Linear-Rational Term-Structure Models

Linear-Rational Term-Structure Models Anders Trolle (joint with Damir Filipović and Martin Larsson) Ecole Polytechnique Fédérale de Lausanne Swiss Finance Institute AMaMeF and Swissquote Conference, September 9, 215

Near-zero short-term interest rates 2/23

Contribution Existing models that respect zero lower bound (ZLB) on interest rates face limitations: Shadow-rate models do not capture volatility dynamics Multi-factor CIR and quadratic models do not easily accommodate unspanned factors and swaption pricing We develop a new class of linear-rational term structure models Respects ZLB on interest rates Easily accommodates unspanned factors affecting volatility and risk premia Admits semi-analytical solutions to swaptions Extensive empirical analysis Parsimonious model specification has very good fit to interest rate swaps and swaptions since 1997 Captures many features of term structure, volatility, and risk premia dynamics. 3/23

Outline The linear-rational framework The Linear-Rational Square-Root (LRSQ) model Empirical analysis 4/23

Outline The linear-rational framework The Linear-Rational Square-Root (LRSQ) model Empirical analysis The linear-rational framework 5/23

Linear-rational framework and bond pricing State-price density, ζ t Π(t, T ) = 1 ζ t E t [ζ T C T ] m-dimensional factor process, Z t, with linear drift given by dz t = κ(θ Z t )dt + dm t, for some κ R m m, θ R m, and some martingale M t ζ t given by ζ t = e αt ( φ + ψ Z t ), for some φ R and ψ R m such that φ + ψ z > for all z E, and some α R Conditional expectations: Price of zero-coupon bond: E t [Z T ] = θ + e κ(t t) (Z t θ) P(t, t + τ) = (φ + ψ θ)e ατ + ψ e (α+κ)τ (Z t θ) The linear-rational framework φ + ψ Z 6/23 t

Interest rates and the zero lower bound Short rate: r t = T log P(t, T ) T =t = α ψ κ(θ Z t ) φ + ψ Z t Define α = sup z ψ κ(θ z) φ + ψ z and α = inf z ψ κ(θ z) φ + ψ z Set α = α so that r t [, α α ] α and α are finite if z R d + and all components of ψ are strictly positive Range is parameter dependent, verify that range is wide enough If eigenvalues of κ have nonnegative real part then α is the infinite-maturity ZCB yield The linear-rational framework 7/23

Interest rate swaps Exchange a stream of fixed-rate for floating-rate payments Consider a tenor structure T < T 1 < < T n, T i T i 1 At T i, i = 1... n: pay k, for fixed rate k 1 receive floating LIBOR L(T i 1, T i ) = P(T i 1,T i ) 1 Value of payer swap at t T n t = P(t, T ) P(t, T n ) k P(t, T i ) }{{} i=1 floating leg }{{} fixed leg Π swap Forward swap rate S t = P(t,T) P(t,Tn) n i=1 P(t,T i ) The linear-rational framework 8/23

Swaptions Payer swaption = option to enter the swap at T paying fixed, receiving floating Payoff at expiry T of the form C T = ( Π swap T ) + = ( n ) + c i P(T, T i ) = 1 p swap (Z T ) + ζ T i= for the explicit linear function p swap (z) = n c i e αt i i= Swaption price at t T is given by ( ) φ + ψ θ + ψ e κ(t i T ) (z θ) Π swaption t = 1 ζ t E[ζ T C T F t ] = 1 ζ t E t [ pswap (Z T ) +] Efficient swaption pricing via Fourier transform...! The linear-rational framework 9/23

Fourier transform Define q(x) = E t [exp (x p swap (Z T ))] for every x C such that the conditional expectation is well-defined Then Π swaption t = 1 ζ t π for any µ > with q(µ) < Re ] [ q(µ + iλ) (µ + iλ) 2 dλ q(x) has semi-analytical solution in LRSQ model The linear-rational framework 1/23

Outline The linear-rational framework The Linear-Rational Square-Root (LRSQ) model Empirical analysis The Linear-Rational Square-Root (LRSQ) model 11/23

Linear-Rational Square-Root (LRSQ) model Objective: A model with joint factor process (Z t, U t ), where Zt : m term structure factors U t : n m USV factors Denoted LRSQ(m,n) Based on a (m + n)-dimensional square-root diffusion process X t taking values in R m+n + of the form ) dx t = (b βx t ) dt + Diag (σ 1 X1t,..., σ m+n Xm+n,t db t, Define (Z t, U t ) = SX t as linear transform of X t with state space E = S(R m+n + ) Need to specify a (m + n) (m + n)-matrix S such that the implied term structure state space is E = R m + the drift of Z t does not depend on U t, while U t feeds into the martingale part of Z t The Linear-Rational Square-Root (LRSQ) model 12/23

Linear-Rational Square-Root (LRSQ) model (cont.) S given by S = ( ) Idm A Id n with A = ( Idn ). β chosen upper block-triangular of the form ( ) ( ) κ β = S 1 κ κa AA A S = κa κa A κa for some κ R m m b given by b = βs 1 ( θ θ U ) = for some θ R m and θ U R n. ( ) κθ AA κaθ U A κaθ U The Linear-Rational Square-Root (LRSQ) model 13/23

Linear-Rational Square-Root (LRSQ) model (cont.) Resulting joint factor process (Z t, U t ): dz t = κ (θ Z t) dt + σ(z t, U t)db t ) du t = A κa (θ U U t) dt + Diag (σ m+1 U1t db m+1,t,..., σ m+n Unt db m+n,t, with dispersion function of Z t given by σ(z, u) = (Id m, A) Diag ( σ 1 z1 u 1,..., σ m+n un ). Example: LRSQ(1,1) dz 1t = κ 11 (θ 1 + θ 2 Z 1t ) dt + σ 1 Z1t U 1t db 1t + σ 2 U1t db 2t du 1t = κ 22 (θ 2 U 1t ) dt + σ 2 U1t db 2t The Linear-Rational Square-Root (LRSQ) model 14/23

Linear-rational vs. exponential-affine framework Exponential-affine Linear-rational Short rate affine LR ZCB price exponential-affine LR ZCB yield affine log of LR Coupon bond price sum of exponential-affines LR Swap rate ratio of sums of exponential-affines LR ZLB ( ) USV ( ) Cap/floor valuation semi-analytical semi-analytical Swaption valuation approximate semi-analytical Linear state inversion ZCB yields bond prices or swap rates Table 1: Comparison of exponential-affine and linear-rational frameworks. In the exponential-affine framework, respecting the zero lower bound (ZLB) on interest rates is only possible if all factors are of the square-root type, and accommodating unspanned stochastic volatility (USV) is only possible if at least one factor is conditionally Gaussian. The Linear-Rational Square-Root (LRSQ) model 15/23

Outline The linear-rational framework The Linear-Rational Square-Root (LRSQ) model Empirical analysis Empirical analysis 16/23

Data and estimation approach Panel data set of swaps and swaptions Swap maturities: 1Y, 2Y, 3Y, 5Y, 7Y, 1Y Swaptions expiries: 3M, 1Y, 2Y, 5Y 866 weekly observations, Jan 29, 1997 Aug 28, 213 Estimation approach: Quasi-maximum likelihood in conjunction with the unscented Kalman Filter.8 Panel A1: Swap data 25 Panel B1: Swaption data.6 2.4.2 15 1 5 Jan97 Jan1 Jan5 Jan9 Jan13 Jan97 Jan1 Jan5 Jan9 Jan13 Empirical analysis 17/23 Panel A2: Swap fit, LRSQ(3,3) Panel B2: Swaption fit, LRSQ(3,3)

Model specifications Model specifications (always 3 term structure factors) LRSQ(3,1): volatility of Z 1t containing an unspanned component LRSQ(3,2): volatility of Z1t and Z 2t containing unspanned components LRSQ(3,3): volatility of term structure factors containing unspanned components α = α and range of r t : LRSQ(3,1) LRSQ(3,2) LRSQ(3,3) Long ZCB yield α 7.46% 6.88% 5.66% Upper bound on r t 2% 146% 72% Empirical analysis 18/23

Level-dependence in factor volatilities Volatility of Z it with USV: Volatility of Z it without USV: σ i Zit σ 2 i Z it + (σ 2 i+3 σ2 i )U it Vol. of Z1,t LRSQ(3,1) 2 1.5 1.5.5 1 1.5 LRSQ(3,2).6.5.4.3.2.1.5 1 LRSQ(3,3).6.5.4.3.2.1.5 1 Vol. of Z2,t.8.6.4.2.5 1.8.6.4.2.5 1 1.5.6.5.4.3.2.1.5 1 Vol. of Z3,t.12.1.8.6.2.15.1.4.5.2.2 Empirical analysis.5 1.5.1.15.1.2.3 19/23.8.6.4

Volatility dynamics near the ZLB Level-dependence in volatility, 3M/1Y swaption IV vs. 1Y swap rate 25 2 3M normal implied volatility, basis points 15 1 5.5.1.15.2.25.3.35.4 Empirical analysis 1Y swap rate 2/23 Figure 1: Level-dependence in volatility of 1-year swap rate

Level-dependence in volatility Regress weekly changes in the 3M swaption IV on weekly changes in the underlying swap rate σ N,t = β + β 1 S t + ɛ t 1 yr 2 yrs 3 yrs 5 yrs 7 yrs 1 yrs Mean Panel A: ˆβ 1 All.18.16.16.16.16.16.16 (2.38) (2.88) (3.31) (4.12) (4.59) (4.97) %-1% 1.2.74.62.48.76 (8.3) (8.79) (8.19) (7.83) 1%-2%.54.64.46.52.45.26.48 (2.7) (6.21) (6.77) (5.2) (5.23) (8.24) 2%-3%.28.11.3.36.4.4.31 (3.1) (1.97) (3.77) (5.8) (5.62) (4.93) 3%-4%.2.11.6.5.11.17.8 (.22) (1.21) (.92) (.8) (1.82) (1.96) 4%-5%.4 (.31).7 (.82).1 (.8).8 (1.59).7 (1.76).7.3 (1.65) Panel B: R 2 All.5.6.8.1.11.1.8 %-1%.52.54.54.44.51 1%-2%.25.49.45.55.55.27.43 2%-3%.16.6.28.37.44.45.29 3%-4%..3.1.1.7.12.4 4%-5%..1..3.3.3.2 Empirical analysis 21/23 Table 4: Level-dependence in volatility.

Level-dependence in volatility, LRSQ(3,3).8 Panel A: ˆβ1 in data.8 Panel B: Model-implied ˆβ1.6.6.4.4.2.2 All %-1% 1%-2% 2%-3% 3%-4% 4%-5% All %-1% All %-1% 1%-2% 2%-3% 3%-4% 4%-5% All %-1% 1%-2% 2%-3% 3%-4% 4%-5% 1%-2% 2%-3% 3%-4% 4%-5% Panel C: R 2 in data Panel D: Model-implied R 2.4.4.2.2 Empirical analysis 22/23

Conclusion Key features of framework: Respects ZLB on interest rates Easily accommodates unspanned factors affecting volatility and risk premia Admits semi-analytical solutions to swaptions Extensive empirical analysis: Parsimonious model specification has very good fit to interest rate swaps and swaptions since 1997 Captures many features of term structure, volatility, and risk premia dynamics. Conclusion 23/23