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European Finance Association Meeting, Frankfurt 2010

Density Approximations for Multivariate Affine JumpDiffusion Processes by Damir Filipovi´c, Eberhard Mayerhofer, and Paul Schneider Discussion by Anna Cie´slak University of Lugano Institute of Finance

August 27, 2010

c EFA 2010 ( 2010 Anna Cie´slak)

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Overview ⊲ This paper Overview Comparison Role Matrix-valued Appendix

This paper... introduces a closed-form density approximation, I call it FMS approximation, for multivariate affine-jump diffusions (AJD) based on polynomial expansion Exploits... the convenient polynomial property of affine processes to develop a mathematically rigorous framework I will focus on applications and discuss the following points... i. Comparison of FMS to alternative multivariate density approximations: saddlepoint ii. Computational advantage when the dimension of the process is larger than two iii. How much we have learnt about transition densities by estimating MV affine processes: a term structure example iv. Matrix-valued extensions

c EFA 2010 ( 2010 Anna Cie´slak)

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This paper

⊲ Overview Literature Idea FMS in action Comparison Role Matrix-valued Appendix

Overview FMS approximation

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Literature on multivariate likelihood approximations This paper Overview Literature Idea FMS in action

⊲

Series expansions for diffusions... [A¨ıt-Sahalia (AS, 2008)]  Reducible diffusion Y transforms into a unit diffusion dYt = µ(Yt , θ)dt + dWt , and Hermite expansion works as in UV case p(y|y0 )

Comparison

=

Role Matrix-valued

ηh (∆, y0 )

Appendix

=

−m/2

∆

N



y − y0 ∆1/2



X

tr(h)≤J

ηh (∆, y0 ) Hh



y − y0 ∆1/2

  1 −1/2 E Hh (∆ (Yt+∆ − y0 ))|Yt = y0 h1 ! . . . hm !



(1)

(2)

 But... Most MV models are irreducible: η’s are functions of expectations of nonlinear moments thus double Taylor expansion in time ∆ and state y − y0 is needed

Alternatives... Saddlepoint approximation works for both reducible and irreducible processes, i.e. jump-diffusions and Levy processes [AS & Yu, 2006] Conventional... QML [Fisher & Gilles, 1996], SML [Brandt & Santa-Clara, 2002], Fourier inversion of characteristic function [Liu, Pan, & Pedersen (2001)], EMM [Gallant & Tauchen, 1996], MCMC [Eraker, Johannes & Polson (2003)]

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Polynomial property of affine processes This paper Overview Literature Idea FMS in action

⊲

Comparison Role Matrix-valued Appendix

Thm 2.3 CKT (2008).⋆ Y is time-homogenous Markov process. Ptm are real-valued polynomials of order ≤ m such that Pt f (y) ∈ Pol≤m Pt f (y) := E [f (Yt |Y0 = y)]

(3)

where f : Rn → R and Pol≤m is the vector space of polynomials up to degree m ≥ 0 in Rn . i. Y is m-polynomial

ii. There exists a linear map A on Pol≤m such that Pt |Pol = eAt iii. The infinitesimal generator of Y , A, solves iv. Af = Af

∂f (y,t) ∂t

= Af (y, t)

Very useful... since we can now easily compute moments of AJD! mk

=

E(Y∆k |Y0 = y)

(4)

=

(0, . . . , 1, . . . , 0) eA∆ (y 0 , . . . , y k , . . . , y m )′

(5)

⋆ See: Cuchiero, Keller-Ressel, Teichmann (2008): Approach works for AJD, Levy, Jacobi processes

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FMS density approximation in action This paper Overview Literature Idea FMS in action

⊲

i. Recenter and rescale Y into wisely chosen W so that common orthogonal polynomials can be used ii. Choose a weight function (leading term), e.g. for positive coordinates of Y use Gamma density:

Comparison

w(y) ∼ Gamma(1 + D, 1),

Role Matrix-valued Appendix

e−w wD pdf: γ(w) = Γ(1 + D)

(6)

iii. Correct the leading term using an expansion in orthogonal polynomials, e.g. for γ-weight:   n i X γ Hn (w(y∆ ))=

(−1)i

i=0 Qn i=1 (i

n+D n−i

(w(y∆ )) , H0γ = 1 i!

(gen. Laguerre poly)

+ D)

, HO0γ = 1 (normalization) n!  γ  E Hn (w(y∆ )) |y0 ⋆ cn (y0 , ∆)= , c0 = 1, c2 = c3 = 0 ⋆ (polynomial property) γ HOn γ HOn =

iv. Approximate the density: pF M S,(J) (y, ∆|y0 ) = γ(w(y))

J X i=0

c EFA 2010 ( 2010 Anna Cie´slak)

ci (y0 , ∆)Hiγ (w(y))

m1 m2 − m21

(7)

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This paper Overview

⊲ Comparison Saddlepoint FMS vs SP Comments Role Matrix-valued Appendix

A comparison Saddlepoint approximation

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Saddlepoint approximation This paper

From Laplace to cumulant transform of an affine process yt :

Overview Comparison Saddlepoint FMS vs SP Comments

⊲

ϕ(∆, u|y0 )

=

eα(∆,u)+β(∆,u)·y0

(8)

K(∆, u|y0 )

=

ln ϕ(∆, u|y0 )

(9)

The saddlepoint u ˆ solves:

Role

u ˆ:

Matrix-valued Appendix

∂K(∆, u|y0 ) =y ∂u

The saddlepoint density approximation is given as (let K(n) = pSP,(0) (∆, y|y0 ) =

(10) ∂ (n) K(∆,u|y0 ) ): ∂un

exp (K (∆, u ˆ|y0 ) − u ˆy) √ 1/2 2πK(2) | {z } leading term

Note: Works for multivariate processes, but here notation is univariate. See: Daniels (1954); Luganani & Rice (1980); Ait-Sahalia & Yu (2006)

c EFA 2010 ( 2010 Anna Cie´slak)

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Saddlepoint approximation This paper

From Laplace to cumulant transform of an affine process yt :

Overview Comparison Saddlepoint FMS vs SP Comments

⊲

ϕ(∆, u|y0 )

=

eα(∆,u)+β(∆,u)·y0

(8)

K(∆, u|y0 )

=

ln ϕ(∆, u|y0 )

(9)

The saddlepoint u ˆ solves:

Role

u ˆ:

Matrix-valued Appendix

∂K(∆, u|y0 ) =y ∂u

(10)

The saddlepoint density approximation is given as (let K(n) =

∂ (n) K(∆,u|y0 ) ): ∂un

"

2

1 K(4) 5 K(3) exp (K (∆, u ˆ|y0 ) − u ˆy) SP,(1) (∆, y|y0 ) = × 1+ − + ... p √ 2 3 1/2 8 K 24 K 2πK(2) (2) (2) | {z } | {z } leading term

1st order

Note: Works for multivariate processes, but here notation is univariate. See: Daniels (1954); Luganani & Rice (1980); Ait-Sahalia & Yu (2006)

c EFA 2010 ( 2010 Anna Cie´slak)

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Saddlepoint approximation This paper

From Laplace to cumulant transform of an affine process yt :

Overview Comparison Saddlepoint FMS vs SP Comments

⊲

ϕ(∆, u|y0 )

=

eα(∆,u)+β(∆,u)·y0

(8)

K(∆, u|y0 )

=

ln ϕ(∆, u|y0 )

(9)

The saddlepoint u ˆ solves:

Role

u ˆ:

Matrix-valued Appendix

∂K(∆, u|y0 ) =y ∂u

(10)

The saddlepoint density approximation is given as (let K(n) =

∂ (n) K(∆,u|y0 ) ): ∂un

"

2

1 K(4) 5 K(3) exp (K (∆, u ˆ|y0 ) − u ˆy) SP,(2) (∆, y|y0 ) = × 1+ − + ... p √ 2 3 1/2 8 K 24 K 2πK(2) (2) (2) | {z } | {z } leading term

1st order

2 2 K 4 # K K K K K K (4) 1 (6) 35 (4) 7 (3) (5) 35 (3) 385 (3) − + + − + 3 4 4 5 6 48 K(2) 384 K(2) 48 K(2) 64 K(2) 1152 K(2) | {z } 2nd order

Note: Works for multivariate processes, but here notation is univariate. See: Daniels (1954); Luganani & Rice (1980); Ait-Sahalia & Yu (2006) c EFA 2010 ( 2010 Anna Cie´slak)

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Saddlepoint approximation This paper Overview Comparison Saddlepoint FMS vs SP Comments

⊲

Role Matrix-valued Appendix

Idea... Taylor-expand the cumulant generating function around the saddlepoint; correct the leading term with higher order terms of the expansion Requirement... Laplace transform is given in closed form e.g. as for affine processes (vector and matrix-valued) Elements... ⋆ saddlepoint equation (10) solved numerically unless α(∆, u), β(∆, u) explicit; ⋆ derivatives of K(∆, u) up to 6th order; ⋆ integrating constant as saddlepoint density does not, in general, integrate to one Attractive features... ⋆ works for multivariate processes both reducible and irreducible; ⋆ fairly accurate in the tails and already at the 1st order; ⋆ positive leading term

See: Glasserman & Kim (2009) for application to AJD

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FMS versus saddlepoint √ I consider a simple example of CIR: dyt = κ(θ − yt )dt + σ yt dBt ∆ = 1/52 60

∆ = 1/12 true FMS 6

∆ = 2/12

30

20

pdf y∆ |y0

15 40

20 10

20

10 5

0 0

0.05 y

0.1

0.2

0 0

1 FMS 4 FMS 6

0.1

0.05

y

0.1

Abs error, pbY − pY

0 0

0.5

0

0

0

−0.1

−0.5

−0.5

0.05 y

0.1

−1 0

0.05

y

0.1

y

0.1

0.15

0.1

0.15

1

0.5

−0.2 0

0.05

−1 0

0.05

y

Note: CIR parameters: θ = 0.07, κ = 0.5, σ = 0.25, y0 = 0.05, FMS 4: 4th order expansion, FMS 6: 6th order; SP 0: leading term in saddlepoint approximation, SP 2: 2nd order saddlepoint approximation c EFA 2010 ( 2010 Anna Cie´slak)

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FMS versus saddlepoint √ I consider a simple example of CIR: dyt = κ(θ − yt )dt + σ yt dBt ∆ = 1/52

pdf y∆ |y0

60

∆ = 1/12 true FMS 6 SP 2

40

∆ = 2/12

30

20 15

20 10

20

10 5

0 0

0.05 y

0.2

0.1

FMS 6 SP 0 SP 2

0.1

0 0

0.4

0.05

y

0.1

Abs error, pbY − pY

0 0

0.5

0

0

0

−0.1

−0.2

−0.5

0.05 y

0.1

−0.4 0

0.05

y

0.1

y

0.1

0.15

0.1

0.15

1

0.2

−0.2 0

0.05

−1 0

0.05

y

Note: CIR parameters: θ = 0.07, κ = 0.5, σ = 0.25, y0 = 0.05, FMS 4: 4th order expansion, FMS 6: 6th order; SP 0: leading term in saddlepoint approximation, SP 2: 2nd order saddlepoint approximation c EFA 2010 ( 2010 Anna Cie´slak)

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FMS versus saddlepoint √ I consider a simple example of CIR: dyt = κ(θ − yt )dt + σ yt dBt ∆ = 1/52

pdf y∆ |y0

60

∆ = 1/12 true FMS 6 SP 2

40

∆ = 2/12

30

20 15

20 10

20

10 5

0 0

0.05 y

0.1

0 0

0.05

y

0.1

Log error, ln pby − ln pY

0 0

2

2

0

0

0

−2

−2

−4

−4

−2

FMS 6 SP 0 SP 2

−4 0

0.05 y

0.1

0

0.05

y

0.1

0.05

y

0.1

0.15

0.1

0.15

2

0

0.05

y

Note: CIR parameters: θ = 0.07, κ = 0.5, σ = 0.25, y0 = 0.05, FMS 4: 4th order expansion, FMS 6: 6th order; SP 0: leading term in saddlepoint approximation, SP 2: 2nd order saddlepoint approximation c EFA 2010 ( 2010 Anna Cie´slak)

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FMS versus saddlepoint √ I consider a simple example of CIR: dyt = κ(θ − yt )dt + σ yt dBt ∆ = 1/52

pdf y∆ |y0

60

∆ = 1/12 true FMS 6 SP 2

40

∆ = 2/12

30

20 15

20 10

20

10 5

0 0

0.05 y

0.1

0 0

0.05

y

0.1

Log error, ln pc Y − ln pY

0 0

2

2

0

0

0

−2

−2

−4

−4

−2 FMS 6 SP 0 SP 2

−4 0

0.05 y

0.1

0

0.05

y

0.1

0.05

y

0.1

0.15

0.1

0.15

2

0

0.05

y

Note: CIR parameters: θ = 0.07, κ = 0.5, σ = 0.25, y0 = 0.05, FMS 4: 4th order expansion, FMS 6: 6th order; SP 0: leading term in saddlepoint approximation, SP 2: 2nd order saddlepoint approximation c EFA 2010 ( 2010 Anna Cie´slak)

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Open questions and remarks This paper Overview Comparison Saddlepoint FMS vs SP Comments

⊲

Role Matrix-valued Appendix

Q1 How many orders in the expansion do we need? How many are feasible? The dimension grows fast... Q2 How does the approximation behave in the tails? [see Rogers & Zane (1999) for SP] Q3 How does the approximation behave as the horizon ∆ expands? R1 Matrix exponential is fully explicit only in special cases:  

When factors interact, its structure can become complex P∞ (At)i At The expansion e = i=0 i! can be numerically unstable [Moler & van Loan, 2003]

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This paper Overview Comparison

⊲ Role Yield curve Details PC vs filter Matrix-valued Appendix

Role for transition density approximation A yield curve example

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Yield curve estimation... This paper Overview Comparison Role Yield curve Details PC vs filter

Is natural area of applications, but... Q: How much can we learn by applying sophisticated methods to standard dynamic term structure models?

⊲

Matrix-valued Appendix

I compare factors obtained from... i. 5-factor Gaussian model estimated with ML plus Kalman filter ii. PC decomposition of the unconditional covariance matrix of yields Why this choice?  Gaussian models are useful to answer Q: ⋆ transition density is exact; ⋆ Kalman filter is optimal; ⋆ seem convenient for term premia modeling  5 factors help detect small components in the transition that are not in the cross-section  5 PCs give a benchmark how well we can do in terms of minimizing pricing errors ⋆ no transition density involved

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Details: Gaussian model This paper Overview

Transition equation for the (5 × 1) state vector: Xt+1 = µ + KXt + Σεt+1

Comparison Role Yield curve Details PC vs filter

⊲

(11)

Measurement equation: a vector of zero yields: yt = A + BXt + et

Matrix-valued Appendix

εt ∼ N (0, I5 )

2 Im ), et ∼ N (0, σM

(12)

Estimation... Kalman filter gives correct conditional means and covariances of the transition density in the Gaussian setting. The likelihood function is set up on the prediction errors et (1:5)

Rotation... Let V ar(yt ) = U ΛU ′ , then for comparison, we can rotate the filtered states as Xt⊥ = U ′ BXt . Standard PCs are given as P Ct = U ′ yt . Data... FB zero yields with maturities 1, 2, 3, 4, 5 years, 3-month T-bill rate from secondary market quotes, sample 1961:01–2007:12 Note: This approach follows Duffee (2009). I do not impose no-arbitrage to obtain A, B. These coefficients and the model’s fit are very close to the no-arbitrage case.

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Filtered states versus PCs Slope

Level 4

4

PC

3 2 2 1

0

0 −2 −1 −2 1960

1970

1980

1990

2000

−4 1960

2010

1970

1980

1990

2000

2010

Curve 4

 The plots compare three standard PCs (cross-section)...

2 0



−2



−4 −6 1960

1970

1980

c EFA 2010 ( 2010 Anna Cie´slak)

1990

2000

2010

15

Filtered states versus PCs Level vs Gaussian , corr=1 4

Slope vs Gaussian , corr=1 PC Gaussian

3

4 2

2 1

0

0 −2 −1 −2 1960

1970

1980

1990

2000

2010

−4 1960

1970

1980

1990

2000

2010

Curve vs Gaussian , corr=0.99 4

 The plots compare three standard PCs (cross-section)...

2 0 −2

 ... and the filtered states at the ML estimates of 5-factor Gaussian model

−4

 correlations exceed 99.9%

−6 1960

1970

1980

c EFA 2010 ( 2010 Anna Cie´slak)

1990

2000

2010

15

Filtered states versus PCs 4th PC

4

5th PC

6 4

2

2

0

0 −2

−2

−4 −6 1960

−4 1970

1980

1990

2000

2010

−6 1960

1970

1980

1990

2000

2010

 The figure compares 4th and 5th standard PCs...  

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Filtered states versus PCs 4

4th PC vs Gaussian, corr=0.92

6

5th PC vs Gaussian, corr=0.81

4

2

2

0

0 −2

−2 PC Gaussian

−4 −6 1960

1970

1980

1990

2000

2010

−4 −6 1960

1970

1980

1990

2000

2010

 The figure compares 4th and 5th standard PCs...  ... and the filtered states at the ML estimates of a Gaussian model 

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Filtered states versus PCs 4

5th PC vs Gaussian, corr=0.92

6

5th PC vs Gaussian, corr=0.81

4

2

2

0

0 −2 −4 −6 1960

−2

PC Gaussian PC 3m MA 1970

1980

1990

2000

−4 2010

−6 1960

1970

1980

1990

2000

2010

 The figure compares 4th and 5th standard PCs...  ... and the filtered states at the ML estimates of a Gaussian model  Kalman filter smoothes through the noise of PCs that can otherwise be read from the cross-section [see: 3-month moving-average of PCs] The power of cross-sectional information propagates onto ⋆ expected excess returns (transition) through affine MPR, ⋆ or, similarly, conditional volatilities in SV models

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This paper Overview Comparison Role

⊲ Matrix-valued Wishart Appendix

Matrix-valued extensions

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Wishart process Let Yt ∼ W (K, M, Σ) be an exact discretization of the n × n continuous time Wishart process with mean reversion matrix A, diffusion parameter Q, and integer dof K. Thm. 10.3.2, Muirhead (1982) The density function of Y is 1 (det Σ(τ ))−K/2 (det Yt+τ )(K−n−1)/2 Kn/2 2 Γ(K/2) h i 1 −1 ′
Yt+τ + M (τ )Yt M (τ ) } × exp{− T r Σ(τ ) 2   K 1 −1 ′ −1 ×0 F1 , Σ(τ ) M (τ )Yt M (τ ) Σ(τ ) Yt+τ 2 4

p(Yt+τ |Yt ) =

where M (τ ) = eAτ , Σ(τ ) =

Rτ 0

1

(13) (14) (15)

′

eAs Q′ QeA s ds, Γ(·) is a multidimensional Gamma function, and 0 F1 (·) is a

hypergeometric function of matrix arguments.

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Wishart process Let Yt ∼ W (K, M, Σ) be an exact discretization of the n × n continuous time Wishart process with mean reversion matrix A, diffusion parameter Q, and integer dof K. Thm. 10.3.2, Muirhead (1982) The density function of Y is 1 (det Σ(τ ))−K/2 (det Yt+τ )(K−n−1)/2 Kn/2 2 Γ(K/2) h i 1 −1 ′
Yt+τ + M (τ )Yt M (τ ) } × exp{− T r Σ(τ ) 2   K 1 −1 ′ −1 ×0 F1 , Σ(τ ) M (τ )Yt M (τ ) Σ(τ ) Yt+τ 2 4

p(Yt+τ |Yt ) =

where M (τ ) = eAτ , Σ(τ ) =

Rτ 0

1

(13) (14) (15)

′

eAs Q′ QeA s ds, Γ(·) is a multidimensional Gamma function, and 0 F1 (·) is a

hypergeometric function of matrix arguments.

Explicit pdf but... i. p Fq (·) involves ∞ expansion and zonal polynomials in eigenvalues of the argument

ii. Difficult to approximate: ⋆ slow convergence thus large truncation parameter needed ⋆ simple evaluation of single polynomial has complexity that grows as O(nm ) ⋆ see Koev and Edelman (2006), and their matlab code

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Wishart process Let Yt ∼ W (K, M, Σ) be an exact discretization of the n × n continuous time Wishart process with mean reversion matrix A, diffusion parameter Q, and integer dof K. Thm. 10.3.2, Muirhead (1982) The density function of Y is 1 (det Σ(τ ))−K/2 (det Yt+τ )(K−n−1)/2 Kn/2 2 Γ(K/2) h i 1 −1 ′
Yt+τ + M (τ )Yt M (τ ) } × exp{− T r Σ(τ ) 2   K 1 −1 ′ −1 ×0 F1 , Σ(τ ) M (τ )Yt M (τ ) Σ(τ ) Yt+τ 2 4

p(Yt+τ |Yt ) =

where M (τ ) = eAτ , Σ(τ ) =

Rτ 0

1

(13) (14) (15)

′

eAs Q′ QeA s ds, Γ(·) is a multidimensional Gamma function, and 0 F1 (·) is a

hypergeometric function of matrix arguments.

Explicit pdf but... i. p Fq (·) involves ∞ expansion and zonal polynomials in eigenvalues of the argument

ii. Difficult to approximate: ⋆ slow convergence thus large truncation parameter needed ⋆ simple evaluation of single polynomial has complexity that grows as O(nm ) ⋆ see Koev and Edelman (2006), and their matlab code Can FMS approximation avoid these problems?

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This paper Overview Comparison Role Matrix-valued

⊲ Appendix SP CIR FMS CIR Literature

c EFA 2010 ( 2010 Anna Cie´slak)

Appendix

19

Example 1: Saddlepoint approximation for the CIR We want to approximate the transition density p(∆, y|y0 ) of the CIR process √ dyt = κ (θ − yt ) dt + σ yt dWt Let c=

2κ , (1 − e−∆κ ) σ 2

b = ce−∆κ y0 ,

q=

2θκ −1 σ2

Then, y∆ has a noncentral χ2 distribution with 2q + 2 degrees of freedom and noncentrality parameter 2b : 2cy∆ ∼ χ2 (2q + 2, 2b)

The cumulant transform has the form:

K(∆, u|y0 ) = ln



u −q−1 exp 1− c



bu c−u



For CIR, the saddlepoint is given as: 2

u ˆ=c−

c EFA 2010 ( 2010 Anna Cie´slak)

q + q + 2q + 4bcy + 1 2y


1/2

+1

20

Example 2: FMS approximation for the CIR Generator for the CIR process: ∂f (y) 1 2 ∂ 2 f (y) Af (y) = κ (θ − y) + σ y ∂y 2 ∂y 2 Polynomial moments up to J-th order:     k k Ak ∆ 0 k J P∆ y = E y∆ |y0 = (0, ..., 1, ..., 0) e y , ..., y , ..., y For the first JL = 6 moments we have: 

    A6 =    

0 κθ 0 0 0 0 0

0 −κ 2 σ + 2κθ 0 0 0 0

0 0 −2κ 2 3σ + 3κθ 0 0 0

0 0 0 −3κ 2 6σ + 4κθ 0 0

0 0 0 0 −4κ 10σ 2 + 5κθ 0

0 0 0 0 0 −5κ 15σ 2 + 6κθ

0 0 0 0 0 0 −6κ

        

(16)

Since A6 is lower triangular, we can obtain closed form expressions for eA6 ∆ .

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Literature This paper Overview Comparison Role Matrix-valued Appendix SP CIR FMS CIR Literature

⊲

A¨ıt-Sahalia, Y. (2008): “Closed-Form Likelihood Expansions for Multivariate Diffusions,” Annals of Statistics, 36, 906–937. A¨ıt-Sahalia, Y., and R. L. Kimmel (2010): “Estimating Affine Multifactor Term Structure Models Using Closed-Form Likelihood Expansions,” Journal of Financial Economics, 98, 113–144. A¨ıt-Sahalia, Y., and J. Yu (2006): “Saddlepoint Approximations for Continuoustime Markov Processes,” Journal of Econometrics, 134, 507–551. Brandt, M., and P. Santa-Clara (2002): “Simulated Likelihood Estimation of Diffusions with an Application to Exchange Rate Dynamics in Incomplete Markets,” Journal of Financial Economics, 116, 259–292. Cuchiero, C., M. Keller-Ressel, and J. Teichmann (2008): “Polynomial Processes and Their Applications to Mathematical Finance,” Working Paper, ETH Zurich and Vienna Institute of Finance. Daniels, H. (1954): “Saddlepoint Approximations in Statistics,” Annals of Mathematical Statistics, 25, 631–650. Duffee, G. R. (2009): “Information in (and Not in) the Term Structure,” Working Paper, Johns Hopkins University. Eraker, B., M. Johannes, and N. Polson (2003): “The Impact of Jumps in Equity Index Volatility and Returns,” Journal of Finance, 58, 1269–1300.

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Literature This paper Overview Comparison Role Matrix-valued Appendix SP CIR FMS CIR Literature

Fisher, M., and C. Gilles (1996): “Estimating Exponential-Affine Models of the Term Structure,” Working Paper, Federal Reserve Bank of Atlanta. Gallant, A., and G. Tauchen (1996): “Which Moments to Match?,” Econometric Theory, 12, 657–681. Glasserman, P., and K.-K. Kim (2009): “Saddlepoint Approximations for Affine Jump-Diffusion Models,” Journal of Economic Dynamics and Control, 33, 15–36. Koev, P., and A. Edelman (2006): “The Efficient Evaluation of the Hypergeometric Function of a Matrix Argument,” Mathematics of Computation, 75, 833–846. Lugannani, R., and S. Rice (1980): “Saddlepoint Approximation for the Distribution of the Sum of Independent Random Variables,” Advances in Applied Probability, 12, 475–490. Moler, C., and C. V. Loan (2003): “Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later,” SIAM Review, 45, 3–000. Muirhead, R. J. (1982): Aspects of Multivariate Statistical Theory. Wiley Series in Probability and Mathematical Statistics. Rogers, L., and O. Zane (1999): “Saddlepoint Approximations to Option Prices,” The Annals of Applied Probability, 9, 493–503.

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