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## Developing a practical robust long term yield curve model

### Citations

1032 | An Equilibrium Characterization of the Term Structure - Vasicek - 1977 |

778 | A new extension of the Kalman Filter to nonlinear systems
- Julier, Uhlmann
- 1997
(Show Context)
Citation Context ... Carlo simulations for calculating the true Black model bond prices. Krippner approximation Christensen and Rudebusch(2013 a,b) apply the Krippner framework to estimate a 3-factor shadow-rate model. They argue that the divergence of Krippner approach from the fully arbitrage-free Black approach is not very significant and well compensated by much greater tractability. Wu and Xia (2014) apply an approach equivalent to the Krippner framework in discrete time. Cumulant approximation Priebsch (2013) proposes to view the quantity log ( ) logE exp max(0, )du . t Q t t u t P s (32) as the value at -1 of conditional cumulant-generating function of the random variable ( ) max(0, )du t t u t S s under Q. It can be expanded as 1 log exp ( ) 1 ! Q j jQ t t j E j S (33) where Qj is the j -th cumulant of ( )tS under Q and an approximation can be computed by taking a finite number of terms in this series. The method of Ichiue and Ueno (2013) is equivalent to using the first term approximation in (28). Priebsch (2013) evaluates both 1- and 2-term approximations by analytically deriving the expression for the first two moments of ( )t S .... |

656 | A yield-factor model of interest rates
- Duffie, Kan
- 1996
(Show Context)
Citation Context ...n the market and B and d are defined above. The centred measurement error process t is a K vector serially independent Gaussian noise with covariance matrix H.6 Given a data series for the observed yields obsty the Kalman filter generates an estimated expected path of the Gaussian state variables, and their conditional 6 But see Dempster and Tang (2011) regarding handling measurement error serial correlation, which we intend to implement in future research. 12 covariance matrix |1t t . The filter is initialized using unconditional moments. Following Harvey (1993) gives 1 0 ˆ : (I A)x c (19) 1 0( ) : (I A A) ( ) ,vec vec G where is the Kronecker product, vec(.) is the operation of writing out a matrix as a vector and G is the covariance matrix of the factor dynamics innovations . The matrix A and the vector c are the entities in the transition equation (17) and the elements of Σ0 can be computed analytically. KF prediction |1 1 ˆ ˆ t t tx Ax c |1 |1 ˆ ˆ t t t ty Bx d (20) |1 1 T t t tA A G KF update |1 |1ˆ ˆ: obs obs t t t t t t tv y y y Bx d |1t t tF B B H (21) 1 |1 |1 ˆ ˆ T t t t t t t tx x B F v 1 |1 |1 |1t t t t... |

607 |
Neural Networks
- Haykin
- 1994
(Show Context)
Citation Context ...crisis of 2008. There are two main reasons for such a timeline. First, the zero-lower bound was not observed in the U.S. from the Great Depression until 2008; only in Japan from the mid-1990s did rates in a major economy start hitting zero. Perhaps more importantly, the implementation of the Black correction is considerably more difficult (both theoretically and computationally) than implementation of the usual affine termstructure models. The main problem is the lack of a closed form formula for the bond price given by ( ) exp max( ,0) . t Q t t u t P E du s (24) 1-factor Black models Gorovoi and Linetsky (2002) showed that for a shadow short rate following a 1- dimensional diffusion process, the zero-coupon bond price can be calculated as the Laplace transform (at the unit value of the transform parameter) of the area functional of the shadow rate process. They applied the method of eigen function expansions (see Linetsky, 2002; Davydov and Linetsky, 2003; Linetsky, 2004) to derive the quasi-analytic formulae (relying on Weber-Hermite parabolic cylinder functions) for the bond price in the Vasicek and shifted CIR process cases. Unfortunately their me... |

586 | Specification analysis of affine term structure models
- QiangDai, Singleton
- 2000
(Show Context)
Citation Context ... (9) Zero coupon bond prices in terms of expectations under the risk-neutral Q measure are given by ( ) exp . t Q t t s t P E r ds (10) Zero coupon bond yields to maturity, termed rates, are linked with the bond prices by ( ) log ( ) / .t ty P (11) There are models that lack affine structure (and thus forfeit simple formulae for bond prices) but a vector of K rates Rt of specified maturities may sometimes still be recovered as the numerical solution of the Ricatti equation ( ) 1 ( ) ( ) ( ) r 1 , 2 t t t t t R R R S R (12) where 1 is the K vector of ones. Dai and Singleton (2000) denote different affine subfamilies by ( )mA n with n the number of factors and m n the number of bounded factors. They perform empirical tests on the different subclasses for n equal to 3. Dempster et al. (2014) also analyze different 3-factor models with requirements similar to ours to uncover a variety of shortcomings with the models evaluated. They were led to introduce a Black-corrected affine model which always produces nonnegative rates.4 In their paper, they summarized in a table the stylized features satisfied by the alternat... |

384 |
An Efficient Method of Finding the Minimum of a Function of Several Variables Without Calculating Derivatives
- Powell
- 1964
(Show Context)
Citation Context ...ISDA** for ISDAfix rates). Yield curve bootstrapping For the short rate maturities (T in years equal to 0.25, 0.5, 1 ) we use only the LIBOR data and the formula (Ron, 2000) ) 22 (T) ln(1 LiborRate(T) 0.01 T) / T .y (45) For the longer maturities we use the following process. If the coupons on the swaps are paid semi-annually( as is the case for USD, GBP and JPY), then we calculate the discount factor for one year as (1) 1/ (1 LiborRate(1) /100) .df (46) If the coupon payments are annual, we use the formula (1) 1/ (1 (0.5) /100 0.5) 1/ (1 (1) /100) .df LiborRate LiborRate (47) We then proceed to calculate the discount factors 1 1 1 1 _ (i) (n) , 1 1 _ n i swap rate df freq df swap rate freq (48) where 1freq for annual coupons and 2freq for semi-annual coupons, _swap rate is the swap coupon paid. Finally, the rates are given by ( ) log(d (T)) / T .y T f (49) In-sample goodness- of-fit First, let us consider statistics for overall goodness-of-fit across the entire sample period for the five currency areas EUR, CHF, GPB, USD and JPY, ordered by average rates in the data period from highest to lowest average rate. Table 2 shows the comparative goodne... |

358 |
Lipschitzian optimization without the Lipschitz constant
- Jones, Stuckman
- 1993
(Show Context)
Citation Context ...er argues that the American option can be approximated by an analytically tractable European one and introduces an auxiliary bond price equation ( , ) ( ) C ( , ;1) ,aux S Et t tP P (29) 15 where C ( , ;1)Et is the value of a European call option at time t with maturity at time t and strike 1 written on a shadow bond maturing at t . Krippner then takes the limit with 0 to obtain the non-negative (due to future currency availability immediately before maturity) instantaneous forward rate as 0 ( ) lim ln ( , ) .auxt tf P (30) The non-negative yield with maturity τ in Krippner's framework is calculated as 0 C ( , ;1)1 1 y ( ) (s)ds ( ) lim ds . ( ) t t E S t t t t tt t f y P s (31) Here ( )Sty are the shadow bond yields. Unfortunately, closed-form analytic expressions for the bond prices and yields are still not available, but they can be evaluated through calculating integrals that are numerically tractable. More importantly, Krippner's approach is not fully arbitrage-free. The short rates are identical under the market measure P in the Black and Krippner ... |

248 |
A New Approach for Filtering Nonlinear Systems
- Julier, Uhlmann, et al.
- 1995
(Show Context)
Citation Context ...gence of Krippner approach from the fully arbitrage-free Black approach is not very significant and well compensated by much greater tractability. Wu and Xia (2014) apply an approach equivalent to the Krippner framework in discrete time. Cumulant approximation Priebsch (2013) proposes to view the quantity log ( ) logE exp max(0, )du . t Q t t u t P s (32) as the value at -1 of conditional cumulant-generating function of the random variable ( ) max(0, )du t t u t S s under Q. It can be expanded as 1 log exp ( ) 1 ! Q j jQ t t j E j S (33) where Qj is the j -th cumulant of ( )tS under Q and an approximation can be computed by taking a finite number of terms in this series. The method of Ichiue and Ueno (2013) is equivalent to using the first term approximation in (28). Priebsch (2013) evaluates both 1- and 2-term approximations by analytically deriving the expression for the first two moments of ( )t S . He shows that this technique is sufficiently fast and accurate to fit the term-structure within a half basis point for a single time step. Priebsch notes that the Krippner approximation tends to underestimate the arbitrage-... |

154 |
Maximum likelihood for incomplete data via the EM algorithm (with discussion
- Dempster, Laird, et al.
- 1977
(Show Context)
Citation Context ... present the original economic factor model (EFM) of the yield curve which we have used previously in a variety of applications in the five principal currency areas with various time steps from daily to quarterly.5 The evolution under the risk-neutral measure Q of the 3 unobservable (i.e. latent) factors of the model is governed by the SDEs , X t X X t X t Y t Y Y t Y t R t t t t R t d X dt d d Y dt d d k X Y R dt d X W Y W R W (13) with fixed pair- wise correlations of the standard Brownian motion innovations given by ( dt, dt, dt) .XY XR YR (14) The stochastic evolution of the factors under the market (i.e. real-world) measure P involving the market prices of risk of the 3 factors is governed by . XX X t X X t X t X YY Y t Y Y t Y t Y RR R t t t t R t d X dt d d Y dt d d k X Y R dt d k X W Y W R W (15) The first factor Xt represents the long rate and the third factor tR the short rate. The second factor Yt represents minus the slope of the yield curve between the long rate and the unobservable instantaneous short rate. Thus the sum of the first t... |

131 |
A History of Interest Rates
- Homer
- 1963
(Show Context)
Citation Context ...mulae (relying on Weber-Hermite parabolic cylinder functions) for the bond price in the Vasicek and shifted CIR process cases. Unfortunately their method works only in the scalar case. Gorovoi and Linetsky applied their method to estimating yield curve models for a single time point. However, their method was used by Ueno et al. (2006) for the Japanese market who applied the method to a dynamic model with a market price of risk. The shadow rate ts in these models follows a diffusion, therefore in state space form the single discretized transition equation takes the form 1 .t t tax c x (25) 14 The mapping that links observed yields and the shadow rate is no longer linear, so it takes the piecewise linear form ( ) .obst thy x (26) Ueno et al.(2006) applied the Kalman filter with conditional linearization of (25) to calibrate the model. However, it was clear from their results that further work in developing the shadow rate models would be needed. For example, the shadow rates in their analysis reach the implausibly low levels of -15%, which suggests model misspecification. 2-factor Black models Both Bomfim (2003) and Kim and Singleton (2011) relied on a numerical (finitedifferen... |

76 |
Interest Rate Modelling
- James, Webber
- 2000
(Show Context)
Citation Context ...n be expressed as ( ) ( ) C ( , ;1) ,S At t tP P (28) where ( )StP is the shadow bond price ( i.e. the price of a bond in a market where currency is not available) and C ( , ;1)At is the value of an American call option at time t with maturity in years and strike 1, written on the shadow bond maturing in years. There is no analytic formula for C ( , ;1)At , but Krippner argues that the American option can be approximated by an analytically tractable European one and introduces an auxiliary bond price equation ( , ) ( ) C ( , ;1) ,aux S Et t tP P (29) 15 where C ( , ;1)Et is the value of a European call option at time t with maturity at time t and strike 1 written on a shadow bond maturing at t . Krippner then takes the limit with 0 to obtain the non-negative (due to future currency availability immediately before maturity) instantaneous forward rate as 0 ( ) lim ln ( , ) .auxt tf P (30) The non-negative yield with maturity τ in Krippner's framework is calculated as 0 C ( , ;1)1 1 y ( ) (s)ds ( ) lim ds . ( ) t t E S t t t t tt t f y P s ... |

55 | Which model for term-structure of interest rates should one use? In: - Rogers - 1995 |

49 |
Interest rates as options.
- Black
- 1995
(Show Context)
Citation Context ...odels To illuminate the analysis that we undertake below for the more complex multi-factor models, we first discuss the characteristics of the simpler one-factor models. The stochastic differential equations (SDEs) governing the evolution of the short rate under the risk-neutral or pricing measure Q for the respective models are:3 1. Vasicek (1977) ( X )t t td dt d X W (1) 2. Dothan (1978) Xt t t td dt X d X W (2) 3. Cox-Ingersoll-Ross (1985) ( X )t t t td dt X d X W (3) 4. Ho-Lee (1986) t t td dt d X W (4) 5. Hull-White (1990) ( X ) .t t t t td dt d X W (5) It is easy to see that Hull-White ( also called extended Vasicek) is the most general of these models. It can fit any term structure exactly, because of the time-dependent 3 We use boldface type in the sequel to denote stochastic entities, here conditionally. 7 equilibrium drift coefficients θ t . However, having such a time-dependent parameter, as in the Ho-Lee and Hull-White models, contradicts our requirements 9-10, i.e. parsimony and time homogeneity. The Ho-Lee model lacks the desired mean-reversion property and the Vasicek, HoLee and Hull-White diffusion models can all produce negative ... |

43 | Measuring the effect of the zero lower bound on medium- and longer-term interest rates. Working Paper, Federal Reserve Bank of San Francisco, - Swanson, Williams - 2013 |

37 | Pricing options on scalar diffusions: An eigenfunction expansion approach.
- Davydov, Linetsky
- 2003
(Show Context)
Citation Context ...nomic factor model discussed in Medova et al. (2006), Yong (2007) and. Dempster et al. (2010). 10 Basic EFM model We first present the original economic factor model (EFM) of the yield curve which we have used previously in a variety of applications in the five principal currency areas with various time steps from daily to quarterly.5 The evolution under the risk-neutral measure Q of the 3 unobservable (i.e. latent) factors of the model is governed by the SDEs , X t X X t X t Y t Y Y t Y t R t t t t R t d X dt d d Y dt d d k X Y R dt d X W Y W R W (13) with fixed pair- wise correlations of the standard Brownian motion innovations given by ( dt, dt, dt) .XY XR YR (14) The stochastic evolution of the factors under the market (i.e. real-world) measure P involving the market prices of risk of the 3 factors is governed by . XX X t X X t X t X YY Y t Y Y t Y t Y RR R t t t t R t d X dt d d Y dt d d k X Y R dt d k X W Y W R W (15) The first factor Xt represents the long rate and the third factor tR the short rate. The second factor Yt represents minus the... |

27 | Interpolation methods for curve construction.
- Hagan, West
- 2006
(Show Context)
Citation Context ...the elements of Σ0 can be computed analytically. KF prediction |1 1 ˆ ˆ t t tx Ax c |1 |1 ˆ ˆ t t t ty Bx d (20) |1 1 T t t tA A G KF update |1 |1ˆ ˆ: obs obs t t t t t t tv y y y Bx d |1t t tF B B H (21) 1 |1 |1 ˆ ˆ T t t t t t t tx x B F v 1 |1 |1 |1t t t t t t t tB F B Quasi MLE parameter estimation Letting denote the 14 SDE model parameters of the transition equation and defining : , H , the log-likelihood is given by 1 1 1 1 1 log L( , ) log 2 logdet . 2 2 2 T T t t t t t t TK H F v F v (22) where K is the total number of maturities used, T is a the number of time steps and v and F are computed using (20). The maximization of the log likelihood is performed in two steps, alternatively optimizing and H to convergence. There are two phases of the numerical optimization: a global phase using the DIRECT global optimization algorithm (Jones et al., 1993) to locate the region of the maximum, followed by a local phase using an approximate conjugate gradient algorithm (Powell, 1963) to locate the maximum itself. Black correction The distribution of the instantaneous short rate tr is Ga... |

27 | Term structure models and the zero bound: An empirical investigation of Japanese yields.
- Kim, Singleton
- 2011
(Show Context)
Citation Context ...ur development. Unscented Kalman filter We initialize the filter at the unconditional mean and variance of the state variables under the P measure in the EFM model. This can be justified by the fact that most of our datasets start before the onset of low-rate regimes. Since only the measurement equation is non-linear, the state prediction step is the same as that of the linear Kalman filter in (20). For the factor path update step of the UKF, the state is first augmented with the expected measurement error (here 0) of the linear KF to give |1 |t 1ˆ , , a t t t tx x E (35) and the state innovation conditional covariance matrix is augmented with the measurement error covariance matrix to give 18 |1 |1 0 . 0 a t t t t H (36) Next, a set of perturbed sigma-points is constructed as 0 |1 |1 |1 |1 |1 |1 |1 |1 ˆ ˆ , 1,...,L ˆ , 1,..., 2L , t t t t j a t t t t t t j j a t t t t t t j L x x L j x L j L (37) where denotes the matrix square root of the symmetric positive definite augmented matrix (36), whose j th column augments the conditional state vector to give the augmented conditi... |

24 | The spectral decomposition of the option value.
- Linetsky
- 2004
(Show Context)
Citation Context ... (40) 2 |t 1 |1 |1 0 ˆ ˆ , t t L j j j x y c t t t t t t j W x y where the weights jW for combining sigma point estimates (predictions), are potentially different for the state vector and the covariance matrices. They are given by 19 0 0 2: : (1 ) 1 : : 1,..., 2 . 2( ) s c j j s c W W L L W W j L L (41) Here β is related to the higher moments of the state vector distribution and is usually set to 2, which is optimal for Gaussian innovations. These results are used to compute UKF Kalman gain 1: , t t t tt x y y y K (42) which defining |1 |1ˆ ˆ: obs obs t t t t t t tv y y y Bx d gives the updated state estimate in observation prediction error feedback form as |t 1ˆ ˆ:t t t tx x K v (43) with updated state covariance matrix |1 .t tt t t t y y tK K (44) Choice of parameters for the UKF As noted above the choice of parameters ( , , ) for the UKF is very important and is not usually detailed in the literature (but see Tuner et al., 2012). Some nonlinear models are known to exhibit UKF algorithm divergence with certain parameter values. The other problem is inefficient estimation bec... |

19 | Black's model of interest rates as options, eigenfunction expansions and Japanese interest rates.
- Gorovoi, Linetsky
- 2004
(Show Context)
Citation Context ...onal moments. Following Harvey (1993) gives 1 0 ˆ : (I A)x c (19) 1 0( ) : (I A A) ( ) ,vec vec G where is the Kronecker product, vec(.) is the operation of writing out a matrix as a vector and G is the covariance matrix of the factor dynamics innovations . The matrix A and the vector c are the entities in the transition equation (17) and the elements of Σ0 can be computed analytically. KF prediction |1 1 ˆ ˆ t t tx Ax c |1 |1 ˆ ˆ t t t ty Bx d (20) |1 1 T t t tA A G KF update |1 |1ˆ ˆ: obs obs t t t t t t tv y y y Bx d |1t t tF B B H (21) 1 |1 |1 ˆ ˆ T t t t t t t tx x B F v 1 |1 |1 |1t t t t t t t tB F B Quasi MLE parameter estimation Letting denote the 14 SDE model parameters of the transition equation and defining : , H , the log-likelihood is given by 1 1 1 1 1 log L( , ) log 2 logdet . 2 2 2 T T t t t t t t TK H F v F v (22) where K is the total number of maturities used, T is a the number of time steps and v and F are computed using (20). The maximization of the log likelihood is performed in two steps, alternatively optimizing and H to convergence. There are... |

16 | Monetary Policy Expectations at the Zero Lower Bound, Federal Reserve Bank of San Francisco Working Paper
- Bauer, Rudebusch
- 2014
(Show Context)
Citation Context ... and multi-factor models. Analysis of one factor short rate models To illuminate the analysis that we undertake below for the more complex multi-factor models, we first discuss the characteristics of the simpler one-factor models. The stochastic differential equations (SDEs) governing the evolution of the short rate under the risk-neutral or pricing measure Q for the respective models are:3 1. Vasicek (1977) ( X )t t td dt d X W (1) 2. Dothan (1978) Xt t t td dt X d X W (2) 3. Cox-Ingersoll-Ross (1985) ( X )t t t td dt X d X W (3) 4. Ho-Lee (1986) t t td dt d X W (4) 5. Hull-White (1990) ( X ) .t t t t td dt d X W (5) It is easy to see that Hull-White ( also called extended Vasicek) is the most general of these models. It can fit any term structure exactly, because of the time-dependent 3 We use boldface type in the sequel to denote stochastic entities, here conditionally. 7 equilibrium drift coefficients θ t . However, having such a time-dependent parameter, as in the Ho-Lee and Hull-White models, contradicts our requirements 9-10, i.e. parsimony and time homogeneity. The Ho-Lee model lacks the desired mean-reversion property and the Vasicek, Ho... |

16 |
Modifying Gaussian term structure models when interest rates are near the zero lower bound. Discussion Paper 2012/02, Reserve Bank of New Zealand.
- Krippner
- 2012
(Show Context)
Citation Context ...be justified by the fact that most of our datasets start before the onset of low-rate regimes. Since only the measurement equation is non-linear, the state prediction step is the same as that of the linear Kalman filter in (20). For the factor path update step of the UKF, the state is first augmented with the expected measurement error (here 0) of the linear KF to give |1 |t 1ˆ , , a t t t tx x E (35) and the state innovation conditional covariance matrix is augmented with the measurement error covariance matrix to give 18 |1 |1 0 . 0 a t t t t H (36) Next, a set of perturbed sigma-points is constructed as 0 |1 |1 |1 |1 |1 |1 |1 |1 ˆ ˆ , 1,...,L ˆ , 1,..., 2L , t t t t j a t t t t t t j j a t t t t t t j L x x L j x L j L (37) where denotes the matrix square root of the symmetric positive definite augmented matrix (36), whose j th column augments the conditional state vector to give the augmented conditional state vector. Here L is the dimension of the augmented state and the scalar parameter is defined as 2: ( ) ,L L (38) where and control the spread of the ... |

13 |
Equilibrium interest rates and the yield curve in a low interest rate environment. Working Paper 07-E-18, Bank of Japan.
- Ichiue, Ueno
- 2007
(Show Context)
Citation Context ...r with conditional linearization of (25) to calibrate the model. However, it was clear from their results that further work in developing the shadow rate models would be needed. For example, the shadow rates in their analysis reach the implausibly low levels of -15%, which suggests model misspecification. 2-factor Black models Both Bomfim (2003) and Kim and Singleton (2011) relied on a numerical (finitedifference) method for solving a 2-dimensional parabolic quasilinear bond price PDE given by 2 1 ( x) max 0, ( ) 0 2 t t t t P P P tr K s x P x x x (27) with boundary condition ( 0, x) 1P . Here the short rate ( )s x is an affine function of the state 2-dimensional state x . Bomfim (2003) estimated the parameters of his model on the subset of data where rates were safely above zero, using an analytical approximation, i.e. the usual affine model. Kim and Singleton (2011) used the extended Kalman filter with quasimaximum likelihood to estimate the parameters. Ueno et al. (2007) performed a sensitivity analysis of the 2-factor Black-corrected model without estimating the parameters. Kim and Singleton and Ueno et al. report superior perform... |

13 | Measuring the macroeconomic impact of monetary policy at the zero lower bound, - Wu, Xia - 2016 |

11 |
Zero-coupon yield curves. Technical Documentation No.25, Bank for International Settlements,
- BIS
- 2005
(Show Context)
Citation Context ...asicek (1977), Dothan (1978), Cox-Ingersoll-Ross (1985), Ho-Lee (1986), Hull-White (1990) and many other one- and multi-factor models. Analysis of one factor short rate models To illuminate the analysis that we undertake below for the more complex multi-factor models, we first discuss the characteristics of the simpler one-factor models. The stochastic differential equations (SDEs) governing the evolution of the short rate under the risk-neutral or pricing measure Q for the respective models are:3 1. Vasicek (1977) ( X )t t td dt d X W (1) 2. Dothan (1978) Xt t t td dt X d X W (2) 3. Cox-Ingersoll-Ross (1985) ( X )t t t td dt X d X W (3) 4. Ho-Lee (1986) t t td dt d X W (4) 5. Hull-White (1990) ( X ) .t t t t td dt d X W (5) It is easy to see that Hull-White ( also called extended Vasicek) is the most general of these models. It can fit any term structure exactly, because of the time-dependent 3 We use boldface type in the sequel to denote stochastic entities, here conditionally. 7 equilibrium drift coefficients θ t . However, having such a time-dependent parameter, as in the Ho-Lee and Hull-White models, contradicts our requirements 9-10, i.e. p... |

8 | Modeling Yields at the Zero Lower Bound: Are Shadow Rates the Solution,
- Christensen, Rudebusch
- 2013
(Show Context)
Citation Context ... form ,t t t td X dt S d X W (6) with X the K-dimensional state vector; W K-dimensional Brownian motion; a fixed point in K-dimensional space; Λ, St and K K matrices and St a diagonal matrix with diagonal elements satisfying 8 1,..., ,t i i tiiS X i K (7) with prime denoting transpose. For an admissable parametrization, the bond prices can be calculated as ( ) ( )( ) ,tA B XtP e (8) where A and B are solutions of certain ODEs (see e.g. James and Webber, 2000). The instantaneous short rate is also an affine function of the state 0 . Q Q t X tr X (9) Zero coupon bond prices in terms of expectations under the risk-neutral Q measure are given by ( ) exp . t Q t t s t P E r ds (10) Zero coupon bond yields to maturity, termed rates, are linked with the bond prices by ( ) log ( ) / .t ty P (11) There are models that lack affine structure (and thus forfeit simple formulae for bond prices) but a vector of K rates Rt of specified maturities may sometimes still be recovered as the numerical solution of the Ricatti equation ( ) 1 ( ) ( ) ( ) r 1 , 2 t t t t t R R R S R (12) where 1... |

8 |
Yield Curve Modeling and Forecasting: The Dynamic Nelson-Siegel Approach.
- Diebold, Rudebusch
- 2013
(Show Context)
Citation Context ...to fit all of the observed rates approximately, in contrast to other approaches which often fit a small number of rates (equal to the number of factors) exactly. KF transition equation Taking the discretization time step Δt := 1, the Euler approximation of the SDEs for the 3 factor state variables becomes the state variable transition equation 1t t tAx c x , (17) where η t ~ N(0, G) is the Gaussian innovation, with A, c and G deterministic matrix or vector-valued functions of the SDE coefficients. KF measurement equation The corresponding measurement equation is obst t tBx d y , (18) where obsty corresponds to the yields observed in the market and B and d are defined above. The centred measurement error process t is a K vector serially independent Gaussian noise with covariance matrix H.6 Given a data series for the observed yields obsty the Kalman filter generates an estimated expected path of the Gaussian state variables, and their conditional 6 But see Dempster and Tang (2011) regarding handling measurement error serial correlation, which we intend to implement in future research. 12 covariance matrix |1t t . The filter is initialized using unconditional moments. Fo... |

8 | The use of the Black model of interest rates as options for monitoring the JGB market expectations. Working Paper 06‐E‐15, Bank of Japan. - Ueno, Baba, et al. - 2006 |

7 | Model based learning of sigma points in unscented Kalman filtering. - Turner, Rasmussen - 2012 |

6 |
Monetary policy and the yield curve at zero interest: The macro-finance model of interest rates as options. Working Paper 06-E-16, Bank of Japan.
- Ichiue, Ueno
- 2006
(Show Context)
Citation Context ...eir method works only in the scalar case. Gorovoi and Linetsky applied their method to estimating yield curve models for a single time point. However, their method was used by Ueno et al. (2006) for the Japanese market who applied the method to a dynamic model with a market price of risk. The shadow rate ts in these models follows a diffusion, therefore in state space form the single discretized transition equation takes the form 1 .t t tax c x (25) 14 The mapping that links observed yields and the shadow rate is no longer linear, so it takes the piecewise linear form ( ) .obst thy x (26) Ueno et al.(2006) applied the Kalman filter with conditional linearization of (25) to calibrate the model. However, it was clear from their results that further work in developing the shadow rate models would be needed. For example, the shadow rates in their analysis reach the implausibly low levels of -15%, which suggests model misspecification. 2-factor Black models Both Bomfim (2003) and Kim and Singleton (2011) relied on a numerical (finitedifference) method for solving a 2-dimensional parabolic quasilinear bond price PDE given by 2 1 ( x) max 0, ( ) 0 2 t t t t P P P tr K s x P x x x... |

6 | A practical guide to swap curve construction. Working Paper 2000-17, Bank of Canada, - Ron - 2000 |

5 |
Time Series Models, Second edition. HarvesterWheatsheaf,
- Harvey
- 1993
(Show Context)
Citation Context ...l, 1963) to locate the maximum itself. Black correction The distribution of the instantaneous short rate tr is Gaussian in most yield curve models, therefore it is easy to see that it can become negative when initialized at a low level. Black (1995) suggested a way of solving this problem. He argued that 13 nominal rates cannot become negative, because there is always the option of investing in the (0-yielding) currency instead. Black started from a process ts which can take negative values, which he called the shadow short rate, and the nominal short rate is then defined as max(0, ) .t tr s (23) This modification makes all the yields calculated through the bond price formula nonnegative. A similar model was independently discussed by Rogers (1995). Unfortunately, the shadow short rates implied by affine models lose their linearity when modified using this idea. This makes the resulting models difficult to calibrate. We shall discuss different approaches to calibration in the next section. 4. Alternative approaches to calibrating Black models Although Black's idea was proposed in the 90s, the first implementation followed seven years later in the work of Gorovoi and Linetsky (2002). A... |

5 |
A Priebsch (2013). Estimation of multi-factor shadow-rate term structure models. Working Paper, Federal Reserve Board,
- Kim, M
- 1013
(Show Context)
Citation Context ...zes the likelihood function by using Powell (1964) local search combined with Nelder-Mead global search. Summary Implementing the Black correction leads to non-linearity of the measurement equation,i.e. of the mapping of factors/states to yields, so that the classical Kalman filter is no longer applicable. Taking account of the information in the literature reviewed above, we will use the unscented Kalman filter (Julier et al.,1995; Julier & Uhlmann, 1997) for our shadow rate model. To calculate the bond prices, we will use the measurement equation approximation 0 ( ) ,obst t tB d y x (34) where denotes coordinate-wise maximum at each step of the UKF dynamics. It will be demonstrated in the sequel that the computational times for our approach are very acceptable relative to those of the basic linear Kalman filtering algorithm and the nonlinear KF alternatives. 5. Unscented Kalman filter EM algorithm for the Black EFM model As stated above, we will use the unscented Kalman filter for our development. Unscented Kalman filter We initialize the filter at the unconditional mean and variance of the state variables under the P measure in the EFM model. This can be justified by the f... |

5 |
A Scheinkman
- Litterman, J
- 1991
(Show Context)
Citation Context ...ally different for the state vector and the covariance matrices. They are given by 19 0 0 2: : (1 ) 1 : : 1,..., 2 . 2( ) s c j j s c W W L L W W j L L (41) Here β is related to the higher moments of the state vector distribution and is usually set to 2, which is optimal for Gaussian innovations. These results are used to compute UKF Kalman gain 1: , t t t tt x y y y K (42) which defining |1 |1ˆ ˆ: obs obs t t t t t t tv y y y Bx d gives the updated state estimate in observation prediction error feedback form as |t 1ˆ ˆ:t t t tx x K v (43) with updated state covariance matrix |1 .t tt t t t y y tK K (44) Choice of parameters for the UKF As noted above the choice of parameters ( , , ) for the UKF is very important and is not usually detailed in the literature (but see Tuner et al., 2012). Some nonlinear models are known to exhibit UKF algorithm divergence with certain parameter values. The other problem is inefficient estimation because of excessive spread of the sigma points. The first issue is not a problem here, but we aim to address the last issue in future research. Quasi maximum likelihood estimation Paramet... |

5 | The limitations of simple two-factor interest rate models. - Rebonato, Cooper - 1995 |

4 |
Estimating term premia at zero bound: An analysis of Japanese,
- Ichiue, Ueno
- 2013
(Show Context)
Citation Context ...e more plausible levels of the shadow rate. However, the analysis of Section 2 suggests that 3 factors would be preferred for realistic modelling. Unfortunately, the alternating direction implicit finite difference scheme used by Kim and Singleton cannot easily be extended to the corresponding PDE in 3 dimensions. Krippner (2013) applies a different method, which can be seen as an approximation to the Black model. The advantage of his method is that the forward rates have closed-form formulae. In the Black model, the price of a bond can be expressed as ( ) ( ) C ( , ;1) ,S At t tP P (28) where ( )StP is the shadow bond price ( i.e. the price of a bond in a market where currency is not available) and C ( , ;1)At is the value of an American call option at time t with maturity in years and strike 1, written on the shadow bond maturing in years. There is no analytic formula for C ( , ;1)At , but Krippner argues that the American option can be approximated by an analytically tractable European one and introduces an auxiliary bond price equation ( , ) ( ) C ( , ;1) ,aux S Et t tP P (29) 15 where C ( , ;1)Et is the value of a European cal... |

4 |
Measuring the stance of monetary policy in conventional and unconventional environments.
- Krippner
- 2014
(Show Context)
Citation Context ...efined as 2: ( ) ,L L (38) where and control the spread of the sigma points in an elliptical configuration around the conditional augmented state vector. The choice of these parameters is very important for the results of the calibration. We used a (NAG) code which sets equal to 1 and equal to 0, but we shall see that this is probably not the best choice. Next, the (here piecewise linear) measurement equation is evaluated at the 2L (= 34) sigma points to obtain 2L estimates of the augmented observations as |1 |1 |1( ) 0 (B d) 1,...,2L. j j j t t t t t th j (39) These 2L sigma point results are then combined to obtain the predicted (here yield) measurements, measurements covariance matrix and predicted state-measurement cross-covariance matrix 2 |1 0 ˆ L j j t t s t j y W 2 |1 |1 0 ˆ ˆ t t L j j j y y c t t t t t t j W y y (40) 2 |t 1 |1 |1 0 ˆ ˆ , t t L j j j x y c t t t t t t j W x y where the weights jW for combining sigma point estimates (predictions), are potentially different for the state vector and the covariance matrices. They are given by 19 0 0 2: : (1 ) 1 : : 1,...,... |

3 |
Exotic spectra.
- Linetsky
- 2002
(Show Context)
Citation Context ...e yield) measurements, measurements covariance matrix and predicted state-measurement cross-covariance matrix 2 |1 0 ˆ L j j t t s t j y W 2 |1 |1 0 ˆ ˆ t t L j j j y y c t t t t t t j W y y (40) 2 |t 1 |1 |1 0 ˆ ˆ , t t L j j j x y c t t t t t t j W x y where the weights jW for combining sigma point estimates (predictions), are potentially different for the state vector and the covariance matrices. They are given by 19 0 0 2: : (1 ) 1 : : 1,..., 2 . 2( ) s c j j s c W W L L W W j L L (41) Here β is related to the higher moments of the state vector distribution and is usually set to 2, which is optimal for Gaussian innovations. These results are used to compute UKF Kalman gain 1: , t t t tt x y y y K (42) which defining |1 |1ˆ ˆ: obs obs t t t t t t tv y y y Bx d gives the updated state estimate in observation prediction error feedback form as |t 1ˆ ˆ:t t t tx x K v (43) with updated state covariance matrix |1 .t tt t t t y y tK K (44) Choice of parameters for the UKF As noted above the choice of parameters ( , , ) for the UKF is very impor... |

3 | Computing arbitrage-free yields in multi-factor Gaussian shadow-rate term structure models. Working Paper
- Priebsch
- 2013
(Show Context)
Citation Context ...e LIBOR data and the formula (Ron, 2000) ) 22 (T) ln(1 LiborRate(T) 0.01 T) / T .y (45) For the longer maturities we use the following process. If the coupons on the swaps are paid semi-annually( as is the case for USD, GBP and JPY), then we calculate the discount factor for one year as (1) 1/ (1 LiborRate(1) /100) .df (46) If the coupon payments are annual, we use the formula (1) 1/ (1 (0.5) /100 0.5) 1/ (1 (1) /100) .df LiborRate LiborRate (47) We then proceed to calculate the discount factors 1 1 1 1 _ (i) (n) , 1 1 _ n i swap rate df freq df swap rate freq (48) where 1freq for annual coupons and 2freq for semi-annual coupons, _swap rate is the swap coupon paid. Finally, the rates are given by ( ) log(d (T)) / T .y T f (49) In-sample goodness- of-fit First, let us consider statistics for overall goodness-of-fit across the entire sample period for the five currency areas EUR, CHF, GPB, USD and JPY, ordered by average rates in the data period from highest to lowest average rate. Table 2 shows the comparative goodness-of-fit, in terms of optimal log likelihood and standard deviation (vol) of the sample measurement errors across all yields at the ... |

2 | On deterministic-shift extensions of the short-rate models.
- Brigo, Mercurio
- 2001
(Show Context)
Citation Context ... to have an affine representation and Dai and Singleton (2000) analyze the different subfamilies of affine term structure models. Dai and Singleton's analysis for the 3- factor case shows that some affine subfamilies explain historical interest rate behaviour better than others. The SDEs for their factors are of the form ,t t t td X dt S d X W (6) with X the K-dimensional state vector; W K-dimensional Brownian motion; a fixed point in K-dimensional space; Λ, St and K K matrices and St a diagonal matrix with diagonal elements satisfying 8 1,..., ,t i i tiiS X i K (7) with prime denoting transpose. For an admissable parametrization, the bond prices can be calculated as ( ) ( )( ) ,tA B XtP e (8) where A and B are solutions of certain ODEs (see e.g. James and Webber, 2000). The instantaneous short rate is also an affine function of the state 0 . Q Q t X tr X (9) Zero coupon bond prices in terms of expectations under the risk-neutral Q measure are given by ( ) exp . t Q t t s t P E r ds (10) Zero coupon bond yields to maturity, termed rates, are linked with the bond prices by ( ) log ( ) / .t ty P (... |

2 |
A regime-switching model of the yield curve at the zero bound. Federal Reserve Bank of San Francisco Working Paper
- Christensen
- 2015
(Show Context)
Citation Context ...ngleton's analysis for the 3- factor case shows that some affine subfamilies explain historical interest rate behaviour better than others. The SDEs for their factors are of the form ,t t t td X dt S d X W (6) with X the K-dimensional state vector; W K-dimensional Brownian motion; a fixed point in K-dimensional space; Λ, St and K K matrices and St a diagonal matrix with diagonal elements satisfying 8 1,..., ,t i i tiiS X i K (7) with prime denoting transpose. For an admissable parametrization, the bond prices can be calculated as ( ) ( )( ) ,tA B XtP e (8) where A and B are solutions of certain ODEs (see e.g. James and Webber, 2000). The instantaneous short rate is also an affine function of the state 0 . Q Q t X tr X (9) Zero coupon bond prices in terms of expectations under the risk-neutral Q measure are given by ( ) exp . t Q t t s t P E r ds (10) Zero coupon bond yields to maturity, termed rates, are linked with the bond prices by ( ) log ( ) / .t ty P (11) There are models that lack affine structure (and thus forfeit simple formulae for bond prices) but a vector of K rates Rt of specified m... |

2 |
Nonlinear Kalman filtering in affine term structure models.
- Christoffersen, Dorion, et al.
- 2014
(Show Context)
Citation Context ...) with prime denoting transpose. For an admissable parametrization, the bond prices can be calculated as ( ) ( )( ) ,tA B XtP e (8) where A and B are solutions of certain ODEs (see e.g. James and Webber, 2000). The instantaneous short rate is also an affine function of the state 0 . Q Q t X tr X (9) Zero coupon bond prices in terms of expectations under the risk-neutral Q measure are given by ( ) exp . t Q t t s t P E r ds (10) Zero coupon bond yields to maturity, termed rates, are linked with the bond prices by ( ) log ( ) / .t ty P (11) There are models that lack affine structure (and thus forfeit simple formulae for bond prices) but a vector of K rates Rt of specified maturities may sometimes still be recovered as the numerical solution of the Ricatti equation ( ) 1 ( ) ( ) ( ) r 1 , 2 t t t t t R R R S R (12) where 1 is the K vector of ones. Dai and Singleton (2000) denote different affine subfamilies by ( )mA n with n the number of factors and m n the number of bounded factors. They perform empirical tests on the different subclasses for n equal to 3. Dempster et al. (2014) also analyze differen... |

2 | A non-linear macroeconomic term structure model. Working Paper, - Richard - 2013 |

1 |
The shadow rate, Taylor rules and monetary policy lift-off. Working Paper, Federal Reserve Bank of San Francisco,
- Bauer, Rudebusch
- 2013
(Show Context)
Citation Context ...86), Hull-White (1990) and many other one- and multi-factor models. Analysis of one factor short rate models To illuminate the analysis that we undertake below for the more complex multi-factor models, we first discuss the characteristics of the simpler one-factor models. The stochastic differential equations (SDEs) governing the evolution of the short rate under the risk-neutral or pricing measure Q for the respective models are:3 1. Vasicek (1977) ( X )t t td dt d X W (1) 2. Dothan (1978) Xt t t td dt X d X W (2) 3. Cox-Ingersoll-Ross (1985) ( X )t t t td dt X d X W (3) 4. Ho-Lee (1986) t t td dt d X W (4) 5. Hull-White (1990) ( X ) .t t t t td dt d X W (5) It is easy to see that Hull-White ( also called extended Vasicek) is the most general of these models. It can fit any term structure exactly, because of the time-dependent 3 We use boldface type in the sequel to denote stochastic entities, here conditionally. 7 equilibrium drift coefficients θ t . However, having such a time-dependent parameter, as in the Ho-Lee and Hull-White models, contradicts our requirements 9-10, i.e. parsimony and time homogeneity. The Ho-Lee model lacks the desired m... |

1 |
Long-term interest rates and consol bond valuation.
- Dempster
- 2010
(Show Context)
Citation Context ...tions of the SDE parameters (see e.g. Medova et al. ( 2006), and Yong (2007) for the factor 5 It is interesting to note that this model originated at Long Term Capital Management and was first brought to our attention by Lehman Brothers under the auspices of Pioneer Investments of UniCredit Bank. It has been attributed to Chi Fu Huang but we have been unable to verify this. 11 covariance matrix). In particular, denoting the 3 latent factors at time t in vector form by : ( , , ) ',t t t tx X Y R the yields of the K different maturity zero coupon bonds are given in affine form by ,t ty Bx d (16) where B and d are closed-form deterministic affine functions (matrix or vectorvalued) of the SDE parameters. Basic EFM calibration Calibration of the EFM model is a non-trivial task even without the Black correction. The parameters of the model are estimated using a version of the expectation-maximization (EM) algorithm (Dempster et al., 1977) which iterates to parameter convergence the Kalman filter (KF) to generate sample paths and maximum likelihood estimation (MLE) of parameters for each path. Given a fixed set of parameters, the Kalman filter produces estimates for the unobserved states ... |

1 |
Estimating exponential affine models with correlated measurement errors: Applications to fixed income and commodities.
- Dempster, Tang
- 2011
(Show Context)
Citation Context ...ed states of the factors and prediction for the yields from (16). These are then used as the observed sample for the next numerical parameter optimization step of MLE. Note that for the EFM model in state space form, MLE is trying to fit all of the observed rates approximately, in contrast to other approaches which often fit a small number of rates (equal to the number of factors) exactly. KF transition equation Taking the discretization time step Δt := 1, the Euler approximation of the SDEs for the 3 factor state variables becomes the state variable transition equation 1t t tAx c x , (17) where η t ~ N(0, G) is the Gaussian innovation, with A, c and G deterministic matrix or vector-valued functions of the SDE coefficients. KF measurement equation The corresponding measurement equation is obst t tBx d y , (18) where obsty corresponds to the yields observed in the market and B and d are defined above. The centred measurement error process t is a K vector serially independent Gaussian noise with covariance matrix H.6 Given a data series for the observed yields obsty the Kalman filter generates an estimated expected path of the Gaussian state variables, and their conditiona... |

1 |
Risk premia, volatilities, and Sharpe ratios in a non-linear term structure model. Working Paper, The Wharton School,University of Pennsylvania.
- Feldhutter, Heyerdahl-Larsen, et al.
- 2013
(Show Context)
Citation Context ... which we intend to implement in future research. 12 covariance matrix |1t t . The filter is initialized using unconditional moments. Following Harvey (1993) gives 1 0 ˆ : (I A)x c (19) 1 0( ) : (I A A) ( ) ,vec vec G where is the Kronecker product, vec(.) is the operation of writing out a matrix as a vector and G is the covariance matrix of the factor dynamics innovations . The matrix A and the vector c are the entities in the transition equation (17) and the elements of Σ0 can be computed analytically. KF prediction |1 1 ˆ ˆ t t tx Ax c |1 |1 ˆ ˆ t t t ty Bx d (20) |1 1 T t t tA A G KF update |1 |1ˆ ˆ: obs obs t t t t t t tv y y y Bx d |1t t tF B B H (21) 1 |1 |1 ˆ ˆ T t t t t t t tx x B F v 1 |1 |1 |1t t t t t t t tB F B Quasi MLE parameter estimation Letting denote the 14 SDE model parameters of the transition equation and defining : , H , the log-likelihood is given by 1 1 1 1 1 log L( , ) log 2 logdet . 2 2 2 T T t t t t t t TK H F v F v (22) where K is the total number of maturities used, T is a the number of time steps and v and F are computed using (20). Th... |

1 |
A Pelsser
- Jong, Driessen
- 2001
(Show Context)
Citation Context ...here C ( , ;1)Et is the value of a European call option at time t with maturity at time t and strike 1 written on a shadow bond maturing at t . Krippner then takes the limit with 0 to obtain the non-negative (due to future currency availability immediately before maturity) instantaneous forward rate as 0 ( ) lim ln ( , ) .auxt tf P (30) The non-negative yield with maturity τ in Krippner's framework is calculated as 0 C ( , ;1)1 1 y ( ) (s)ds ( ) lim ds . ( ) t t E S t t t t tt t f y P s (31) Here ( )Sty are the shadow bond yields. Unfortunately, closed-form analytic expressions for the bond prices and yields are still not available, but they can be evaluated through calculating integrals that are numerically tractable. More importantly, Krippner's approach is not fully arbitrage-free. The short rates are identical under the market measure P in the Black and Krippner frameworks but different under the risk-neutral measure Q. Krippner's approach is extendible to 3- factors. 3-factor Black models There are several approaches to calibrating 3-factor shadow rate models. Most of the ... |

1 |
A L Vladu
- Lemke
- 2014
(Show Context)
Citation Context ...o 0, but we shall see that this is probably not the best choice. Next, the (here piecewise linear) measurement equation is evaluated at the 2L (= 34) sigma points to obtain 2L estimates of the augmented observations as |1 |1 |1( ) 0 (B d) 1,...,2L. j j j t t t t t th j (39) These 2L sigma point results are then combined to obtain the predicted (here yield) measurements, measurements covariance matrix and predicted state-measurement cross-covariance matrix 2 |1 0 ˆ L j j t t s t j y W 2 |1 |1 0 ˆ ˆ t t L j j j y y c t t t t t t j W y y (40) 2 |t 1 |1 |1 0 ˆ ˆ , t t L j j j x y c t t t t t t j W x y where the weights jW for combining sigma point estimates (predictions), are potentially different for the state vector and the covariance matrices. They are given by 19 0 0 2: : (1 ) 1 : : 1,..., 2 . 2( ) s c j j s c W W L L W W j L L (41) Here β is related to the higher moments of the state vector distribution and is usually set to 2, which is optimal for Gaussian innovations. These results are used to compute UKF Kalman gain 1: , t t t tt x y y y K (42) whi... |

1 |
Modelling the long term dynamics of yield curves.
- Medova, Rietbergen, et al.
- 2005
(Show Context)
Citation Context ...ven by 19 0 0 2: : (1 ) 1 : : 1,..., 2 . 2( ) s c j j s c W W L L W W j L L (41) Here β is related to the higher moments of the state vector distribution and is usually set to 2, which is optimal for Gaussian innovations. These results are used to compute UKF Kalman gain 1: , t t t tt x y y y K (42) which defining |1 |1ˆ ˆ: obs obs t t t t t t tv y y y Bx d gives the updated state estimate in observation prediction error feedback form as |t 1ˆ ˆ:t t t tx x K v (43) with updated state covariance matrix |1 .t tt t t t y y tK K (44) Choice of parameters for the UKF As noted above the choice of parameters ( , , ) for the UKF is very important and is not usually detailed in the literature (but see Tuner et al., 2012). Some nonlinear models are known to exhibit UKF algorithm divergence with certain parameter values. The other problem is inefficient estimation because of excessive spread of the sigma points. The first issue is not a problem here, but we aim to address the last issue in future research. Quasi maximum likelihood estimation Parameters estimates in the approximate Black corrected EM algorithm are calibrated... |

1 |
Review of Yield Curve Modelling and Forecasting - The Dynamic Nelson-Siegel Approach by
- Rebonato
- 2015
(Show Context)
Citation Context ...e swaps are paid semi-annually( as is the case for USD, GBP and JPY), then we calculate the discount factor for one year as (1) 1/ (1 LiborRate(1) /100) .df (46) If the coupon payments are annual, we use the formula (1) 1/ (1 (0.5) /100 0.5) 1/ (1 (1) /100) .df LiborRate LiborRate (47) We then proceed to calculate the discount factors 1 1 1 1 _ (i) (n) , 1 1 _ n i swap rate df freq df swap rate freq (48) where 1freq for annual coupons and 2freq for semi-annual coupons, _swap rate is the swap coupon paid. Finally, the rates are given by ( ) log(d (T)) / T .y T f (49) In-sample goodness- of-fit First, let us consider statistics for overall goodness-of-fit across the entire sample period for the five currency areas EUR, CHF, GPB, USD and JPY, ordered by average rates in the data period from highest to lowest average rate. Table 2 shows the comparative goodness-of-fit, in terms of optimal log likelihood and standard deviation (vol) of the sample measurement errors across all yields at the data maturities and all observations, of three models: the affine EFM estimated with both the Kalman and unscented Kalman filters and the Black EFM estimated with the UKF. ... |

1 | Economic factor model of the term structure. Internal Research Note, - Villaverde - 2003 |

1 | Scenario Generation for Dynamic Fund Management. - Yong - 2007 |