General Short-Rate Model: Theory & Applications

• General short-rate models are stochastic frameworks where the instantaneous risk-free rate evolves via an affine structure driven by diffusion, jumps, or mixed processes.
• They enable analytical tractability in pricing derivatives and modeling yield curves through closed-form exponential-affine bond pricing formulas satisfying the HJM condition.
• Extensions to multidimensional and non-Gaussian settings enhance calibration accuracy to market phenomena such as mean reversion, volatility smiles, and heavy-tailed risks.

A general short-rate model is a term used for any stochastic model in which the instantaneous risk-free interest rate—referred to as the short rate—evolves according to a time-homogeneous Markov process, often but not exclusively in the form of a diffusion, jump process, or their mixture. Such models serve as the basis for derivative pricing, yield curve modeling, and risk management. Their appeal lies in the analytical tractability, the close alignment with the Heath-Jarrow-Morton (HJM) framework, and the ability to generate a wide range of empirically observed market behaviors, including mean reversion, volatility smiles, jumps, memory effects, and "higher-for-longer" rate regimes.

1. General Structure and Mathematical Formulation

The canonical structure of a general short-rate model is given by a stochastic differential equation (SDE) or a more general stochastic equation: $$dR(t) = F(R(t))\,dt + G(R(t{-}))\,dZ(t), \qquad R(0) = x \ge 0,$$ where $F$ is the drift (commonly affine: $F(x) = ax + b$), $G$ the state-dependent volatility or jump amplitude, and $Z$ a (possibly multidimensional) Lévy martingale or Brownian motion, possibly with jumps or heavy tails (Barski et al., 2019, Barski et al., 2022).

The discounted price of a zero-coupon bond,

$$P(t,T) = \mathbb{E}\left[\,\exp\Bigl(-\int_t^T R_s\,ds\Bigr) \,\big|\, \mathcal{F}_t\right],$$

is typically required to be a local martingale under the risk-neutral measure, enforcing the so-called HJM (Heath–Jarrow–Morton) condition, which restricts the permissible choices for $F$, $G$, and the law of $Z$ (Barski et al., 2019).

Affinely structured models remain a primary class: $P(t,T) = \exp\bigl(-A(T-t) - B(T-t) R(t)\bigr).$ Here $A(\tau)$ and $B(\tau)$ are determined by Riccati-type ODEs driven by the model's coefficients and the Laplace exponent $J$ of $Z$ (Barski et al., 2019). Extensions exist for multidimensional ($d \ge 1$) or state-space–enlarged specifications, as in the Wishart and quadratic Gaussian models (Gnoatto, 2012, Grbac et al., 2015).

2. Classification According to the Driving Process

2.1. Diffusive and Stable Models

When $Z$ is a Brownian motion and $G(x) = c\sqrt{x}$, one recovers the Cox–Ingersoll–Ross (CIR) model. More generally, when $Z$ is an α-stable Lévy process ($\alpha\in(1,2]$) and $G(x) = cx^{1/\alpha}$, one has

$$dR(t) = (aR(t) + b)dt + cR(t-)^{1/\alpha}dZ_\alpha(t), \qquad b\ge 0$$

where the Laplace exponent $J(z) = -c_\alpha z^\alpha$ captures heavy tails and jump risk. This generalizes CIR to permit non-Gaussian stable fluctuations (Barski et al., 2019, Barski et al., 2022).

2.2. Pure-Jump and Mixed Processes

If $G$ is constant and $Z$ possesses only positive jumps with bounded variation (no Gaussian part), then $R$ admits a generalization of the Vasicek model with strictly nonnegative paths: $$dR(t) = (aR(t) + b)dt + \omega dZ(t), \qquad b \geq \omega \int_0^\infty y\,v(dy)$$ where $v(dy)$ is the Lévy measure (Barski et al., 2019). Models where the driving noise is a sum of compound Poisson subordinators (pure-jump OU processes) support affine bond pricing, fast computations of characteristic functions, and explicit option pricing via Fourier methods (Hess, 2020).

2.3. Multivariate and Spherical Lévy Noise

The multidimensional extension considers $R$ driven by several independent or dependent Lévy processes: $$dR(t) = F(R(t))\,dt + \sum_{i=1}^{d}G_i(R(t{-}))\,dZ_i(t)$$ with $Z=(Z_1, \ldots, Z_d)$ possibly independent (with regularly varying Laplace exponents) or "spherical" (Lévy measure of stable-type, with a radial part of general form) (Barski et al., 2022). In both cases, affine structure and nonnegativity force the generator of $R$ to be "stable-type"—a generalized version of CIR—a result shown to be robust under mild conditions such as infinite variation or $G(0)=0$. Any such multidimensional equation is equivalent (in law) to one driven by independent stable coordinates (Barski et al., 2022).

3. Affine Structure and the HJM Condition

The HJM no-arbitrage argument imposes a functional relationship: $$J\bigl(G(x)B(v)\bigr) = -A'(v) - [B'(v)-1]x + B(v)F(x),\quad\forall x,\ \forall v\ge 0,$$ where $J$ is the Laplace exponent of $Z$, and $A, B$ are the functions in the exponential-affine bond price formula (Barski et al., 2019). Solving these equations shows that, except for special degenerate noise structures, $F$ must be affine, $G$ must be a power function, and $Z$ must exhibit α-stable or pure-positive jump characteristics. This structure ensures closed-form pricing and explicit Riccati ODEs for $A, B$.

The bond price generator in the multidimensional case includes

$$A f(x) = c x f''(x) + (a x + b + \text{jump drift}) f'(x) + \text{jump integral terms},$$

with jump measures decomposed into a part acting at $x=0$ and an $x$-linear stable component (Barski et al., 2022).

4. Solution Properties and Model Implications

A key finding is that any such general equation—against the imposed affine structure and nonnegativity—produces a unique law for $R(t)$, regardless of the detailed construction of $Z$ or $G$. For example, different pairs $(G, Z)$ (even when $Z$ is not stable) can induce the same generator and thus the same law for $R$ (Barski et al., 2022). This result enables flexibility in model construction, allowing the practitioner to select $Z$ and $G$ pairs best suited for calibration or simulation without altering the economic implications.

Empirically, the introduction of jump and stable components provides for fat-tailed risk, skewness, and realistic jump-to-default or severe event risk, not available in purely diffusive (Gaussian) models. This is crucial in modern post-crisis fixed income markets, which routinely display heavy tails and discontinuous rate movements.

5. Classical Models as Special Cases

The classical CIR and Vasiček models are obtained as limiting cases:

• CIR: $Z$ is Brownian ($\alpha=2$), $G(x)=c\sqrt{x}$, $F(x)=a x + b$.
• Generalized CIR: $Z$ α-stable ($1 < \alpha < 2$), $G(x)=c x^{1/\alpha}$.
• Nonnegative Vasicek: $G(x) = \omega$ (constant), $Z$ has positive jumps, bounded variation, and the drift compensated to prevent explosion.

These results generalize and unify the theory, making clear that the familiar affine pricing formulas have broader validity: not only for models with Brownian drivers but for a wide range of Lévy-driven type processes, including multidimensional and spherically symmetric cases.

6. Practical Consequences and Calibration

• Tractability: Models retain exponential-affine bond prices, ensuring analytic tractability for zero-coupon bonds, caps, swaptions, and more (Barski et al., 2019, Hess, 2020).
• Flexibility: Multidimensional and spherical noise allows matching of empirically observed return distributions and dynamic features such as volatility clustering and jumps (Barski et al., 2022).
• Calibration: Non-Gaussian and jump components improve calibration to market prices, allowing fit to both yield curves and implied volatility smiles or skews (Grzelak, 2022).
• Risk Management: Explicit generator characterization gives robust tools for sensitivity analysis and scenario testing. Jump and stable-process components can be tuned to model extreme market moves, relevant for stress testing and risk assessment.

7. Summary Table—General Short-Rate Model Classification

Noise Class	$G(x)$ Form	$F(x)$ Form	Key Requirements / Features
α-stable ($\alpha\in(1,2]$)	$c x^{1/\alpha}$	$a x + b$, $b\ge 0$	$Z$ stable, supports heavy tails
Positive jumps, BV	constant $\omega$	$a x + b$, $b\ge \omega\int y v(dy)$	$Z$: no Gaussian part, positive jumps
Spherical/Multivariate	$G_i(x)$ as above per coord	as above	Lévy measure stable/infinite variation
Brownian (CIR)	$c \sqrt{x}$	$a x + b$	$\alpha=2$, classic CIR

• (Barski et al., 2019) On generalized CIR equations
• (Barski et al., 2022) CIR equations with multivariate Lévy noise
• (Hess, 2020) A pure-jump mean-reverting short rate model
• (Grzelak, 2022) Randomization of Short-Rate Models, Analytic Pricing and Flexibility in Controlling Implied Volatilities

Conclusion

The general short-rate framework, as formalized through affine SDEs with appropriately chosen drift and driving noise, unifies and extends classical short-rate theory. By specifying $F, G,$ and $Z$ to comply with the no-arbitrage HJM condition, tractable pricing and risk management tools are preserved, while the space of admissible models is broadened to encompass heavy tails, jumps, and multidimensional structure. These results are robust—different specifications with the same generator yield indistinguishable laws—making the general short-rate model a versatile platform for both theory and practice in modern fixed-income analysis.