3.9.3 Short-Rate Models

Basic Prerequisites: 02 · Bonds, Yield Curve & Forward Rates and Stochastic Calculus & Itô’s Lemma.

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1. Intuition & Practical Objective

A short-rate model picks the dynamics of the instantaneous rate $r_t$ directly, and then derives the whole bond/forward structure as a consequence. The objective of this page: understand the three canonical models - Vasicek (Gaussian, mean-reverting, negative rates possible), CIR (square-root diffusion, non-negative rates), and Hull–White (time-dependent drift for exact fit to today’s curve) - and the one idea that connects them to prices: the market price of risk.

The key structural idea that has no analogue in a constant-$r$ BSM world: bond prices are not uniquely determined by the physical ($\mathbb{P}$) dynamics of $r$. Because there is only one random source ($dW$) but the market of bonds has infinitely many maturities, one bond cannot hedge another unless you know how much risk the market prices. That missing quantity is the market price of risk $\lambda$ - the excess return per unit of volatility. Different $\lambda$ give different risk-neutral measures $\mathbb{Q}$ and hence different bond prices (Björk Prop 23.1–23.3).

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2. Mathematical Ground Truth & Derivations

2.1 The term-structure equation and the market price of risk (Björk Ch23)

Let $dr=\mu (t,r)dt+\sigma (t,r)d\bar{W}$ under $\mathbb{P}$. A portfolio of a $S$-bond and a $T$-bond kills the $dW$ term, which forces the market price of risk to be the same for all maturities:

$$\frac{{\alpha }_T(t)-r(t)}{{\sigma }_T(t)}=\lambda (t)\forall T\text{(Bjo¨rk Prop 23.1)}.$$

$F(t,r;T)=P(t,T)$ then satisfies the term-structure equation (Björk Prop 23.2):

$$F_t+(\mu -\lambda \sigma )F_r+\frac{1}{2}{\sigma }^2{F}_{rr}-rF=0,F(T,r;T)=1.$$

Under the risk-neutral measure the $r$-dynamics become $dr=(\mu -\lambda \sigma )dt+\sigma dW$ and (Björk Prop 23.3)

$$P(t,T)={\mathbb{E}}_{t,r}^{\mathbb{Q}}\left[e^{-{\int }_t^T{r}_{s}ds}\right].$$

$\lambda$ is exogenous - not pinned by the model, but calibrated to (or assumed for) the market. It is the single lever connecting real-world and risk-neutral drift.

2.2 Vasicek (BM Ch3; Björk Prop 24.3; Hull Ch31)

$$dr=a(b-r)dt+\sigma dW.$$

Gaussian, mean-reverting to $b$ with speed $a$. The affine bond price $P(t,T)=A(t,T)e^{-B(t,T)r}$ has closed forms

$$B(t,T)=\frac{1-e^{-a(T-t)}}{a},A(t,T)=\exp \left\{\frac{(B-(T-t))(a^{2}b-\frac{1}{2}{\sigma }^2)}{a^2}-\frac{{\sigma }^2{B}^2}{4a}\right\}.$$

Flaw: $r$ is Gaussian, so $\mathbb{P}(r<0)>0$ - nominal rates can go negative.

2.3 Cox–Ingersoll–Ross (BM Ch3; Björk Prop 24.6; Shreve Ch31)

$$dr=a(b-r)dt+\sigma \sqrt{r}dW.$$

The square-root diffusion keeps $r\ge 0$; the Feller condition $2ab\ge {\sigma }^2$ makes $r>0$ strictly and $r$ unattainable at 0. Closed-form bond (Björk Prop 24.6, $h=\sqrt{a^2+2{\sigma }^2}$):

$$B(t,T)=\frac{2(e^{h\tau }-1)}{(h+a)(e^{h\tau }-1)+2h},A(t,T)={\left[\frac{2h{e}^{(a+h)\tau /2}}{(h+a)(e^{h\tau }-1)+2h}\right]}^{\frac{2ab}{{\sigma }^2}},\tau =T-t.$$

$r(t)$ has a non-central chi-square transition density; the stationary density is Gamma (Shreve Ch31).

2.4 Hull–White extended Vasicek (BM Ch3 3.33–3.34; Björk Prop 24.8; Hull Ch32; Shreve Ch30)

$$dr=[\theta (t)-ar]dt+\sigma dW,\theta (t)=\frac{\partial }{\partial T}{f}^M(0,t)+a{f}^M(0,t)+\frac{{\sigma }^2}{2a}\left(1-e^{-2at}\right).$$

The time-dependent drift $\theta (t)$ is chosen so the model reproduces today’s market forward curve exactly (no yield-curve inversion - this is why HW is the workhorse). With constant $a,\sigma$, $r$ is Gaussian (Shreve Ch30), ${\int }_0^{T}rdt$ is normal, and bond prices and bond-options have closed forms. Ho-Lee is the $a=0$ special case; a two-factor Hull–White adds a second process for the humped vol structure.

2.5 Affine term structure (ATS) - the unifying machinery (Björk Prop 24.2; BM Ch3)

If $\mu (t,r)=\alpha (t)r+\beta (t)$ and ${\sigma }^2(t,r)=\gamma (t)r+\delta (t)$ (affine drift & variance), then $P=A(t,T)e^{-B(t,T)r}$ with $B,A$ solving Riccati ODEs:

$$B_t+\alpha B-\frac{1}{2}\gamma B^2=-1,B(T,T)=0;A_t=\beta B-\frac{1}{2}\delta B^2,A(T,T)=0.$$

All of Vasicek, CIR, Ho-Lee, Hull–White are affine; Dothan and Black-Derman-Toy are not.

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3. Computational Implementation - closed forms vs MC, and the exact-fit HW

Verify the closed-form bonds against Monte Carlo of $r$, and confirm Hull–White’s exact fit. Stdlib only.

Closed forms and MC agree to $O(10^{-4})$ (MC sampling error). The Hull–White $\theta (t)$ reconstruction reproduces the target curve - the model fits today’s forwards by construction, which is its reason for being.

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4. Failure Modes & First-Principles Breakdowns

1. Vasicek negative rates. Gaussian $r$ means $\mathbb{P}(r<0)>0$; for low-rate or negative-rate regimes this is more than a theoretical blemish. CIR fixes it (at a cost: heavier calibration); negative-rate regimes force shifted-lognormal/Bachelier handling anyway (05).
2. Market price of risk is not identified by the model. Bond prices depend on $\lambda$ only through the $\mathbb{Q}$-drift $\mu -\lambda \sigma$. From market bond prices alone you cannot separate $\lambda$ from the real-world drift - this is a fundamental identification problem, not a numerical one.
3. Equilibrium vs no-arbitrage models. Vasicek/CIR are equilibrium models: their implied curve rarely matches today’s market curve. Hull–White is a no-arbitrage model: $\theta (t)$ forces exact fit. Using Vasicek to price a swap against the live curve misprices the residual.
4. HW calibration differentiation instability. Recovering $\theta (t)$ needs numerical derivatives of the market forward curve (up to 3rd order in Shreve Ch30 Remark 30.1); noisy or coarsely sampled curves make the fit numerically unstable.

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5. References

• Brigo–Mercurio, Interest Rate Models
• Björk, Arbitrage Theory in Continuous Time
• Shreve, Stochastic Calculus for Finance I
• Hull, Options, Futures, and Other Derivatives

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6. Connected Graph Bridges

• Back: 02 · Bonds, Yield Curve & Forward Rates
• Forward: 04 · Numeraire, HJM & Market Models · Index Hub
• Base: Stochastic Calculus & Itô