[0.5cm] Affine multiple yield curve models - ESC Lyon, … · A ne multiple yield curve models ......

Affine multiple yield curve models

Claudio Fontana

(based on a joint work with C. Cuchiero and A. Gnoatto)

Laboratoire de Probabilites et Modeles AleatoiresUniversite Paris Diderot

EMLYON QUANT 12 workshop:quantitative approaches in management and economics

November 26-27, 2015

Claudio Fontana (LPMA) Affine multiple yield curve models Lyon, November 26, 2015 1 / 19

Introduction

Multiple yield curves

The Libor/Euribor rate for some interval [T ,T + δ], where the tenorδ is typically 1M, 2M, 3M, 6M or 12M, is the underlying of basicinterest rate products, such as FRAs, swaps, caps/floors, swaptions...

Since the last crisis, due to credit and liquidity risk in the interbanksector, Libor rates cannot be considered risk-free any longer.

Emergence of spreads in fixed income markets, notably spreadsbetween Libor rates and Overnight Indexed Swap (OIS) rates.

For every tenor δ ∈ δ1, . . . , δm, a different yield curve can beconstructed from traded assets that depend on the Libor rateassociated with the tenor δ.

⇒ Term structure models for multiple yield curves are needed.


Introduction

An (incomplete) overview of the modeling approaches

Short rate approaches:Kijima et al. (2009), Kenyon (2010), Filipovic and Trolle (2013),Morino and Runggaldier (2014), Grasselli & Miglietta (2015);

LIBOR market model approaches:Mercurio (2010,...,2013), Grbac et al. (2015);

HJM approaches:Moreni and Pallavicini (2010), Pallavicini and Tarenghi (2010), Fujiiet al. (2010,2011), Crepey et al. (2012,2015); Cuchiero et al. (2015);

Rational models: Nguyen and Seifried (2015), Crepey et al. (2015).

Recent books: Henrard (2014) and Grbac & Runggaldier (2015).


The multi-curve setting Modeling the post-crisis interest rate market

Libor rates and OIS rates

Lt(t, t + δ): Libor rate at date t for the period [t, t + δ];we consider a finite set of tenors δ1 < . . . < δm, for m ∈ N.

OIS rate: market swap rate of an Overnight Indexed Swap (best proxyfor a risk-free rate and typically used as collateral rate).

By bootstrapping techniques, OIS rates allow to recoverI the term structure of OIS zero-coupon bond prices: T 7→ B(t,T );I simply comp. OIS forward rates

LOISt (T ,T + δ) :=

1

δ

(B(t,T )

B(t,T + δ)− 1

).

In the post-crisis market setting, Lt(t, t + δ) 6= LOISt (t, t + δ).



Euribor - OIS spreads

Additive spreads between Euribor and simply comp. OIS spot ratesLt(t, t + δ)− LOIS

t (t, t + δ) from Jan. 2007 to Sept. 2013 forδ ∈ 1/12, 3/12, 6/12, 1.



Spot multiplicative spreads

As main modeling quantity, we consider spot multiplicative spreads:

Sδi (t, t) :=1 + δiLt(t, t + δi )

1 + δiLOISt (t, t + δi )

, for i = 1, . . . ,m.

Directly observable from market quotes of Libor and OIS rates;

can be interpreted in terms of foreign exchange premia between OISbonds and artificial “risky” bonds associated to Libor rates;

typical requirements:I Sδi (t, t) ≥ 1, for all i = 1, . . . ,m;I Sδi (t, t) ≤ Sδj (t, t), for all i , j = 1, . . . ,m such that i < j .



The basic assets traded in the market

Let T <∞ denote a time horizon. As basic traded assets, we consider:

OIS zero-coupon bonds for all maturities T ∈ [0,T];

FRA contracts for all maturities T ∈ [0,T] and tenors δ1, . . . , δm.

ΠFRAt (T ,T + δi ,K ) denotes the price at date t of a FRA contract

with maturity T + δi and rate K > 0, written on the Libor rateLT (T ,T + δi ).

The fair FRA rate at date t is denoted by Lt(T ,T + δi ).

We then define forward multiplicative spreads by

Sδi (t,T ) :=1 + δiLt(T ,T + δi )

1 + δiLOISt (T ,T + δi )

, for all i = 1, . . . ,m.



Absence of arbitrage

Definition

Let Q be a probability measure on (Ω,F) and B = (Bt)0≤t≤T a strictlypositive (Ft)-adapted process such that B0 = 1. We say that the couple(B,Q) is a numeraire - martingale measure couple if the B-discountedprice of every basic traded asset is a martingale on (Ω, (Ft)0≤t≤T,Q).

The existence of a couple (B,Q) suffices to ensure absence ofarbitrage in the sense of no asymptotic free lunch with vanishing risk(see Cuchiero et al. 2014).

Standing assumption

There exists a numeraire - martingale measure couple (B,Q).

The typical specification:I the numeraire is given by the OIS bank account Bt = exp(

∫ t

0rsds),

with the process (rt)0≤t≤T denoting the OIS short rate;I Q is the spot martingale measure.



OIS bond prices, fair FRA rates and multiplicative spreads

Proposition

1 OIS zero-coupon bond prices satisfy

B(t,T ) = EQ [Bt/BT |Ft ] ;

2 for every i = 1, . . . ,m, the fair FRA rate satisfies

Lt(T ,T + δi ) = EQT+δi [LT (T ,T + δi )|Ft ];

3 for every i = 1, . . . ,m, the multiplicative spread satisfies

Sδi (t,T ) = EQT[Sδi (T ,T )|Ft ].

Here, for every T ∈ [0,T], the measure QT denotes the T -forwardmartingale measure with density dQT/dQ = 1/(BTB(0,T )).


The modeling framework Multi-curve models based on affine processes

An overview of the modeling approach

By the previous proposition, it suffices to model, under a given measure Q,

1 the numeraire process B = (Bt)0≤t≤T;

2 the spot multiplicative spreads (Sδi (t, t))0≤t≤T; i = 1, . . . ,m.

A tractable model for the above two ingredients leads to:

an arbitrage-free model for all the basic traded assets;

an easy characterization of order relations between spreads associatedto different tenors;

immediate valuation formulae for all linear interest rate derivatives;

depending on the model specification, tractable valuation formulae fornon-linear interest rate derivatives.



Preliminaries on affine processes

(Ω,F , (Ft)0≤t≤T,Q): filtered probability space;

X = (Xt)0≤t≤T: time-homogeneous affine process, taking values in aconvex state space D ⊆ V , with initial state X0 = x ;

UT :=ζ ∈ V + iV : E

[e〈ζ,Xt〉

]<∞, for all t ∈ [0,T ]

.

The affine property

There exist functions φ and ψ such that, for any t ∈ [0,T] and u ∈ Ut ,

EQx

[e〈u,Xt〉] = eφ(t,u)+〈ψ(t,u),x〉.

The characteristic exponents φ and ψ are solutions of Riccati ODEs and arein a one-to-one relation with the semimartingale characteristics of X .



Affine multi-curve models

Definition

Let u = (u0, u1, . . . , um) be a family of functions ui : [0,T]→ V ,i = 0, 1, . . . ,m, such that u0(T ) ∈ UT and ui (T ) + u0(T ) ∈ UT , forevery T ∈ [0,T] and i = 1, . . . ,m;

let v = (v0, v1, . . . , vm) be a family of functions vi : [0,T]→ R,i = 0, 1, . . . ,m.

We say that the triplet (X ,u, v) is an affine multi-curve model if

1 the numeraire Bt satisfies

logBt = −v0(t)− 〈u0(t),Xt〉, for all t ∈ [0,T];

2 the spot multiplicative spreads Sδi (t, t); i = 1, . . . ,m satisfy

log Sδi (t, t) = vi (t) + 〈ui (t),Xt〉, for all t ∈ [0,T], i = 1, . . . ,m.



Affine multi-curve modelsDepending on the specification of the couple (B,Q) and of the triplet(X ,u, v), different modeling approaches can be recovered:

OIS specification: B is the OIS bank account and Q is thecorresponding spot martingale measure;

affine Libor models: B is the OIS zero-coupon bond with maturity Tand Q = QT is the T-forward measure;

real-world approach: B is the growth-optimal portfolio and Q = P isthe physical probability measure.

Proposition

(X ,u, v) is an affine multi-curve model, for some families of functions uand v, if and only if B-discounted bond prices and multiplicative spreadsadmit the representations, for all 0 ≤ t ≤ T ≤ T and i = 1, . . . ,m,

B(t,T )/Bt = exp(A0(t,T ) + 〈B0(t,T ),Xt〉

),

Sδi (t,T ) = exp(Ai (t,T ) + 〈Bi (t,T ),Xt〉

).



Ordered spreads and fitting the initial term structuresPropositionLet (X ,u, v) be an affine multi-curve model, with X taking values in a cone CX .If vi ≥ 0 and that ui takes values in the dual cone C∗

X , for all i = 1, . . . ,m, thenSδi (t,T ) ≥ 1 for all 0 ≤ t ≤ T ≤ T and i = 1, . . . ,m. Moreover, if in addition

v1(t) ≤ . . . ≤ vm(t) and u1(t) ≺ . . . ≺ um(t), for all t ∈ [0,T],

with ≺ denoting the partial order on C∗X , then it holds that

Sδ1 (t,T ) ≤ . . . ≤ Sδm(t,T ), for all 0 ≤ t ≤ T ≤ T.

Proposition

(X ,u, v) achieves an exact fit to the initially observed term structures if and onlyif the family of functions v = (v0, v1, . . . , vm) satisfies

v0(t) = logBM(0, t)− logB0(0, t), for all t ∈ [0,T],

vi (t) = log SM,δi (0, t)− log S0,δi (0, t), for all t ∈ [0,T] and i = 1, . . . ,m,

with B0(0, t) and S0,δi (0, t) denoting the theoretical bond prices and spreadscomputed according to the affine multi-curve model (X ,u, 0).



Affine multi-curve models based on the OIS short rateSuppose that

1 the numeraire is Bt := exp(∫ t

0 rsds), where rs is the OIS short rate;

2 Q is the corresponding spot martingale measure.

Definition

Let ` ∈ R, λ ∈ VX , c = (c1, . . . , cm) ∈ Rm and γ = (γ1, . . . , γm) ∈ VmX .

The tuple (X , `, λ, c,γ) is an affine short rate multi-curve model if

rt = `+ 〈λ,Xt〉, for all t ∈ [0,T],

log Sδi (t, t) = ci + 〈γi ,Xt〉, for all t ∈ [0,T] and i = 1, . . . ,m.

Any affine short rate multi-curve model is an affine multi-curve model(⇒ exponentially affine bond prices and spreads, ordered spreads);

negative OIS rates together with positive spreads are possible;

allows for a deterministic shift extension.



Pricing applications and simple specifications

Direct valuation of linear interest rate derivatives:for instance, the price of a FRA contract (unitary notional) is given by

ΠFRAt (T ,T + δi ,K ) = B(t,T )Sδi (t,T )− (1 + δiK )B(t,T + δi );

general semi-closed formula for caplet prices (by Fourier techniques);

approximation formula for swaption prices (in the spirit of Caldana etal., 2015).

Simple and tractable specifications:

OIS short rate setup;

let X be an affine process of the form X = (X 0,Y ), where1 X 0 is a multi-dimensional CIR and Y a gamma subordinator;2 X 0 is a Wishart process and Y a gamma subordinator;

calibration to market data on caplet prices shows that these simplespecifications perform reasonably well.



A tractable specification based on a Wishart process

Let X be a Wishart process in S+2 of the form

dXt = (κQ>Q + MXt + XtM>)dt +

√XtdWtQ + Q>dW>

t

√Xt ,

let rt = λ2X22,t , with λ2 > 0, and log Sδ(t, t) = u1X11,t , with u1 > 0.

X11 and X22 are stochastically correlated CIR processes: naturalextension of the classical CIR model to the multiple curve setting.

Proposition

In the above setting, the price of caplet is given by

ΠCPLT0 = Sδ(0,T )B(0,T )

(1− FT (C )

)− (1 + δK )B(0,T + δ)

(1−FT (C )

),

with C := log(1 + δK ) + φ(0,−λ2), where φ is the first characteristicexponent of the affine process (X ,

∫ ·0 X22,sds) and where FT and FT

denote the cumulative distribution functions of a weighted sum of twoindependent non-central χ2-distributed random variables.



Conclusions

A multi-curve setup based on affine processes, whereI the numeraire andI spot multiplicative spreads between Libor rates and OIS rates

are modeled as functions of a common affine process.

Nice features: ordered spreads, exact fit to the initially observed termstructures and direct pricing of all linear derivatives.

This framework embeds most of the existing multi-curve modelsbased on affine processes (both short rate and Libor models).

Tractable pricing and calibration procedures where the multi-curvesetting does not lead to a higher computational complexity.

Fourier pricing methods to value non-linear derivatives and evenclosed form expressions in the case of Wishart processes.



Thank you for your attention!


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