Interest rates stochastic models Ioane Muni Toke Ecole Centrale - - PowerPoint PPT Presentation

Interest rates stochastic models

Ioane Muni Toke

Ecole Centrale Paris Option Math´ ematiques Appliqu´ ees Majeure Math´ ematiques Financi` eres

December 2010 - January 2011

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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, Soci´ et´ e G´ en´ erale) Lecture 5 Practical aspects of market models - II (E.Durand, Soci´ et´ e G´ en´ erale)

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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’int´ erˆ et, Economica. Shreve S. (2004). Stochastic Calculus for Finance II: Continuous-Time Models, Springer. Original research papers (references below).

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Part I Basic concepts

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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) = −ln P(t, T) T − t i.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)

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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

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Forward interest rates

Forward-rate agreement : exchange of a fixed-rate payment and a floating-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) = −∂ ln P(t, T) ∂T i.e. P(t, T) = exp

• −

T

t

f (t, u)du

• (5)

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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+1 1 1+τjL(t,Tj−1,Tj)

β

i=α+1 τi

i

j=α+1 1 1+τjL(t,Tj−1,Tj)

(8)

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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 at time Ti Nτi [L(Ti−1, Ti) − K]+ . (9) Swaption : A European payer swaption is an option giving the right to enter a payer swap (α, β) at maturity T, i.e. contract with payoff at time T if T = Tα N

• β
• i=α+1

τiP(Tα, Ti) [L(Tα, Ti−1, Ti) − K] + . (10)

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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

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The Vasicek model

Model definition

Original paper Vasicek, O. (1977). “An equilibrium characterization of the term structure”, 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.

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The Vasicek model

Dynamics of the short rate

Proposition In the Vasicek model, the SDE defining the short rate dynamics can be integrated to obtain r(t) = r(s)e−κ(t−s) + θ(1 − e−κ(t−s)) + σ t

s

e−κ(t−u)dWu (12) Short rate r(t) is normally distributed conditionally on Fs.

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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.

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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)

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The CIR model

Table of contents

1

The Vasicek model

2

The CIR model

3

The Hull-White (extended Vasicek) model

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The CIR model

Model definition

Original paper Cox J.C., Ingersoll J.E. and Ross S.A. (1985). ”A Theory of the Term Structure 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.

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The CIR model

Dynamics of the short rate

Proposition In the CIR model, the short rate r(t) follows a noncentral χ2 distribution

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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) =

• 2γe

κ+γ 2

(T−t)

2γ + (κ + γ)(eγ(T−t) − 1) 2κθ

σ2

B(t, T) = 2(eγ(T−t) − 1) 2γ + (κ + γ)(eγ(T−t) − 1) γ =

• κ2 + 2σ2

(19)

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The Hull-White (extended Vasicek) model

Table of contents

1

The Vasicek model

2

The CIR model

3

The Hull-White (extended Vasicek) model

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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.

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The Hull-White (extended Vasicek) model

Model definition

Proposition This model can exactly fit the term-structure observed on the market by setting 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

s

e−a(t−u)dWu (22) with α(t) = f M(0, t) + σ2

2a2 (1 − e−at)2.

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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)

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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 C HW

ZC

= P(t, S)N(q1) − KP(t, T)N(q2) (25) with        σT

r

= σ

• 1−e−2a(T−t)

2a

B(T, S) q1 =

1 σT

r ln

P(t,S) KP(t,T) + σT

r

2

q2 = q1 − σT

r

(26)

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Part III From short rate models to HJM framework

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Table of contents

4

Multifactor models

5

The HJM framework

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Multifactor models

Table of contents

4

Multifactor models

5

The HJM framework

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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))

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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 dW1, W2t = ρ and φ is a deterministic function such that φ(0) = r0.

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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 the T-maturity zero-coupon bond can be written P(t, T) = exp{− T

0 φ(u)du − 1−e−a(T−t) a

x(t)− 1−e−b(T−t)

b

y(t)+ 1

2V (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+ρση ab (1−eaT )(1−ebT ) (30)

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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 the T-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.

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The HJM framework

Table of contents

4

Multifactor models

5

The HJM framework

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The HJM framework

Framework definition

Original paper Heath D., Jarrow R. and Morton A.(1992). ”Bond Pricing and the Term Structure 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)

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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 is solution 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

t

α(t, u)du, σ∗(t, T) = T

t

σ(t, u)du. (35)

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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-neutral measure Q : df (t, T) = σ(t, T)σ∗(t, T)dt + σ(t, T)dW Q

t

dP(t, T) = rtP(t, T)dt − σ∗(t, T)P(t, T)dW Q

t

(37)

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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 σ(u, t) t

u

σ(u, s)ds du + t σ(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−

T

t κ(u)du Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models

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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 derivative product is completely determined by a two-dimensional Markov process if and only if σ(t, T) = η(t)e− T

t

κ(u)du where η is an adapted process and κ

is a deterministic and integrable process.

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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 with an affine term structure P(t, T) = A(t, T)e−B(t,T)r(t), then this model belongs 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 are

• ne-dimensional no-arbitrage HJM models.

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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)

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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

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Table of contents

6

Change of numeraire

7

The Black Formula

8

The BGM market model

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Change of numeraire

Table of contents

6

Change of numeraire

7

The Black Formula

8

The BGM market model

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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.

Xt Mt = E QM XT MT

• Ft
• .

Let (Nt)t≥0 be a numeraire. Then there exists an equivalent probability measure QN such that the price of any traded asset X “discounted” by N is a QN-martingale, i.e.

Xt Nt = E QN XT NT

• Ft
• .

QN is defined by the Radon-Nikodym derivative dQN dQM

• FT

= NT N0 M0 MT . (43)

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Change of numeraire

Changing from risk-neutral measure

Proposition Let Q be the risk-neutral measure associated with a riskless numeraire β(t) = e

t

0 rudu. Let X be a traded asset with Q-dynamics

dXt = rtXtdt + σX(t, Xt)dW Q

t

(44) Let N be another traded asset : dNt = rtNtdt + σN(t, Nt)dW Q

t

(45) Then Xt

Nt is a QN-martingale with dynamics :

d Xt Nt

• = Xt

Nt

• σX(t, Xt) − σN(t, Nt)
• σN(t, Nt)dW QN

t

(46) where dW QN

t

= dW Q

t − σN(t, Nt)dt is a QN-brownian motion.

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The Black Formula

Table of contents

6

Change of numeraire

7

The Black Formula

8

The BGM market model

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The Black Formula

The Black formula

Proposition The Black formula for a caplet (maturity T, strike K) on the θ-tenor Libor L(., . + 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 ln L(t, T, T + θ) K + 1 2σ √ T − t d2 = d1 − σ √ T − t (48) When is this formula justified ?

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The BGM market model

Table of contents

6

Change of numeraire

7

The Black Formula

8

The BGM market model

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The BGM market model

Model definition (I)

Original paper Brace A., Gatarek D. and Musiela M. (1997). “The Market Model of Interest 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 tenor θi = Ti − Ti−1. Notation : ∀i = 1, . . . , M, Li(t) = L(t, Ti−1, Ti)

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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 the associated forward measure : dLi(t) Li(t) = γi(t)dW i,QTi (t) (49) where γi(t) is a deterministic function.

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The BGM market model

No arbitrage condition

Proposition In the BGM model, the no arbitrage condition gives the following relationship between the volatilities γi of the forward Libor and the volatilities Γi of the zero-coupon bonds P(t, Ti): γi(t) = 1 + θiLi(t) θiLi(t) [Γi(t) − Γi−1(t)] . (50)

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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 v ln Li(0) K + 1 2v d2 = d1 − v v = Ti−1 γ2

i (t)dt

(52) The volatility implied by the Black formula is then σBlack

imp (Li) =

• 1

Ti−1 Ti−1 γ2

i (t)dt

(53)

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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

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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

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The BGM market model

Specifying Libor volatilities (III)

Simpler : piecewise-constant volatility that depends only on the time to maturity ∀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

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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)

• r

γi(t) = ηi

• [a(Ti−1 − t) + d] e−b(Ti−1−t) + c
• (58)

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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) under the 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.

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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), the dynamics 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)

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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 the SDE : 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

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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., Teichmann

• J. (2009). “A new approach to LIBOR modeling”)

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