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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)

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).

Part I

Basic concepts

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 + 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 + Y (t,T ))(T−t)

Term structure of interest rates

Reproduced from “Danish Government Borrowing and Debt 1998”, Danmarks National Bank, 1999.

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)

f (t,T ) : Instantaneous forward interest rate

f (t,T ) = −∂ lnP(t,T )

∂Ti.e. P(t,T ) = exp

−∫ T

tf (t, u)du

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

i=α+1

τiKP(t,Ti)

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 )

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

i=α+1

τiP(Tα,Ti ) [L(Tα,Ti−1,Ti )− K ]

. (10)

Part II

Short-rate models

Table of contents

1 The Vasicek model

2 The CIR model

3 The Hull-White (extended Vasicek) model

The Vasicek model

Table of contents

1 The Vasicek model

2 The CIR model

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.

The Vasicek model

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

se−κ(t−u)dWu (12)

Short rate r(t) is normally distributed conditionally on Fs .

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)

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.

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)+

4κ3(T − t)(1−e−κ(T−t))2 (15)

R∞ = θ − σ2

2κ2(16)

The CIR model

Table of contents

1 The Vasicek model

2 The CIR model

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.

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.

The CIR model

Proposition

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

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)

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

The Hull-White (extended Vasicek) model

Table of contents

1 The Vasicek model

2 The CIR model

Model definition

Original paper

J Hull and A White (1990). ”Pricing interest-rate-derivative securities”.The Review of Financial Studies 3, 573–592.

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.

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) +

2a(1− e−2at) (21)

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

se−a(t−u)dWu (22)

with α(t) = f M(0, t) + σ2

2a2(1− e−at)2.

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)

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))

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)

σTr = σ

1−e−2a(T−t)

2a B(T ,S)

q1 = 1σTrln P(t,S)

KP(t,T ) +σTr2

q2 = q1 − σTr

Part III

From short rate models to HJM framework

Table of contents

4 Multifactor models

5 The HJM framework

Multifactor models

Table of contents

5 The HJM framework

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))

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,

with d〈W1,W2〉t = ρ and φ is a deterministic function such thatφ(0) = r0.

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 )}

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+

2b2(1−ebT )2+ρ

ab(1−eaT )(1−ebT )

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)

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)

First step allowing the modeling of correlations.

The HJM framework

Table of contents

5 The HJM framework

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)

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 ) =

tα(t, u)du,

σ∗(t,T ) =

tσ(t, u)du.

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

The HJM framework

Markovianity in the HJM framework (I)

In a no arbitrage HJM framework, the short rate can be written

r(t) = f (0, t) +

0σ(u, t)

uσ(u, s)ds du +

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

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.

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.

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)

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)

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(σ∗(u,T + θ)− σ∗(u,T ))2du

Part IV

Libor market models

Table of contents

6 Change of numeraire

7 The Black Formula

8 The BGM market model

Change of numeraire

Table of contents

7 The Black Formula

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[

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[

QN is defined by the Radon-Nikodym derivative

MT. (43)

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 :

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.

The Black Formula

Table of contents

7 The Black Formula

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)

σ√T − t

lnL(t,T ,T + θ)

2σ√T − t

d2 = d1 − σ√T − t

When is this formula justified ?

The BGM market model

Table of contents

7 The Black Formula

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 )

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.

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)

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)

d2 = d1 − v

v =∫ Ti−1

0 γ2i (t)dt

The volatility implied by the Black formula is then

σBlackimp (Li ) =

Ti−1

∫ Ti−1

0γ2i (t)dt (53)

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

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

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

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]

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∑

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∑

ρijγi(t)γj (t)θjLj(t)

1 + θjLj(t)dt.

Introducing the spot Libor measure

Definition

The spot Libor numeraire is defined as :

B(t) =P(t,Tβ(t)−1)

β(t)−1∏

P(Tj−1,Tj)

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

j=β(t)

ρijγi(t)γj(t)θjLj (t)

1 + θjLj(t)dt. (60)

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

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”)

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