Affine multiple yield curve models Claudio Fontana (based on a joint work with C. Cuchiero and A. Gnoatto) Laboratoire de Probabilit´ es et Mod` eles Al´ eatoires Universit´ e 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
27
Embed
[0.5cm] Affine multiple yield curve models - ESC Lyon, … · A ne multiple yield curve models ... The Libor/Euribor rate for some interval [T;T + ], ... By bootstrapping techniques,
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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
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.
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.
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.
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).
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).
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).
The multi-curve setting Modeling the post-crisis interest rate market
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.
The multi-curve setting Modeling the post-crisis interest rate market
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.
The modeling framework Multi-curve models based on affine processes
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.
The modeling framework Multi-curve models based on affine processes
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,
The modeling framework Multi-curve models based on affine processes
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,
The modeling framework Multi-curve models based on affine processes
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).
The modeling framework Multi-curve models based on affine processes
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).
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.
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.