Quadratic Term Structure Models

• Quadratic term structure models are defined by specifying the short rate as a quadratic function of underlying factors, leading to exponential-quadratic bond prices that capture yield curve curvature.
• The framework employs Riccati-type equations to derive closed-form or semi-closed-form solutions for pricing instruments like zero-coupon bonds, FRAs, caplets, floors, and swaptions.
• Extensions via Gaussian constructions, regime-switching, and polynomial jump-diffusions enhance flexibility, tractability, and calibration efficiency in modeling interest rate dynamics.

Searching arXiv for recent and foundational papers on quadratic term structure models. Quadratic term structure models (QTSMs) are term-structure models in which the instantaneous short rate is specified as a quadratic function of an underlying factor process, and zero-coupon bond prices consequently take an exponential-quadratic form in the state variables. In continuous time, the factor dynamics are typically linear or Ornstein–Uhlenbeck type, while in discrete time they may be regime dependent; across these settings, the defining analytical mechanism is a system of Riccati-type equations or backward recursions for the coefficients in the bond-price exponent. The framework has been developed for zero-coupon bonds, FRAs, caplets, floors, swaptions, futures and forwards, as well as multi-curve, regime-switching, and polynomial jump-diffusion extensions (Hyndman et al., 2014, Grbac et al., 2015, Lorig et al., 2022).

1. Core specification and exponential-quadratic bond pricing

A standard continuous-time QTSM starts from a linear factor diffusion under a risk-neutral measure. In Hyndman–Zhou, the factor process $X_t \in \mathbb R^n$ satisfies

$dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$

and the short rate is

$r_t = r(X_t) = X_t^{\top}\Gamma X_t + R X_t + k,$

where $\Gamma$ is symmetric and positive semidefinite, and $k$ is chosen so that $r(x)\ge 0$ for all $x$. The zero-coupon bond price is then

$$P(t,T)=E^Q\!\Bigl[\exp\!\bigl(-\int_t^T r(X_u)\,du\bigr)\Bigm|\mathcal F_t\Bigr].$$

An equivalent generic specification in Lorig–Suaysom uses an Ornstein–Uhlenbeck-type state variable $Y_t\in\mathbb R^d$,

$dY_t =(a + B\,Y_t)\,dt + \Sigma\,dW_t,$

with eigenvalues of $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$0 having negative real part, and a quadratic short rate

$dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$1

so that $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$2 (Hyndman et al., 2014, Lorig et al., 2022).

The bond-pricing implication is the exponential-quadratic ansatz. In the $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$3-formulation,

$dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$4

where $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$5 solve Riccati-type ODEs. In the $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$6-formulation, the corresponding representation is

$dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$7

The matrix Riccati equation for $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$8, the linear equation for $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$9, and the scalar integral equation for $r_t = r(X_t) = X_t^{\top}\Gamma X_t + R X_t + k,$0 are obtained by matching quadratic, linear, and constant terms after applying Itô’s formula. Hyndman–Zhou further show that the solvability of these Riccati equations is linked to linear-quadratic control, and that under $r_t = r(X_t) = X_t^{\top}\Gamma X_t + R X_t + k,$1 the matrix Riccati equation admits a unique continuous solution on $r_t = r(X_t) = X_t^{\top}\Gamma X_t + R X_t + k,$2 (Hyndman et al., 2014).

A characteristic feature of QTSMs is that the state dependence of $r_t = r(X_t) = X_t^{\top}\Gamma X_t + R X_t + k,$3 is quadratic rather than affine. This distinguishes them from affine term structure models even when both model classes are driven by linear Gaussian state processes. The additional quadratic term produces nonlinear Riccati structure and allows nontrivial curvature in yields while preserving closed-form or semi-closed-form bond pricing (Hyndman et al., 2014).

2. Gaussian exponentially-quadratic short-rate constructions

A particularly explicit specification is the Gaussian exponentially-quadratic multi-curve model of Crépey, Grbac, Ngor, and Skovmand. Under the risk-neutral OIS-bank-account measure $r_t = r(X_t) = X_t^{\top}\Gamma X_t + R X_t + k,$4, the three-dimensional factor process $r_t = r(X_t) = X_t^{\top}\Gamma X_t + R X_t + k,$5 satisfies

$r_t = r(X_t) = X_t^{\top}\Gamma X_t + R X_t + k,$6

with

$r_t = r(X_t) = X_t^{\top}\Gamma X_t + R X_t + k,$7

and independent Wiener components, so that

$r_t = r(X_t) = X_t^{\top}\Gamma X_t + R X_t + k,$8

The short rate and spread are specified as second-order polynomials of Gaussian factors: $r_t = r(X_t) = X_t^{\top}\Gamma X_t + R X_t + k,$9 with

$\Gamma$0

and

$\Gamma$1

The parameter $\Gamma$2 introduces instantaneous correlation between $\Gamma$3 and $\Gamma$4 via the shared factor $\Gamma$5 (Grbac et al., 2015).

In this Gaussian EQ class, the OIS bond price is

$\Gamma$6

Because $\Gamma$7 and $\Gamma$8 are diagonal and the coefficient structure is sparse, only selected entries are nonzero. The explicit solution includes

$\Gamma$9

and

$k$0

with

$k$1

Hence

$k$2

The synthetic Libor-curve bond

$k$3

has the same exponential-quadratic structure, with

$k$4

and $k$5 determined by an analogous ODE (Grbac et al., 2015).

This construction is notable because the factors remain Gaussian while the short rate and spread are quadratic. The resulting model is explicitly non-affine at the short-rate level, yet analytically tractable at the bond-pricing level.

3. Pricing FRAs, caplets, floors, and swaptions

In the multi-curve formulation, the single-curve FRA fair rate is

$k$6

The post-crisis multi-curve FRA rate is

$k$7

A central identity is the factorization

$k$8

where

$k$9

and the deterministic residual is

$r(x)\ge 0$0

The adjustment factor provides an explicit route from pre-crisis single-curve values to post-crisis multi-curve values (Grbac et al., 2015).

For optional claims, the same Gaussian structure yields semi-analytic formulas. A caplet on $r(x)\ge 0$1 is priced as

$r(x)\ge 0$2

Under $r(x)\ge 0$3, $r(x)\ge 0$4 is Gaussian; the three-dimensional integral is reduced to explicit one-dimensional normal-cdf expressions by introducing the set

$r(x)\ge 0$5

identifying the two roots $r(x)\ge 0$6 of

$r(x)\ge 0$7

and evaluating the resulting Gaussian integrals in closed form. Swaption pricing proceeds analogously under $r(x)\ge 0$8, using a region

$r(x)\ge 0$9

and two critical values $x$0 solving $x$1 (Grbac et al., 2015).

Lorig and Suaysom develop a complementary asymptotic methodology for caplet implied volatilities in generic QTSMs. Under the $x$2-forward measure, the caplet forward price $x$3 solves

$x$4

where $x$5 is Markov with generator

$x$6

They expand both the generator and the solution in powers of a book-keeping parameter $x$7,

$x$8

derive explicit formulas for $x$9, and prove that in the short-maturity/parabolic regime

$$P(t,T)=E^Q\!\Bigl[\exp\!\bigl(-\int_t^T r(X_u)\,du\bigr)\Bigm|\mathcal F_t\Bigr].$$0

In numerical experiments for the one-factor Quadratic Ornstein–Uhlenbeck model

$$P(t,T)=E^Q\!\Bigl[\exp\!\bigl(-\int_t^T r(X_u)\,du\bigr)\Bigm|\mathcal F_t\Bigr].$$1

the second-order approximation captures the level, slope and curvature of the exact smile extremely well for all four maturities considered, especially near-ATM, and the absolute relative errors remain below a few tenths of one percent in a neighborhood of $$P(t,T)=E^Q\!\Bigl[\exp\!\bigl(-\int_t^T r(X_u)\,du\bigr)\Bigm|\mathcal F_t\Bigr].$$2 and small $$P(t,T)=E^Q\!\Bigl[\exp\!\bigl(-\int_t^T r(X_u)\,du\bigr)\Bigm|\mathcal F_t\Bigr].$$3 (Lorig et al., 2022).

4. Regime-switching and polynomial jump-diffusion generalizations

A discrete-time extension is the regime-switching QTSM of Goutte. Time is indexed by $$P(t,T)=E^Q\!\Bigl[\exp\!\bigl(-\int_t^T r(X_u)\,du\bigr)\Bigm|\mathcal F_t\Bigr].$$4, and the model includes a finite-state Markov chain $$P(t,T)=E^Q\!\Bigl[\exp\!\bigl(-\int_t^T r(X_u)\,du\bigr)\Bigm|\mathcal F_t\Bigr].$$5 with time-dependent transition matrices $$P(t,T)=E^Q\!\Bigl[\exp\!\bigl(-\int_t^T r(X_u)\,du\bigr)\Bigm|\mathcal F_t\Bigr].$$6, together with a regime-dependent factor process

$$P(t,T)=E^Q\!\Bigl[\exp\!\bigl(-\int_t^T r(X_u)\,du\bigr)\Bigm|\mathcal F_t\Bigr].$$7

when $$P(t,T)=E^Q\!\Bigl[\exp\!\bigl(-\int_t^T r(X_u)\,du\bigr)\Bigm|\mathcal F_t\Bigr].$$8. The short rate in regime $$P(t,T)=E^Q\!\Bigl[\exp\!\bigl(-\int_t^T r(X_u)\,du\bigr)\Bigm|\mathcal F_t\Bigr].$$9 is

$Y_t\in\mathbb R^d$0

The zero-coupon bond price remains exponential-quadratic: $Y_t\in\mathbb R^d$1 with coefficient families initialized at maturity by

$Y_t\in\mathbb R^d$2

and propagated backward through explicit recursions involving

$Y_t\in\mathbb R^d$3

Regularity requires $Y_t\in\mathbb R^d$4 to be invertible and positive-definite, ensuring finiteness of the Gaussian mgf (Goutte, 2013).

A broader continuous-time extension is the polynomial jump-diffusion framework of Filipović–Willems. The state vector $Y_t\in\mathbb R^d$5 evolves on a convex domain $Y_t\in\mathbb R^d$6 with generator $Y_t\in\mathbb R^d$7 satisfying $Y_t\in\mathbb R^d$8. A maximally general specification is

$Y_t\in\mathbb R^d$9

so the drift is affine and diffusion-jump coefficients are at most quadratic in $dY_t =(a + B\,Y_t)\,dt + \Sigma\,dW_t,$0. With a quadratic short rate

$dY_t =(a + B\,Y_t)\,dt + \Sigma\,dW_t,$1

bond prices again take an exponential-quadratic form

$dY_t =(a + B\,Y_t)\,dt + \Sigma\,dW_t,$2

and $dY_t =(a + B\,Y_t)\,dt + \Sigma\,dW_t,$3 satisfy generalized Riccati ODEs that incorporate diffusion and jump contributions. The same machinery supports a quadratic dividend specification,

$dY_t =(a + B\,Y_t)\,dt + \Sigma\,dW_t,$4

closed-form prices for dividend futures, and a present-value formula for the stock price. For derivatives with non-exponential-quadratic payoffs, the paper proposes a moment-based approximation: compute conditional moments, fit a maximum-entropy density matching those moments, and evaluate the price by numerical quadrature. The paper also records positivity and mgf-existence restrictions, including $dY_t =(a + B\,Y_t)\,dt + \Sigma\,dW_t,$5 for nonnegative short rates and light-tail conditions on the jump measure $dY_t =(a + B\,Y_t)\,dt + \Sigma\,dW_t,$6 (1803.02249).

5. Affine comparison and the statistical-consistency obstruction

QTSMs are closely related to affine term structure models (ATSMs), but the relationship is not identity. In an ATSM, one posits an affine short rate $dY_t =(a + B\,Y_t)\,dt + \Sigma\,dW_t,$7 and obtains

$dY_t =(a + B\,Y_t)\,dt + \Sigma\,dW_t,$8

with linear Riccati equations. In a QTSM, the exponent of the bond price contains a quadratic state term and the Riccati system acquires the nonlinear contribution

$dY_t =(a + B\,Y_t)\,dt + \Sigma\,dW_t,$9

Hyndman–Zhou also note that QTSMs can be embedded into ATSMs on an augmented state space of dimension $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$00 by adjoining pairwise products $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$01, and that if $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$02 then the model collapses to the affine case with $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$03 (Hyndman et al., 2014).

A more restrictive result is provided by Benth and Detering in the HJM-diffusion setting. They consider forward curves in Musiela parametrisation,

$dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$04

with finite-dimensional realisation

$dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$05

and absence of arbitrage encoded by the pointwise PDE

$dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$06

They introduce the statistical consistency condition (SCC): for every constant deterministic $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$07, there exists a corresponding drift process $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$08 such that the resulting model remains risk-neutral with the same parametrisation $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$09. Under finite-dimensional realisation, risk-neutrality, and SCC, they prove that $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$10 must be affine in the factor variable: $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$11 and that $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$12 and each component of $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$13 are quasi-exponential in $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$14 (Krühner et al., 2023).

This theorem directly constrains genuine quadratic term-structure parametrisations. For

$dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$15

the gradient and Hessian are

$dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$16

so the Hessian is independent of $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$17 while the gradient is affine in $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$18. The paper shows that the compatibility demanded by SCC can hold for all $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$19 only if $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$20. Hence any genuine quadratic term structure with $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$21 fails SCC. The paper’s stated conclusion is that a non-degenerate quadratic term structure model cannot be both arbitrage-free and statistically consistent in the diffusion setting; possible remedies are to abandon SCC and fix a single diffusion law, or to allow highly constrained state-dependent volatility, which generically forces the model back toward affine form or degeneracy (Krühner et al., 2023).

6. Tractability, calibration, and modeling trade-offs

The principal tractability advantage emphasized in the Gaussian EQ literature is that Gaussian factors remain Gaussian under any forward-measure change, since the change only induces a linear drift shift. This has several direct consequences: log-bond and option prices reduce to normal-cdf expressions, and calibration and simulation are simpler than in exponentially affine models driven by square-root factors, where more involved distributions appear (Grbac et al., 2015).

At the same time, QTSMs involve model-specific admissibility conditions. In the Gaussian EQ multi-curve model, $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$22 and $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$23 can go negative with small probability, though they remain well behaved for realistic $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$24’s and $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$25’s. The same framework allows a deterministic shift extension $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$26 to match the initial term structure exactly while preserving Gaussianity. For optional claims, conditions such as

$dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$27

are imposed so that certain quadratic forms inside Gaussian integrals stay positive (Grbac et al., 2015).

Implementation strategies depend on the extension under consideration. In polynomial jump-diffusion specifications, the paper recommends choosing $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$28 small, for example $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$29–$dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$30 factors, numerically integrating the Riccati ODEs for $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$31, inverting the bond and dividend-future formulas to recover $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$32, and matching option prices by moment fits. The same paper states that matching $dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$33–$dX_t = (A X_t + B)\,dt + \sigma\,dW_t,$34 moments often yields errors well below typical bid–ask spreads. In the regime-switching discrete-time case, the one-factor special case collapses to scalar recursions, and calibration may be performed with maximum-likelihood or the extended Kalman-filter / EM algorithm under hidden regimes (1803.02249, Goutte, 2013).

Taken together, these results place QTSMs in a distinctive position within term-structure theory. They retain closed-form or semi-closed-form pricing through exponential-quadratic transforms and Riccati systems, accommodate multi-curve spreads, caplet and swaption pricing, regime changes, and polynomial jump-diffusion effects, and permit explicit implied-volatility asymptotics. At the same time, the affine-geometry theorem shows that, once statistical consistency is imposed in a diffusion-based HJM setting, genuine quadratic yield-curve parametrisations are no longer compatible with the required no-arbitrage geometry (Hyndman et al., 2014, Krühner et al., 2023).