a tractable multi-factor dynamic term-structure model for risk...

A tractable multi-factor dynamic term-structure modelfor risk management

Roland C. Seydeljoint work with Michael Henseler and Christoph Peters

German Finance Agency, Frankfurt

2013-06-12

(1) German Finance Agency 2013-06-12 1 / 35

Overview

1 The dynamic term-structure model

2 Fitted curves

3 Principal componentsDefinition of principal componentsPseudo principal components

4 Pricing


The dynamic term-structure model

Overview


2 Fitted curves


4 Pricing



Two motivations for two purposes

1 Statistical analysis: Yield curve movements can be “explained” byonly a few factors / principal components.Linear structure necessary for stable PCs ⇒ model yield curve by“deterministic” + “few factors”, e.g. inst. forward rates

f (t, t + τ) := A(τ) + M>(τ)u(t),

A(τ) deterministic function of time to maturity τ > 0u(t) ∈ Rn factor weights at time tM(τ) ∈ Rn basis functions

2 Pricing derivatives: Need no-arbitrage ⇒ choose M as exponential:Assume A = 0 and deterministic market. Then

f (t, t+τ+∆) = M>(τ+∆)u(t)!

= M>(τ)u(t+∆) = f (t+∆, t+∆+τ)

One solution is Mi (τ) = exp(−αiτ), then Mi (τ + ∆) = Mi (∆)Mi (τ).



The model at a glance

Instantaneous forward rates, seen from time t for the maturity t + τmodeled as

f (t, t + τ) := A(τ) + u1(t)e−α1τ + . . .+ un(t)e−αnτ (1)

Fixed deterministic exponents αn > . . . > α1 > 0

Deterministic affine term A(τ)

Stochastic weights u(t) = (u1(t), . . . , un(t))> follow themean-reverting process

du(t) = (diag(−α)u(t) + R)dt + σdWt , (2)

where σ ∈ Rn×n, R ∈ Rn, and W an n-dimensional Brownian motionunder the real-world measure.



Why...?

Why this model? Very straightforward linear specifcation (easiestarbitrage-free affine model)

Why linear combination of exponentials? Under any (affine-)linearspecification of rates and SDE, the function basis M has to be anexponential one because of no-arbitrage conditions

Why dynamics under the physical measure? Because we see it from arisk management perspective – if you don’t then ignore the riskpremium term R

Why diag(−α) in dynamics of weights u? To ease presentation –actually diag(−α) could be replaced by any matrix S , implying anaffine-linear risk premium



Properties of the model

Free of arbitrage if for some b0, the affine term isA(τ) = A(0)− M>(τ)(σb0 − R)− 1

2 M>(τ)σσ>M(τ) for

Mi (τ) =∫ τ

0 Mi (τ)ds

Affine model in Heath-Jarrow-Morton framework in the sense of[Duffie and Kan(1996)]

HJM representation of the class of Gaussian short-rate models:

Short-rate models: Special case for n = 1 Vasicek, case n = 2 is knownas Hull-White 2-factor or G2++ [Brigo and Mercurio(2006)]HJM representation for arbitrary n theoretically derived in[El Karoui and Lacoste(1995)]

Gaussian model ⇒ easy to simulate, closed-form solutions,well-behaved

Negative rates possible! (Disadvantage?)



A normal model: Distributional properties

The SDE (2) has an explicit known solution (multi-dimensionalOrnstein-Uhlenbeck).Starting in t = t0, the weights u(T ) at time T > t0 are normallydistributed with (for S = diag(−α))

u(T ) ∼ N(exp(S(T − t0))u(t0) + S−1(exp(S(T − t0))− 1)R, Σt0,T

)(3)

where the covariance matrix is given by

(Σt0,T )jk = −exp(−(αj + αk)(T − t0))− 1

αj + αk(σσ>)jk . (4)

Therefore, all (inst.) forward and zero rates are normal too (as linearcombinations of normal distribution)!


Fitted curves

Overview


2 Fitted curves


4 Pricing


Fitted curves

Some calibrated curves: Pre financial crisis

All curves displayed were fitted to quotes of German government bondsusing a historical calibration. n = 5 stochastic factors were chosen with anexponent α = 1

12 (1, 2, 3, 4, 5)>.

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

0 5 10 15 20 25 30Time to maturity [in years]

Zero Rates (5 factors)

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%


Quoted Yields

Implied Yields (5 factors)

Figure: Fitted zero curve on 10/31/2005 (left), and quoted and implied yields forthis date (right). Smooth market data result in a very good fit.


Fitted curves

Some calibrated curves: Financial crisis

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%


Zero Rates (5 factors)

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%


Quoted Yields

Implied Yields (5 factors)

Figure: Fitted zero curve on 10/31/2008 (left), and quoted and implied yields forthis date (right). n = 5 parameters are not sufficient for a perfect fit, but impliedyields are still reasonably close to quoted yields.


Fitted curves

Calibrated curves: Error over time

0

5

10

15

20

2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

RM

SE

of

yie

lds

[in

bp

]

Date

NSS

DTSM (5 factors)

DTSM (2 factors)

Figure: Root mean squared error between quoted and implied yields from 2002 to2012, for the model presented with n = 5 stochastic parameters (blue solid line),with n = 2 (green dashed line), and for the benchmark NSS parameterization (reddashed line).


Principal components

Overview


2 Fitted curves


4 Pricing


Principal components Definition of principal components

Recalling principal components

We are given a statistical covariance estimate Σ

The eigendecomposition Σ = ΓΛΓ> with Γ>Γ = I , for Λ = diag(λi )with λ1 ≥ . . . ≥ λd yields the principal components: If a randomvector X has covariance matrix Σ, then

Cov(Γ>X ) = Γ>ΣΓ = Λ,

i.e., the random variables (Γ>X )i are uncorrelated with each other

The columns of Γ as normed eigenvectors are called principalcomponents

If the vector X is composed of rates with different tenors, then theprincipal components can be nicely plotted in time

Typically the largest three eigenvalues account for > 95% of yieldcurve movements as measured by the proportion of “variability”∑3

i=1 λi/∑d

i=1 λi



Principal components: problems for our model

But:

Depends on chosen discretization! Although typically tenors areregularly spaced, this arbitrary choice appears nowhere in an explicitfashion!

Discretizing not necessary for a parametric model where rates arecontinuously defined

Discretizing a yield curve on the interval [0,∞): have to cutsomewhere

General solution

Consider a weighted function space L2(ρ) on [0,∞).



PCs in the yield curve space L2(ρ)

Let ρ : [0,∞)→ R+0 be a weighting function defining the weighted Hilbert

space L2(ρ) together with 〈f , g〉L2(ρ) :=∫∞

0 f (s)g(s)ρ(s) ds.

Idea

Instead of operating on the rates, transform the “monomials” M directly.We represent the result of the transformation by P(τ) = Γ>M(τ) forΓ ∈ Rn×n.

The weights wrt P(τ) denoted by w(t) := Γ−1u(t) should beuncorrelated, i.e., have a diagonal covariance matrix:

ΛM := Γ−1ΣΓ−> (5)

where Σ = σσ>.



PCs in the yield curve space L2(ρ) (II)

We require orthonormality in L2(ρ), i.e. 〈Pi ,Pj〉L2(ρ) = δij fori , j = 1, . . . , n or

Γ>〈M,M>〉L2(ρ)Γ = In, (6)

where 〈M,M>〉L2(ρ) := (〈Mi ,Mj〉L2(ρ))i ,j ∈ Rn×n. In this case, for a

diagonal matrix ΛM = diag(λi ) = Γ−1ΣΓ−>,

Cov(df (t, t + τ1), df (t, t + τ2)) = Cov(M>(τ1)du(t),M>(τ2)du(t))

= dt ·M>(τ1)ΣM(τ2)

= dt ·M>(τ1)ΓΓ−1ΣΓ−>Γ>M(τ2)

= dt ·M>(τ1)ΓΛMΓ>M(τ2)

= dt ·n∑

i=1

λi (Γ>M(τ1))i (Γ>M(τ2))i (7)

The instantaneous forward rate covariance is preserved and can berepresented by the principal component polynomials Pi (τ) = (Γ>M(τ))i .



PCs in the yield curve space L2(ρ): Back to Rn

1 Let D ∈ Rn×n be a matrix with DD> = 〈M,M>〉L2(ρ), e.g., from aCholesky decomposition.

2 Find a Γ = D>Γ with Γ>Γ = In, such that

Γ>D>ΣDΓ (= Γ−1D>ΣDΓ−> = ΛM) (8)

be diagonal (eigenvalue decomposition).

3 The diagonal of (8) contains the sought eigenvalues, and the principalcomponent polynomials result by transforming the monomials usingΓ = D−>Γ.



Example of principal components

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 5 10 15 20 25 30

Time to maturity [in years]

PC 1

PC 2

PC 3

Figure: Example of the first three principal components of zero rates, representingthe typical yield curve movements shift, steepening and curvature.


Principal components Pseudo principal components

Pseudo principal components: motivation

Pseudo principal components are polynomials (linear combinations ofmonomials Mi ) with could be principal components, but are actuallyinvented.

Why pseudo?

The model specification (1) is transparent, but at the same timeexhibits unintuitive weights u, e.g. u = (−1, 1.5,−2, 2.5,−3)>

Need to compute sensitivities wrt observables of the yield curve (e.g.,modified duration as relative sentivity to a parallel shift)

Approach 1: Linearly transform such that the yield curve can be specifiedin terms of zero rates at n different nodes. Disadvantage: Changing onezero rate affects all other zero rates too (except at the nodes)

Solution: Use pseudo principal components!



Pseudo principal components: construction

The following construction principle can be used:

1 Construct the first (forward rate) pseudo PC P1 ∈ L2(ρ) as the linearcombination of M1, . . . ,Mn closest (in the norm of L2(ρ)) to aconstant function, normalize afterwards such that 〈P1,P1〉 = 1.

2 Construct the l-th pseudo PC orthonormal to the existing ones aslinear combination of M1, . . . ,Ml , e.g.,

P2(τ) = p(2)1 M1(τ) + p

(2)2 M2(τ) and 〈P1,P2〉 = 0, 〈P2,P2〉 = 1.

3 Any need to further adjust the pseudo PCs? Use orthonormal rotationmatrices to rotate the basis set.



Pseudo principal components: Example

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20 25 30

Time to maturity [in years]

Zeros …

Figure: Example of the first three pseudo principal components of forward rates,representing the typical yield curve movements shift, steepening and curvature.


Pricing

Overview


2 Fitted curves


4 Pricing


Pricing

Pricing zero bonds

Given the HJM parameterization of the forward rate curve, the price attime t of a zero bond with maturity T is

P(t,T ) = exp(−A(T − t)− M>(T − t)u(t)

), (9)

for A(τ) =∫ τ

0 A(s)ds and

Mi (τ) =

∫ τ

0M(s)ds =

1

αi(1− exp(−αiτ)) .


Pricing

Pricing zero bond options (and thus caps/floors)

Recall Σt,T ∈ Rn×n from (4) with

(Σt,T )jk = −exp(−(αj + αk)(T − t))− 1

αj + αk(σσ>)jk .

Theorem (Price of a European call option on a zero bond P(t,S))

The price of the call at time t with expiry T < S and strike K is

ZBC(t,T ,S ,K ) = P(t, S)Φ

(κ+ 1

2 C 2t,T ,S

Ct,T ,S

)− P(t,T )K Φ

(κ− 1

2 C 2t,T ,S

Ct,T ,S

),

where κ = ln P(t,S)P(t,T )K and C 2

t,T ,S = M>(S − T )Σt,T M(S − T ).

Proof: Using SDE of quotient process d P(t,V )P(t,T ) , and distribution of u(T ).


Pricing

Pricing zero bond options: G2++ formula

For the two-factor model, [Brigo and Mercurio(2006)] compute explicitly

Ct,T ,S =σ2

2α31

[1− e−α1(S−T )

]2 [1− e−2α1(T−t)

]+

η2

2α32

[1− e−α2(S−T )

]2 [1− e−2α2(T−t)

]+ 2ρ

ση

α1α2(α1 + α2)

[1− e−α1(S−T )

] [1− e−α2(S−T )

]·[

1− e−(α1+α2)(T−t)]

(σ, ρ, η have different meanings there)

Imagine the size of the formula for n = 5 parameters. . .


Conclusion

Conclusion

Tractable, transparent dynamic term-structure model as sum ofexponentials with arbitrary degrees of freedom ideal for riskmanagement

Identified principal components in parametric affine models usingweighted function spaces

Pseudo principal components

. . . add intuitive meaning to the weights / factor loadings, and

. . . allow for sensitivities extending well-known ones (e.g. duration);these model-based sensitivities can be useful for hedging and portfoliomanagement.

Pricing caps and floors with transparent matrix-based formulas


Conclusion

References

D. Brigo and F. Mercurio.Interest rate models – theory and practice. With smile, inflation andcredit. 2nd ed.Berlin: Springer, 2006.

D. Duffie and R. Kan.A yield-factor model of interest rates.Math. Finance, 6(4):379–406, 1996.

N. El Karoui and V. Lacoste.Multifactor models of the term structure of interest rates.In AFFI conference proceedings, 1995.

M. Henseler, C. Peters, and R. C. Seydel.A tractable multi-factor dynamic term-structure model for riskmanagement.ssrn.com, February 2013.


SDE solution of u

Overview

5 SDE solution of u

6 Calibration

7 Volatility and correlation


SDE solution of u

SDE solution of u

For S = diag(−α), the stochastic dynamics of the weights in (2), startedin t0 with u(t0), specifies an Ornstein-Uhlenbeck process with the solution

u(T ) = exp(S(T − t0))u(t0) + S−1(exp(S(T − t0))− 1)R

+

∫ T

t0

exp(S(T − v))σ dW (v), (10)

for any time T ≥ t0.


Calibration

Overview

5 SDE solution of u

6 Calibration



Calibration

Calibration: How to get the numbers

Different parameters to be chosen / estimated / calibrated:

Model specification: n, α ∈ Rn

Static curve specification: u ∈ Rn

Specification of dynamics: R ∈ Rn, σ ∈ Rn×n

There are two alternatives: implied calibration to market data andhistorical calibration, each one with its (dis)advantages.


Calibration

Calibration: How to get the numbers (II)

Implied calibration to market data(e.g., for G2++):

1 Determine and interpolateyield curve ; all zero bondprices

2 Fix n, set u = 0, drop R.Adding time-dependentparameters ensures perfect fitof yield curve for any α, u, σ

3 Calibrate to cap volatilitysurface ; α, σ

Historical calibration (no perfectfit of yield curve!):

1 Fix n, α

2 Determine best fit of yieldcurve using (1) byminimizing an error function(e.g., RMSE) ; u(i) fori = 1, . . . ,N

3 Estimate R, σ from curvetime series u(i) ; R, σ,affine term A(τ)

Problem: Interpolation has tobe done beforehand; today’smodel inconsistent with yester-day’s model!

Problem: Historical calibrationneeds A(τ) in step 2, which de-pends on σ. ⇒ Use iterative pro-cedure!


Volatility and correlation

Overview

5 SDE solution of u

6 Calibration





0.0%

0.2%

0.4%

0.6%

0.8%

1.0%


Volatitliy

0

10

20

30

0

10

20

30−0.5

0

0.5

1

Time to maturity [years]

Correlation of zero rates

Time to maturity [years]

Co

rre

latio

n

Figure: Estimated volatility and correlation of annualized zero rates, based onGerman government bonds in the time period from Jan. 2002 to Dec. 2011.


a tractable multi-factor dynamic term-structure model for risk...

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