
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 14 / 61
The Vasicek model
Price of zero-coupon bonds
Proposition
In the Vasicek model, the price of a zero-coupon bond is given by
P(t,T ) = A(t,T )e−B(t,T )r(t) (13)
with
A(t,T ) = exp
[
(θ − σ2
2κ2)(B(t,T )− (T − t))− σ2
4κB(t,T )2
]
B(t,T ) =1− e−κ(T−t)
κ(14)
Explicit pricing formula.
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 15 / 61
The Vasicek model
Continuously-compounded spot rate
Proposition
In the Vasicek model, the continuously-compounded spot rate is written
R(t,T ) = R∞+(rt−R∞)1− e−κ(T−t)
κ(T − t)+
σ2
4κ3(T − t)(1−e−κ(T−t))2 (15)
with
R∞ = θ − σ2
2κ2(16)
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 16 / 61
The CIR model
Table of contents
1 The Vasicek model
2 The CIR model
3 The Hull-White (extended Vasicek) model
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 17 / 61
The CIR model
Model definition
Original paper
Cox J.C., Ingersoll J.E. and Ross S.A. (1985). ”A Theory of the TermStructure of Interest Rates”. Econometrica 53, 385–407.
Dynamics of the short rate
Short rate is given by the following SDE
drt = κ[θ − rt ]dt + σ√rtdWt (17)
with κ, θ, σ positive constants satisfying σ2 < 2κθ.
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 18 / 61
The CIR model
Dynamics of the short rate
Proposition
In the CIR model, the short rate r(t) follows a noncentral χ2 distribution
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 19 / 61
The CIR model
Pricing of zero-coupon bonds
Proposition
In the CIR model, the price of a zero-coupon bond is
P(t,T ) = A(t,T )e−B(t,T )r(t) (18)
with
A(t,T ) =
[
2γeκ+γ2
(T−t)
2γ + (κ+ γ)(eγ(T−t) − 1)
]2κθσ2
B(t,T ) =2(eγ(T−t) − 1)
2γ + (κ+ γ)(eγ(T−t) − 1)
γ =√
κ2 + 2σ2
(19)
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 20 / 61
The Hull-White (extended Vasicek) model
Table of contents
1 The Vasicek model
2 The CIR model
3 The Hull-White (extended Vasicek) model
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 21 / 61
The Hull-White (extended Vasicek) model
Model definition
Original paper
J Hull and A White (1990). ”Pricing interest-rate-derivative securities”.The Review of Financial Studies 3, 573–592.
Dynamics of the short rate
Short rate is given by the following SDE
drt = [b(t)− art ]dt + σdWt (20)
with a and σ positive constants.
Non-time-homogeneous extension of the Vasicek model.
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 22 / 61
The Hull-White (extended Vasicek) model
Model definition
Proposition
This model can exactly fit the term-structure observed on the market bysetting
b(t) =∂f M
∂T(0, t) + af M(0, t) +
σ2
2a(1− e−2at) (21)
Dynamics of the short rate
The short rate SDE can then be integrated to obtain :
r(t) = r(s)e−a(t−s) + α(t)− α(s)e−a(t−s) + σ
∫ t
se−a(t−u)dWu (22)
with α(t) = f M(0, t) + σ2
2a2(1− e−at)2.
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 23 / 61
The Hull-White (extended Vasicek) model
Price of zero-coupon bonds
Proposition
In the Hull-White model, the price of a zero-coupon bond is given by
P(t,T ) = A(t,T )e−B(t,T )r(t) (23)
with
A(t,T ) =PM(0,T )
PM(0, t)exp
[
B(t,T )f M(0, t)− σ2
4a(1− e−2at)B(t,T )2
]
B(t,T ) =1
a(1− e−a(T−t))
(24)
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 24 / 61
The Hull-White (extended Vasicek) model
Price of options on a zero-coupon bond
Proposition
In the Hull-White model, the price of a european call option, with strike K
and maturity T , on a zero-coupon bond of maturity S > T , can be written
CHWZC = P(t,S)N (q1)− KP(t,T )N (q2) (25)
with
σTr = σ
√
1−e−2a(T−t)
2a B(T ,S)
q1 = 1σTrln P(t,S)
KP(t,T ) +σTr2
q2 = q1 − σTr
(26)
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Part III
From short rate models to HJM framework
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 26 / 61
Table of contents
4 Multifactor models
5 The HJM framework
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 27 / 61
Multifactor models
Table of contents
4 Multifactor models
5 The HJM framework
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 28 / 61
Multifactor models
Motivations
Empirical studies : correlations of interests rates by maturity
Reproduced from Martellini, Priaulet, 2000.
Empirical studies : PCA on correlation matrix
Reproduced from Jamshidian, Zhu, (1997), Finance and Stochastics 1(1), 43–67.
Correlation in one-factor affine term structure models
Arbitrage in short/long rates models (Dybvig, Ingersoll, Ross, (1996))
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 29 / 61
Multifactor models
A Gaussian two-factor model (I)
Model definition
The short rate r(t) is written
r(t) = x(t) + y(t) + φ(t), r(0) = r0, (27)
where the two factors x and y are solutions of the following SDEs :
{
dx(t) = −ax(t)dt + σdW1(t), x(0) = 0,dy(t) = −by(t)dt + ηdW2(t), y(0) = 0,
(28)
with d〈W1,W2〉t = ρ and φ is a deterministic function such thatφ(0) = r0.
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 30 / 61
Multifactor models
A Gaussian two-factor model (II)
Price of a zero-coupon bond
In the Gaussian two-factor model, the price P(t,T ) at time t of theT -maturity zero-coupon bond can be written
P(t,T ) = exp{−∫ T0 φ(u)du− 1−e−a(T−t)
a x(t)− 1−e−b(T−t)
b y(t)+ 12V (t,T )}
(29)
Fitting the observed term structure
The Gaussian two-factor model fits the observed term structure PM(0,T )if and only if φ is written
φ(T ) = f M(0,T )+σ2
2a2(1−eaT )2+
η2
2b2(1−ebT )2+ρ
ση
ab(1−eaT )(1−ebT )
(30)
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 31 / 61
Multifactor models
A Gaussian two-factor model (III)
Pricing of zero-coupon bond
In the Gaussian two-factor model, the price P(t,T ) at time t of theT -maturity zero-coupon bond can be written
P(t,T ) = A(t,T ) exp{−Ba(t,T )x(t)− Bb(t,T )y(t)} (31)
where
A(t,T ) =PM(0,T )
PM(0, t)exp{1
2(V (t,T )− V (0,T ) + V (0, t))},
Bi(t,T ) =1− e−i(T−t)
i.
(32)
First step allowing the modeling of correlations.
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 32 / 61
The HJM framework
Table of contents
4 Multifactor models
5 The HJM framework
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 33 / 61
The HJM framework
Framework definition
Original paper
Heath D., Jarrow R. and Morton A.(1992). ”Bond Pricing and the TermStructure of Interest Rates: A New Methodology”. Econometrica 60,77–105.
Forward dynamics
The instantaneous forward rates dynamics is given by the following SDE:
{
df (t,T ) = α(t,T )dt + σ(t,T )dWt ,
f (0,T ) = f M(0,T ).(33)
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 34 / 61
The HJM framework
No arbitrage condition (I)
Dynamics of the zero-coupon bond prices
In a HJM framework, the price of the T -maturity zero-coupon bond issolution of the following SDE:
dP(t,T ) = P(t,T )
[
rt − α∗(t,T ) +1
2σ∗(t,T )2
]
dt−σ∗(t,T )P(t,T )dWt ,
(34)where
α∗(t,T ) =
∫ T
tα(t, u)du,
σ∗(t,T ) =
∫ T
tσ(t, u)du.
(35)
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 35 / 61
The HJM framework
No arbitrage condition (II)
Dynamics of the zero-coupon bond prices
In a HJM framework, there is no arbitrage if there exists a process(θt)0≤t≤T satisfying
α(t,T ) = σ(t,T ) [σ∗(t,T ) + θ(t)] (36)
In this case, dynamics in the model can be rewritten under a risk-neutralmeasure Q :
df (t,T ) = σ(t,T )σ∗(t,T )dt + σ(t,T )dWQt
dP(t,T ) = rtP(t,T )dt − σ∗(t,T )P(t,T )dWQt
(37)
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 36 / 61
The HJM framework
Markovianity in the HJM framework (I)
Dynamics of the short rate
In a no arbitrage HJM framework, the short rate can be written
r(t) = f (0, t) +
∫ t
0σ(u, t)
∫ t
uσ(u, s)ds du +
∫ t
0σ(u, t)dWu (38)
Choice of volatilities to get a markovian model:
Separation of variables : σ(t,T ) = ξ(t)φ(T )
Ritchken and Sankarasubramanian (1995) : σ(t,T ) = η(t)e−∫ Tt κ(u)du
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 37 / 61
The HJM framework
Markovianity in the HJM framework (II)
Ritchken and Sankarasubramanian volatility (1995)
In a 1D HJM framework with σ(t,T ) T -differentiable, every derivativeproduct is completely determined by a two-dimensional Markov process if
and only if σ(t,T ) = η(t)e−∫ Tt
κ(u)du where η is an adapted process and κ
is a deterministic and integrable process.
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 38 / 61
The HJM framework
Link with affine models
Proposition
If the SDE drt = b(t, rt)dt + γ(t, rt)dWt defines a short rate model withan affine term structure P(t,T ) = A(t,T )e−B(t,T )r(t), then this modelbelongs to the HJM framework with σ(t,T ) = ∂
∂T B(t,T )γ(t, rt).
One can check that the Vasicek, CIR and Hull-White models areone-dimensional no-arbitrage HJM models.
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 39 / 61
The HJM framework
Choices of volatility
Ho-Lee
σ(t,T ) = σ (constant) (39)
Vasicek/Hull-White
σ(t,T ) = γ(t)e−λ(T−t) (40)
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 40 / 61
The HJM framework
Pricing of caplet
Proposition
In a no-arbitrage HJM framework, the price of a caplet of maturity T ,strike K , paid in T + θ is written :
C (t,T ,K , θ) = P(t,T +θ) [(1 + θL(t,T ,T + θ)N (d1)− (1 + θK )N (d0)](41)
where
d0 =1
Σ(t,T )ln
1 + θL(t,T ,T + θ)
1 + θK− 1
2Σ2(t,T )
d1 = d0 +Σ(t,T )
Σ(t,T ) =
∫ T
t(σ∗(u,T + θ)− σ∗(u,T ))2du
(42)
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Part IV
Libor market models
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 42 / 61
Table of contents
6 Change of numeraire
7 The Black Formula
8 The BGM market model
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 43 / 61
Change of numeraire
Table of contents
6 Change of numeraire
7 The Black Formula
8 The BGM market model
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 44 / 61
Change of numeraire
General change of measure
Theorem
Assume there exists a numeraire (Mt)t≥0 and an equivalent measure QM
such that the price of any traded asset X “discounted” by the process M
is a QM -martingale, i.e. XtMt
= EQM[
XTMT
∣
∣
∣Ft
]
.
Let (Nt)t≥0 be a numeraire.Then there exists an equivalent probability measure QN such that theprice of any traded asset X “discounted” by N is a QN -martingale, i.e.XtNt
= EQN[
XTNT
∣
∣
∣Ft
]
.
QN is defined by the Radon-Nikodym derivative
dQN
dQM
∣
∣
∣
FT
=NT
N0
M0
MT. (43)
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 45 / 61
Change of numeraire
Changing from risk-neutral measure
Proposition
Let Q be the risk-neutral measure associated with a riskless numeraireβ(t) = e
∫ t0rudu . Let X be a traded asset with Q-dynamics
dXt = rtXtdt + σX (t,Xt)dWQt (44)
Let N be another traded asset :
dNt = rtNtdt + σN(t,Nt)dWQt (45)
Then XtNt
is a QN -martingale with dynamics :
d
(
Xt
Nt
)
=Xt
Nt
(
σX (t,Xt)− σN(t,Nt))
σN(t,Nt)dWQN
t (46)
where dWQN
t = dWQt − σN(t,Nt)dt is a QN -brownian motion.
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 46 / 61
The Black Formula
Table of contents
6 Change of numeraire
7 The Black Formula
8 The BGM market model
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 47 / 61
The Black Formula
The Black formula
Proposition
The Black formula for a caplet (maturity T , strike K ) on the θ-tenor LiborL(., . + θ and paying at date T + θ is written :
C (t,T ,K , θ) = P(t,T + θ) [L(t,T ,T + θ)N (d1)− KN (d2)] (47)
where
d1 =1
σ√T − t
lnL(t,T ,T + θ)
K+
1
2σ√T − t
d2 = d1 − σ√T − t
(48)
When is this formula justified ?
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 48 / 61
The BGM market model
Table of contents
6 Change of numeraire
7 The Black Formula
8 The BGM market model
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 49 / 61
The BGM market model
Model definition (I)
Original paper
Brace A., Gatarek D. and Musiela M. (1997). “The Market Model ofInterest Rate Dynamics”, Mathematical Finance, 7, 127–155.
Assumptions ans notations
Calendar 0 < T0 < T1 < . . . < TM .
M forward Libor rates (L(t,T0,T1), . . . , L(t,TM−1,TM) with tenorθi = Ti − Ti−1.
Notation : ∀i = 1, . . . ,M, Li (t) = L(t,Ti−1,Ti )
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 50 / 61
The BGM market model
Model definition (II)
Dynamics of the forward Libor rates
Each forward Libor is assumed to be a martingale with respect to theassociated forward measure :
dLi (t)
Li(t)= γi (t)dW
i ,QTi (t) (49)
where γi (t) is a deterministic function.
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 51 / 61
The BGM market model
No arbitrage condition
Proposition
In the BGM model, the no arbitrage condition gives the followingrelationship between the volatilities γi of the forward Libor and thevolatilities Γi of the zero-coupon bonds P(t,Ti):
γi (t) =1 + θiLi (t)
θiLi (t)[Γi (t)− Γi−1(t)] . (50)
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 52 / 61
The BGM market model
Pricing caplets
Proposition
In the BGM model, the price at time 0 of a post-paid caplet (strike K ,maturity Ti−1 on a Libor rate L(Ti−1,Ti ) is given by:
C (0,Ti−1,K , θi ) = P(0,Ti ) [Li (0)N (d1)− KN (d2)] (51)
where
d1 =1
vln
Li(0)
K+
1
2v
d2 = d1 − v
v =∫ Ti−1
0 γ2i (t)dt
(52)
The volatility implied by the Black formula is then
σBlackimp (Li ) =
√
1
Ti−1
∫ Ti−1
0γ2i (t)dt (53)
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 53 / 61
The BGM market model
Specifying Libor volatilities (I)
Simple choice : constant volatilities
∀i = 1, . . . ,M, γi (t) = γi constant. (54)
[0,T0] ]T0,T1] ]T1,T2] . . . ]TM−2,TM−1]
L1(t) γ1 dead dead . . . dead
L2(t) γ2 γ2 dead . . . dead...
......
......
...
LM(t) γM γM γM . . . γM
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 54 / 61
The BGM market model
Specifying Libor volatilities (II)
Another simple choice : piecewise-constant volatilities
∀i = 1, . . . ,M, γi (t) = γi ,β(t) constant. (55)
[0,T0] ]T0,T1] ]T1,T2] . . . ]TM−2,TM−1]
L1(t) γ1,1 dead dead . . . dead
L2(t) γ2,1 γ2,2 dead . . . dead...
......
......
...
LM(t) γM,1 γM,2 γM,3 . . . γM,M
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 55 / 61
The BGM market model
Specifying Libor volatilities (III)
Simpler : piecewise-constant volatility that depends only on the time tomaturity
∀i = 1, . . . ,M, γi (t) = γi ,β(t) = ηi−(β(t)−1) constant. (56)
[0,T0] ]T0,T1] ]T1,T2] . . . ]TM−2,TM−1]
L1(t) η1 dead dead . . . dead
L2(t) η2 η1 dead . . . dead...
......
......
...
LM(t) ηM ηM−1 ηM−2 . . . η1
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 56 / 61
The BGM market model
Specifying Libor volatilities (IV)
Parametric choices
One may also define Libor volatilities with ∀i = 1, . . . ,M, :
γi (t) = [a(Ti−1 − t) + d ] e−b(Ti−1−t) + c (57)
orγi (t) = ηi
[
[a(Ti−1 − t) + d ] e−b(Ti−1−t) + c]
(58)
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 57 / 61
The BGM market model
Dynamics of the forward Libor rates under a unique
forward measure
Proposition
Let i ∈ {1, . . . ,M}. The dynamics of the forward Libor rates Li(t) underthe forward measure QTk , k = 1, . . . ,M is given by the following SDE:
if k < i ,dLi (t)
Li (t)= γi(t)dW
i ,QTk −i∑
j=k
ρijγi(t)γj (t)θjLj(t)
1 + θjLj(t)dt,
if k = i ,dLi (t)
Li (t)= γi(t)dW
i ,QTi
if k > i ,dLi (t)
Li (t)= γi(t)dW
i ,QTk +k∑
j=i+1
ρijγi(t)γj (t)θjLj(t)
1 + θjLj(t)dt.
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 58 / 61
The BGM market model
Introducing the spot Libor measure
Definition
The spot Libor numeraire is defined as :
B(t) =P(t,Tβ(t)−1)
β(t)−1∏
j=0
P(Tj−1,Tj)
(59)
Proposition
Under the spot Libor measure QB associated with the numeraire B(t), thedynamics of the foward Libor Li(t) is written:
dLi (t)
Li (t)= γi (t)dW
QB+
i∑
j=β(t)
ρijγi(t)γj(t)θjLj (t)
1 + θjLj(t)dt. (60)
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 59 / 61
The BGM market model
Swap market model
Original paper
Jamshidian F. (1997). “LIBOR and swap market models and measures”,Finance and Stochastics, 1 (4), 293–330.)
Model definition
A swap market models assumes that the swap rate Sα,β is solution of theSDE :
dSα,β(t)
Sα,β(t)= γα,β(t)dW
Qα,β
(t), (61)
where Qα,β is the measure linked with numeraire∑β
i=α+1 τiP(t,Ti).
Compatibility with the Black formula for swaption
Theoretical inconsistency with the BGM market model
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 60 / 61
The BGM market model
Other LIBOR approaches
Markov-functional Libor models (Hunt P., Kennedy J. and Pelsser A.(2000) “ Markov-functional interest rate models ”, Finance and
Stochastics, 4 (4), 391–408.)
Affine Libor Models (Keller-Ressel M., Papapantoleon A., TeichmannJ. (2009). “A new approach to LIBOR modeling”)
Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 61 / 61