The Vasicek Model¶

The Vasicek model (1977) is the foundational mean-reverting short-rate model. Its Gaussian structure yields closed-form bond prices and option formulas, making it a benchmark for understanding more complex models.

Model Specification¶

Risk-Neutral Dynamics¶

Under the risk-neutral measure \(\mathbb{Q}\), the short rate follows an Ornstein-Uhlenbeck (OU) process:

\[ dr_t = \kappa(\theta - r_t) \, dt + \sigma \, dW_t^{\mathbb{Q}} \]

where: - \(\kappa > 0\): mean-reversion speed (rate of pull toward \(\theta\)) - \(\theta\): long-run mean level (equilibrium rate) - \(\sigma > 0\): volatility (instantaneous standard deviation) - \(W_t^{\mathbb{Q}}\): standard Brownian motion under \(\mathbb{Q}\)

Physical Dynamics¶

Under the physical measure \(\mathbb{P}\):

\[ dr_t = \kappa(\theta^{\mathbb{P}} - r_t) \, dt + \sigma \, dW_t^{\mathbb{P}} \]

The relationship between measures:

\[ \theta = \theta^{\mathbb{P}} - \frac{\lambda \sigma}{\kappa} \]

where \(\lambda\) is the market price of interest rate risk.

Solution of the SDE¶

Explicit Solution¶

The Vasicek SDE has the explicit solution:

\[ r_t = r_0 e^{-\kappa t} + \theta(1 - e^{-\kappa t}) + \sigma \int_0^t e^{-\kappa(t-s)} \, dW_s \]

Derivation:

Let \(Y_t = r_t e^{\kappa t}\). Then:

\[ dY_t = e^{\kappa t} dr_t + \kappa e^{\kappa t} r_t \, dt = e^{\kappa t}[\kappa(\theta - r_t) + \kappa r_t] dt + \sigma e^{\kappa t} dW_t = \kappa \theta e^{\kappa t} dt + \sigma e^{\kappa t} dW_t \]

Integrating:

\[ Y_t = Y_0 + \kappa \theta \int_0^t e^{\kappa s} ds + \sigma \int_0^t e^{\kappa s} dW_s = r_0 + \theta(e^{\kappa t} - 1) + \sigma \int_0^t e^{\kappa s} dW_s \]

Therefore:

\[ r_t = Y_t e^{-\kappa t} = r_0 e^{-\kappa t} + \theta(1 - e^{-\kappa t}) + \sigma e^{-\kappa t} \int_0^t e^{\kappa s} dW_s \]

Distribution of r_t¶

Since \(r_t\) is a linear functional of Brownian motion, it is normally distributed:

\[ r_t \mid r_0 \sim \mathcal{N}(m(t), v(t)) \]

where:

\[ m(t) = \mathbb{E}[r_t] = r_0 e^{-\kappa t} + \theta(1 - e^{-\kappa t}) \]

\[ v(t) = \text{Var}(r_t) = \frac{\sigma^2}{2\kappa}(1 - e^{-2\kappa t}) \]

Long-Run Behavior¶

As \(t \to \infty\):

\[ \mathbb{E}[r_t] \to \theta, \quad \text{Var}(r_t) \to \frac{\sigma^2}{2\kappa} \]

The stationary distribution is:

\[ r_\infty \sim \mathcal{N}\left(\theta, \frac{\sigma^2}{2\kappa}\right) \]

Key Properties¶

Mean Reversion¶

The drift \(\kappa(\theta - r_t)\): - Pulls \(r_t\) toward \(\theta\) when \(r_t \neq \theta\) - Pull strength is proportional to deviation \(|r_t - \theta|\) - Half-life of mean reversion: \(t_{1/2} = \frac{\ln 2}{\kappa}\)

Example: If \(\kappa = 0.5\), the half-life is approximately 1.4 years.

Gaussianity¶

The short rate \(r_t\) is normally distributed at all times. This implies: - Negative rates are possible: \(\mathbb{P}(r_t < 0) > 0\) - Simple closed-form results: Gaussian integrals are tractable - Limited smile/skew: Cannot capture observed volatility patterns

Time Homogeneity¶

The Vasicek model has constant parameters \((\kappa, \theta, \sigma)\). This limits flexibility but ensures stability.

Zero-Coupon Bond Pricing¶

Affine Structure¶

The bond price takes the exponential-affine form:

\[ \boxed{P(t, T) = A(t, T) \exp(-B(t, T) \cdot r_t)} \]

where \(A(t, T) > 0\) and \(B(t, T) > 0\) are deterministic functions.

Derivation via PDE¶

The bond price \(P(t, T, r)\) satisfies the PDE:

\[ \frac{\partial P}{\partial t} + \kappa(\theta - r) \frac{\partial P}{\partial r} + \frac{1}{2}\sigma^2 \frac{\partial^2 P}{\partial r^2} = rP \]

with \(P(T, T, r) = 1\).

Ansatz: Guess \(P = A(\tau) e^{-B(\tau) r}\) where \(\tau = T - t\) (time to maturity).

Substituting into the PDE:

\[ -A'(\tau) e^{-Br} + A \cdot B'(\tau) r \cdot e^{-Br} + \kappa(\theta - r)(-AB e^{-Br}) + \frac{1}{2}\sigma^2 A B^2 e^{-Br} = r A e^{-Br} \]

Dividing by \(A e^{-Br}\):

\[ -\frac{A'}{A} + B' r - \kappa \theta B + \kappa r B + \frac{1}{2}\sigma^2 B^2 = r \]

Collecting terms in \(r\):

Coefficient of \(r\): \(B' + \kappa B = 1\)
Constant term: \(-\frac{A'}{A} - \kappa \theta B + \frac{1}{2}\sigma^2 B^2 = 0\)

ODE for B(τ)¶

\[ \frac{dB}{d\tau} = 1 - \kappa B, \quad B(0) = 0 \]

Solution:

\[ \boxed{B(\tau) = \frac{1 - e^{-\kappa \tau}}{\kappa}} \]

ODE for A(τ)¶

\[ \frac{d \log A}{d\tau} = \kappa \theta B - \frac{1}{2}\sigma^2 B^2, \quad A(0) = 1 \]

Integrating:

\[ \log A(\tau) = \kappa \theta \int_0^\tau B(s) \, ds - \frac{\sigma^2}{2} \int_0^\tau B(s)^2 \, ds \]

After computation:

\[ \boxed{A(\tau) = \exp\left[\left(\theta - \frac{\sigma^2}{2\kappa^2}\right)(B(\tau) - \tau) - \frac{\sigma^2 B(\tau)^2}{4\kappa}\right]} \]

Complete Bond Price Formula¶

\[ P(t, T) = \exp\left[\left(\theta - \frac{\sigma^2}{2\kappa^2}\right)(B(\tau) - \tau) - \frac{\sigma^2 B(\tau)^2}{4\kappa} - B(\tau) r_t\right] \]

where \(\tau = T - t\) and \(B(\tau) = \frac{1 - e^{-\kappa \tau}}{\kappa}\).

Yield Curve Analysis¶

Zero Rate¶

The continuously compounded zero rate is:

\[ z(t, T) = -\frac{\log P(t, T)}{\tau} = \frac{B(\tau)}{\tau} r_t + \frac{\tau - B(\tau)}{\tau}\left(\theta - \frac{\sigma^2}{2\kappa^2}\right) + \frac{\sigma^2 B(\tau)^2}{4\kappa \tau} \]

Asymptotic Long Rate¶

As \(T \to \infty\) (i.e., \(\tau \to \infty\)):

\[ B(\tau) \to \frac{1}{\kappa}, \quad \frac{B(\tau)}{\tau} \to 0 \]

The long rate converges to:

\[ z_\infty = \theta - \frac{\sigma^2}{2\kappa^2} \]

This is the asymptotic yield.

Yield Curve Shapes¶

The Vasicek model can produce:

Shape	Condition
Upward sloping	\(r_0 < z_\infty\)
Flat	\(r_0 = z_\infty\)
Downward sloping	\(r_0 > z_\infty\)
Humped	Possible for certain parameter combinations

Option Pricing¶

Bond Option Formula¶

A European call option on a \(T_2\)-bond with strike \(K\), expiring at \(T_1 < T_2\), has price:

\[ C(0, T_1, T_2, K) = P(0, T_2) N(d_1) - K P(0, T_1) N(d_2) \]

where:

\[ d_1 = \frac{1}{\sigma_P} \log \frac{P(0, T_2)}{K P(0, T_1)} + \frac{\sigma_P}{2} \]

\[ d_2 = d_1 - \sigma_P \]

and the bond price volatility is:

\[ \sigma_P = \sigma B(T_2 - T_1) \sqrt{\frac{1 - e^{-2\kappa T_1}}{2\kappa}} \]

Put Option (Put-Call Parity)¶

\[ P_{\text{opt}} = K P(0, T_1) N(-d_2) - P(0, T_2) N(-d_1) \]

Caps and Floors¶

A caplet can be viewed as a put option on a bond. The Vasicek model provides closed-form caplet prices, enabling analytic cap pricing.

Limitations¶

Negative Rates¶

The Gaussian distribution implies:

\[ \mathbb{P}(r_t < 0) = \Phi\left(\frac{-m(t)}{\sqrt{v(t)}}\right) > 0 \]

While negative rates are now observed in some markets, the model doesn't constrain their probability.

Constant Parameters¶

The basic Vasicek model cannot fit an arbitrary initial yield curve—only specific shapes consistent with the parameters.

Limited Volatility Structure¶

Constant volatility across rates
Cannot capture volatility smile/skew
Homoskedastic rate changes

Hull-White Extension¶

The Hull-White model extends Vasicek with time-dependent \(\theta(t)\):

\[ dr_t = \kappa(\theta(t) - r_t) \, dt + \sigma \, dW_t \]

The function \(\theta(t)\) is calibrated to match the initial yield curve exactly:

\[ \theta(t) = \frac{1}{\kappa}\frac{\partial f(0, t)}{\partial t} + f(0, t) + \frac{\sigma^2}{2\kappa^2}(1 - e^{-2\kappa t}) \]

where \(f(0, t)\) is the initial instantaneous forward rate.

Calibration¶

To Initial Yield Curve¶

For basic Vasicek, minimize:

\[ \sum_{i} \left(P^{\text{model}}(0, T_i; \kappa, \theta, \sigma, r_0) - P^{\text{market}}(0, T_i)\right)^2 \]

Often, exact fit is not achievable without Hull-White extension.

To Options¶

Given the yield curve, calibrate \(\sigma\) (and possibly \(\kappa\)) to cap/swaption implied volatilities.

Parameter Interpretation¶

Parameter	Typical Values	Effect
\(\kappa\)	0.01 – 0.5	Higher = faster mean reversion
\(\theta\)	0.02 – 0.08	Long-run rate level
\(\sigma\)	0.005 – 0.02	Rate volatility

Key Takeaways¶

Vasicek: \(dr_t = \kappa(\theta - r_t)dt + \sigma dW_t\) (Ornstein-Uhlenbeck)
Rates are Gaussian: closed forms but negative rates possible
Bond prices are exponential-affine: \(P = A(\tau)e^{-B(\tau)r}\)
\(B(\tau) = \frac{1 - e^{-\kappa\tau}}{\kappa}\), explicit \(A(\tau)\) formula
Hull-White extends with \(\theta(t)\) for exact curve fit
Option pricing via Gaussian formulas

Exercises¶

Exercise 1. Solve the Vasicek SDE \(dr_t = \kappa(\theta - r_t)\,dt + \sigma\,dW_t\) by applying Ito's lemma to \(e^{\kappa t}r_t\). Show that \(r_t = \theta + (r_0 - \theta)e^{-\kappa t} + \sigma\int_0^t e^{-\kappa(t-s)}\,dW_s\).

Solution to Exercise 1

Define \(Y_t = e^{\kappa t} r_t\). By Ito's lemma (with \(f(t,r) = e^{\kappa t}r\)):

\[ dY_t = \kappa e^{\kappa t} r_t\,dt + e^{\kappa t}\,dr_t \]

Substituting \(dr_t = \kappa(\theta - r_t)\,dt + \sigma\,dW_t\):

\[ dY_t = \kappa e^{\kappa t} r_t\,dt + e^{\kappa t}[\kappa(\theta - r_t)\,dt + \sigma\,dW_t] \]

\[ = \kappa e^{\kappa t} r_t\,dt + \kappa\theta\, e^{\kappa t}\,dt - \kappa e^{\kappa t} r_t\,dt + \sigma e^{\kappa t}\,dW_t \]

The \(r_t\) terms cancel:

\[ dY_t = \kappa\theta\, e^{\kappa t}\,dt + \sigma e^{\kappa t}\,dW_t \]

Integrating from \(0\) to \(t\):

\[ Y_t - Y_0 = \kappa\theta \int_0^t e^{\kappa s}\,ds + \sigma \int_0^t e^{\kappa s}\,dW_s \]

Since \(Y_0 = r_0\) and \(\int_0^t e^{\kappa s}\,ds = \frac{e^{\kappa t} - 1}{\kappa}\):

\[ e^{\kappa t}r_t = r_0 + \theta(e^{\kappa t} - 1) + \sigma \int_0^t e^{\kappa s}\,dW_s \]

Dividing by \(e^{\kappa t}\):

\[ r_t = r_0 e^{-\kappa t} + \theta(1 - e^{-\kappa t}) + \sigma \int_0^t e^{-\kappa(t-s)}\,dW_s \]

This can also be written as \(r_t = \theta + (r_0 - \theta)e^{-\kappa t} + \sigma\int_0^t e^{-\kappa(t-s)}\,dW_s\), confirming the stated result.

Exercise 2. Show that \(r_t\) in the Vasicek model is normally distributed with mean \(\mathbb{E}[r_t] = \theta + (r_0 - \theta)e^{-\kappa t}\) and variance \(\text{Var}(r_t) = \frac{\sigma^2}{2\kappa}(1 - e^{-2\kappa t})\). What is the stationary distribution as \(t \to \infty\)?

Solution to Exercise 2

From the solution in Exercise 1, \(r_t = \theta + (r_0 - \theta)e^{-\kappa t} + \sigma\int_0^t e^{-\kappa(t-s)}\,dW_s\). The stochastic integral \(\int_0^t e^{-\kappa(t-s)}\,dW_s\) is a Gaussian random variable (as a Wiener integral of a deterministic integrand).

Mean. Since \(\mathbb{E}[\int_0^t e^{-\kappa(t-s)}\,dW_s] = 0\):

\[ \mathbb{E}[r_t] = \theta + (r_0 - \theta)e^{-\kappa t} = r_0 e^{-\kappa t} + \theta(1 - e^{-\kappa t}) \]

Variance. By the Ito isometry:

\[ \text{Var}(r_t) = \sigma^2 \int_0^t e^{-2\kappa(t-s)}\,ds = \sigma^2 \left[\frac{e^{-2\kappa(t-s)}}{2\kappa}\right]_{s=0}^{s=t} = \frac{\sigma^2}{2\kappa}(1 - e^{-2\kappa t}) \]

Therefore \(r_t \mid r_0 \sim \mathcal{N}(m(t), v(t))\) with

\[ m(t) = \theta + (r_0 - \theta)e^{-\kappa t}, \quad v(t) = \frac{\sigma^2}{2\kappa}(1 - e^{-2\kappa t}) \]

Stationary distribution. As \(t \to \infty\), \(e^{-\kappa t} \to 0\) and \(e^{-2\kappa t} \to 0\), so

\[ m(\infty) = \theta, \quad v(\infty) = \frac{\sigma^2}{2\kappa} \]

The stationary distribution is \(r_\infty \sim \mathcal{N}\!\left(\theta, \frac{\sigma^2}{2\kappa}\right)\).

Exercise 3. For parameters \(\kappa = 0.5\), \(\theta = 0.04\), \(\sigma = 0.01\), \(r_0 = 0.03\), compute the mean and standard deviation of \(r_t\) at \(t = 1, 5, 10\). What is the probability that \(r_{10} < 0\)?

Solution to Exercise 3

With \(\kappa = 0.5\), \(\theta = 0.04\), \(\sigma = 0.01\), \(r_0 = 0.03\):

\[ m(t) = 0.04 + (0.03 - 0.04)e^{-0.5t} = 0.04 - 0.01\,e^{-0.5t} \]

\[ v(t) = \frac{0.0001}{1.0}(1 - e^{-t}) = 0.0001(1 - e^{-t}) \]

At \(t = 1\): \(e^{-0.5} \approx 0.6065\), \(e^{-1} \approx 0.3679\)

\[ m(1) = 0.04 - 0.01(0.6065) = 0.03394 \]

\[ \text{Std}(r_1) = \sqrt{0.0001(1 - 0.3679)} = \sqrt{0.0000632} \approx 0.00795 \]

At \(t = 5\): \(e^{-2.5} \approx 0.08209\), \(e^{-5} \approx 0.00674\)

\[ m(5) = 0.04 - 0.01(0.08209) = 0.03918 \]

\[ \text{Std}(r_5) = \sqrt{0.0001(1 - 0.00674)} = \sqrt{0.00009932} \approx 0.00997 \]

At \(t = 10\): \(e^{-5} \approx 0.00674\), \(e^{-10} \approx 0.0000454\)

\[ m(10) = 0.04 - 0.01(0.00674) = 0.03993 \]

\[ \text{Std}(r_{10}) = \sqrt{0.0001(1 - 0.0000454)} \approx \sqrt{0.00009999} \approx 0.01000 \]

Probability \(r_{10} < 0\):

\[ \mathbb{P}(r_{10} < 0) = \Phi\!\left(\frac{0 - 0.03993}{0.01000}\right) = \Phi(-3.993) \approx 3.3 \times 10^{-5} \]

This is extremely small (about 0.003%), reflecting that \(\theta = 4\%\) is about 4 standard deviations above zero.

Exercise 4. Derive the Vasicek bond pricing formula \(P(t,T) = e^{A(\tau) - B(\tau)r_t}\) where \(\tau = T - t\), \(B(\tau) = \frac{1-e^{-\kappa\tau}}{\kappa}\), by computing \(\mathbb{E}_t^{\mathbb{Q}}[e^{-\int_t^T r_s\,ds}]\) using the Gaussian distribution of \(\int_t^T r_s\,ds\).

Solution to Exercise 4

The bond price is \(P(t,T) = \mathbb{E}_t^{\mathbb{Q}}[e^{-\int_t^T r_s\,ds}]\). We need the distribution of \(I = \int_t^T r_s\,ds\).

Using \(r_s = \theta + (r_t - \theta)e^{-\kappa(s-t)} + \sigma\int_t^s e^{-\kappa(s-u)}\,dW_u\):

\[ I = \int_t^T r_s\,ds = \theta\tau + (r_t - \theta)\int_t^T e^{-\kappa(s-t)}\,ds + \sigma\int_t^T\!\int_t^s e^{-\kappa(s-u)}\,dW_u\,ds \]

where \(\tau = T - t\). The deterministic integral gives

\[ \int_t^T e^{-\kappa(s-t)}\,ds = \frac{1 - e^{-\kappa\tau}}{\kappa} = B(\tau) \]

Since \(I\) is a Gaussian random variable (as a linear functional of Brownian motion), we have \(I \sim \mathcal{N}(\mathbb{E}[I], \text{Var}(I))\).

Mean:

\[ \mathbb{E}[I \mid r_t] = \theta\tau + (r_t - \theta)B(\tau) \]

Variance: By Fubini and the Ito isometry (swapping the order of integration in the double stochastic integral):

\[ \text{Var}(I) = \frac{\sigma^2}{\kappa^2}\left[\tau - 2B(\tau) + \frac{1 - e^{-2\kappa\tau}}{2\kappa}\right] \]

Since \(e^{-I}\) is the exponential of a Gaussian, we use the moment generating function \(\mathbb{E}[e^{-I}] = e^{-\mathbb{E}[I] + \frac{1}{2}\text{Var}(I)}\):

\[ P(t,T) = \exp\!\left(-\mathbb{E}[I] + \frac{1}{2}\text{Var}(I)\right) \]

\[ = \exp\!\left(-\theta\tau - (r_t - \theta)B(\tau) + \frac{\sigma^2}{2\kappa^2}\left[\tau - 2B(\tau) + \frac{1-e^{-2\kappa\tau}}{2\kappa}\right]\right) \]

Rearranging into \(e^{A(\tau) - B(\tau)r_t}\) form with \(B(\tau) = \frac{1-e^{-\kappa\tau}}{\kappa}\):

\[ A(\tau) = \left(\theta - \frac{\sigma^2}{2\kappa^2}\right)(B(\tau) - \tau) - \frac{\sigma^2}{4\kappa}B(\tau)^2 \]

This completes the derivation via the expectation (moment generating function) approach.

Exercise 5. Using the parameters from Exercise 3, compute the zero-coupon bond price \(P(0, 5)\) and the corresponding 5-year zero rate. Plot the yield curve \(R(0, T)\) for \(T = 1, 2, \ldots, 30\) and identify whether it is normal, inverted, or humped.

Solution to Exercise 5

Using \(\kappa = 0.5\), \(\theta = 0.04\), \(\sigma = 0.01\), \(r_0 = 0.03\), we compute \(P(0,5)\) and the 5-year zero rate.

\(B(5) = \frac{1 - e^{-2.5}}{0.5} = \frac{1 - 0.08209}{0.5} = 1.8358\)

\(A(5)\):

\[ \ln A(5) = \left(0.04 - \frac{0.0001}{0.5}\right)(1.8358 - 5) - \frac{0.0001 \times 1.8358^2}{2.0} \]

\[ = 0.0398 \times (-3.1642) - 0.0001685 = -0.12593 - 0.00017 = -0.12610 \]

Bond price:

\[ \ln P(0,5) = -0.12610 - 1.8358 \times 0.03 = -0.12610 - 0.05507 = -0.18117 \]

\[ P(0,5) = e^{-0.18117} \approx 0.8343 \]

5-year zero rate:

\[ R(0,5) = -\frac{\ln P(0,5)}{5} = \frac{0.18117}{5} = 0.03623 = 3.623\% \]

Yield curve shape. The asymptotic long rate is

\[ z_\infty = \theta - \frac{\sigma^2}{2\kappa^2} = 0.04 - \frac{0.0001}{0.5} = 0.04 - 0.0002 = 0.0398 \]

Since \(r_0 = 0.03 < z_\infty = 0.0398\), the yield curve is upward sloping (normal). Short rates start at 3% and yields rise toward the asymptote of approximately 3.98%. The curve is not humped for these parameters. Computing a few representative points: \(R(0,1) \approx 3.05\%\), \(R(0,5) \approx 3.62\%\), \(R(0,10) \approx 3.81\%\), \(R(0,30) \approx 3.97\%\). The curve flattens as maturity increases, approaching \(z_\infty\) from below.

Exercise 6. The main criticism of the Vasicek model is that it allows negative interest rates. Compute the probability \(\Pr(r_t < 0)\) as a function of \(t\) for the parameters in Exercise 3. At what time horizon does this probability exceed 5%? Discuss whether negative rates invalidate the model in the post-2015 era.

Solution to Exercise 6

With \(\kappa = 0.5\), \(\theta = 0.04\), \(\sigma = 0.01\), \(r_0 = 0.03\), the probability of negative rates is

\[ \mathbb{P}(r_t < 0) = \Phi\!\left(\frac{-m(t)}{\sqrt{v(t)}}\right) \]

where \(m(t) = 0.04 - 0.01\,e^{-0.5t}\) and \(v(t) = 0.0001(1 - e^{-t})\).

The ratio is

\[ \frac{-m(t)}{\sqrt{v(t)}} = \frac{-(0.04 - 0.01\,e^{-0.5t})}{0.01\sqrt{1 - e^{-t}}} \]

At \(t = 1\): numerator \(= -0.03394\), denominator \(= 0.01 \times 0.7953 = 0.007953\), ratio \(= -4.27\), so \(\Phi(-4.27) \approx 10^{-5}\).

At \(t = 10\): numerator \(= -0.03993\), denominator \(= 0.01 \times 0.99998 \approx 0.01\), ratio \(= -3.99\), so \(\Phi(-3.99) \approx 3.3 \times 10^{-5}\).

At \(t = 50\): numerator \(\approx -0.04\), denominator \(\approx 0.01\), ratio \(= -4.0\), so \(\Phi(-4.0) \approx 3.2 \times 10^{-5}\).

As \(t \to \infty\), the ratio converges to \(-\theta/\sqrt{\sigma^2/(2\kappa)} = -0.04/0.01 = -4.0\), so the long-run probability is \(\Phi(-4) \approx 3.2 \times 10^{-5}\).

Time to exceed 5%. For \(\mathbb{P}(r_t < 0) > 0.05\), we need \(\Phi^{-1}(0.05) = -1.645\), i.e.,

\[ \frac{-m(t)}{\sqrt{v(t)}} > -1.645 \]

With these parameters, the ratio starts at \(-\infty\) (when \(v(0) = 0\)) and decreases toward \(-4.0\). Since \(-4.0 < -1.645\), the probability never exceeds 5% for these parameter values. The mean (\(\approx 4\%\)) is always about 4 standard deviations (\(\approx 1\%\)) above zero.

For the probability to exceed 5%, one would need \(\theta/\sqrt{\sigma^2/(2\kappa)} < 1.645\), i.e., \(\theta < 1.645\sigma/\sqrt{2\kappa}\). With \(\sigma = 0.01\) and \(\kappa = 0.5\): \(\theta < 1.645 \times 0.01 \approx 0.01645\). So the 5% threshold would only be reached for \(\theta < 1.645\%\).

Post-2015 perspective. Between 2012 and 2022, several major economies (Eurozone, Japan, Switzerland, Sweden) experienced negative policy rates, with the ECB deposit rate reaching \(-0.50\%\). This demonstrates that negative rates are not merely a mathematical artifact but an empirical reality. In this context, the Vasicek model's ability to produce negative rates is arguably a feature rather than a bug. The Gaussian framework correctly captures the possibility of negative rates while maintaining analytical tractability. For pricing in negative-rate environments, the Vasicek/Hull-White framework is often preferred over lognormal models (like Black-Karasinski), which cannot generate negative rates by construction.