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Michael Brenndoerfer

Calibration and Parameter Estimation in Models › Calibrating Short-Rate Models to the Yield Curve

Michael BrenndoerferDecember 13, 20258 min read

Quantitative FinanceAll chapters

Part 8 of 15Econometrics and Statistical Modeling

Chapter 45Calibration and Parameter Estimation in Models › Calibrating Short-Rate Models to the Yield Curve

Reading section 7 of 17 · Calibrating Short-Rate Models to the Yield Curve · 8 minSection 7 of 17 · 8 min

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Short-rate models specify the dynamics of the instantaneous interest rate. For market-consistent pricing, fit the current term structure as closely as the chosen family permits; constant-parameter Vasicek generally cannot match an arbitrary initial curve exactly. In this example the Vasicek coefficients are under the risk-neutral pricing measure $Q$ , rather than historical estimates under $P$ . The bond pays one default-free unit at maturity.

The Vasicek model is one of the foundational short-rate models, valued for its analytical tractability and intuitive interpretation. It specifies the evolution of the instantaneous short rate through the following stochastic differential equation:

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

where:

$r_t$ : instantaneous short rate at time $t$
$\kappa$ : mean reversion speed, determining how quickly the rate returns to equilibrium
$\theta$ : long-term mean rate, the equilibrium level toward which rates are pulled
$\sigma$ : volatility of the rate, controlling the magnitude of random fluctuations
$dW_t$ : standard Brownian motion driving the random component

For $\kappa>0$ , the drift $\kappa(\theta-r_t)$ pulls the conditional mean toward $\theta$ : it is negative above that level and positive below it. Larger $\kappa$ gives faster conditional-mean reversion. This is the model's idealization, not a claim that market rates or Gaussian sample paths stay in a bounded range.

Under the Vasicek model, zero-coupon bond prices have a closed-form solution. This analytical tractability is one of the model's key advantages, enabling rapid calibration and pricing:

P(t, T) = A(t, T) e^{-B(t, T) r_t}

where:

$P(t, T)$ : price at time $t$ of a bond maturing at $T$
$r_t$ : instantaneous short rate at time $t$
$t$ : current time
$T$ : maturity time
$\kappa, \theta, \sigma$ : Vasicek model parameters
$B(t, T) = \frac{1 - e^{-\kappa(T-t)}}{\kappa}$ : duration-like factor that captures how bond prices respond to rate changes
$A(t, T) = \exp\left[\left(\theta - \frac{\sigma^2}{2\kappa^2}\right)(B(t, T) - (T-t)) - \frac{\sigma^2}{4\kappa}B(t, T)^2\right]$ : exponential-affine prefactor incorporating the expected mean-reverting rate path and variance-induced convexity

The function $B(t, T)$ resembles modified duration in some ways, measuring the sensitivity of the bond price to changes in the short rate. The function $A(t, T)$ incorporates convexity effects and the impact of the long-term mean on bond pricing.

The continuously compounded yield can be derived from the bond price formula:

\begin{aligned} R(t, T) &= -\frac{\ln P(t, T)}{T - t} \\ &= -\frac{\ln\left(A(t, T) e^{-B(t, T) r_t}\right)}{T - t} && \text{(substitute bond price)} \\ &= \frac{B(t, T) r_t - \ln A(t, T)}{T - t} && \text{(simplify)} \end{aligned}

where:

$R(t, T)$ : continuously compounded yield for maturity $T$
$P(t, T)$ : zero-coupon bond price
$A(t, T), B(t, T)$ : Vasicek bond price coefficients defined above
$r_t$ : instantaneous short rate
$t$ : current time
$T$ : maturity time

This relationship translates the pricing-measure parameters into observable yields. For stable numerical evaluation, the code uses the Gaussian integral $I=\int_0^T r_s\,ds$ . Its mean is $m=r_0B+\theta(T-B)$ and its variance is $v=\sigma^2\int_0^T[(1-e^{-\kappa u})/\kappa]^2\,du$ , so $\log P=-m+v/2$ . At $\kappa=0$ , this becomes $-r_0T+\sigma^2T^3/6$ , with a positive variance correction. Near zero, a series for the integrated variance avoids subtracting large cancelling terms; yields use log prices directly.

In[14]:

Code

from math import expm1, factorial, fsum


def vasicek_log_bond_price(r0, T, kappa, theta, sigma):
    """Log price of a unit zero-coupon bond under risk-neutral Vasicek."""
    if not all(np.isfinite(x) for x in (r0, T, kappa, theta, sigma)):
        raise ValueError("Vasicek inputs must be finite")
    if T <= 0 or kappa < 0 or sigma < 0:
        raise ValueError("Require T > 0, kappa >= 0 and sigma >= 0")
    x = kappa * T
    if not np.isfinite(x):
        raise ValueError("kappa * T is outside the supported range")
    if x < 0.1:
        b = fsum((-x) ** n / factorial(n + 1) for n in range(13))
        one_minus_b = fsum(
            (-1) ** (n + 1) * x**n / factorial(n + 1) for n in range(1, 13)
        )
        j = fsum(
            (-1) ** n * (2 ** (n + 2) - 2) * x**n / (factorial(n + 2) * (n + 3))
            for n in range(13)
        )
    else:
        b = -expm1(-x) / x
        one_minus_b = 1 - b
        b2 = -expm1(-2 * x) / (2 * x)
        j = fsum((1.0, -2 * b, b2)) / x**2
    mean_integral = T * (r0 * b + theta * one_minus_b)
    variance_integral = sigma**2 * T**3 * j
    log_price = -mean_integral + variance_integral / 2
    if not np.isfinite(log_price):
        raise ValueError("Unrepresentable Vasicek log price")
    return log_price


def vasicek_bond_price(r0, T, kappa, theta, sigma):
    """Unit bond price, with an explicit finite-representation check."""
    with np.errstate(over="raise", under="raise"):
        return float(np.exp(vasicek_log_bond_price(r0, T, kappa, theta, sigma)))


def vasicek_yield(r0, T, kappa, theta, sigma):
    """Continuously compounded yield, evaluated directly from log price."""
    return -vasicek_log_bond_price(r0, T, kappa, theta, sigma) / T

Let's create a synthetic market yield curve and calibrate the Vasicek model to it:

In[15]:

Code

# Market yield curve (maturities in years)
maturities = np.array([0.25, 0.5, 1, 2, 3, 5, 7, 10])
market_yields = np.array([0.02, 0.022, 0.025, 0.028, 0.030, 0.033, 0.035, 0.037])


def vasicek_calibration_objective(params, maturities, market_yields):
    """Squared yield errors in basis points."""
    maturities = np.asarray(maturities, dtype=float)
    market_yields = np.asarray(market_yields, dtype=float)
    if (
        maturities.ndim != 1
        or market_yields.ndim != 1
        or maturities.size == 0
        or maturities.shape != market_yields.shape
        or not np.all(np.isfinite(maturities))
        or not np.all(maturities > 0)
        or not np.all(np.isfinite(market_yields))
    ):
        raise ValueError("Require paired finite yields and positive maturities")
    params = np.asarray(params, dtype=float)
    if params.shape != (4,) or not np.all(np.isfinite(params)):
        return np.inf
    r0, kappa, theta, sigma = params
    if kappa < 0 or sigma < 0:
        return np.inf
    try:
        model = np.array(
            [vasicek_yield(r0, T, kappa, theta, sigma) for T in maturities]
        )
    except (ValueError, FloatingPointError, OverflowError):
        return np.inf
    if not np.all(np.isfinite(model)):
        return np.inf
    return float(np.sum(((model - market_yields) * 10000) ** 2))


# Initial guess
x0_vasicek = [0.02, 0.5, 0.03, 0.01]

# Bounds
bounds_vasicek = [(0.001, 0.10), (0.01, 2.0), (0.01, 0.10), (0.001, 0.10)]

# Calibrate
result_vasicek = minimize(
    vasicek_calibration_objective,
    x0_vasicek,
    args=(maturities, market_yields),
    method="L-BFGS-B",
    bounds=bounds_vasicek,
    options={"ftol": 1e-12, "gtol": 1e-6, "maxiter": 2000},
)

require_successful_fit(result_vasicek, "Vasicek calibration")
r0_cal, kappa_cal, theta_cal, sigma_cal = result_vasicek.x

Out[16]:

Console

Vasicek Calibration Results:
----------------------------------------
r0 (short rate):     0.0189 (1.89%)
kappa (mean rev.):   0.4418
theta (long-term):   0.0494 (4.94%)
sigma (volatility):  0.0591

Calibration SSE (bps squared): 112.64440100
RMSE (yield): 3.75 bps

Out[18]:

Visualization

Scatter plot of market yield observations overlaid with the calibrated Vasicek yield curve across maturities from 0.25 to 10 years. — Vasicek fit to a synthetic yield curve. Compare the fitted curve with its target yields and inspect the residuals and optimizer diagnostics before assessing adequacy.

Out[20]:

Visualization

Bar chart of Vasicek model calibration residuals in basis points across maturities from 0.25 to 10 years, showing small positive and negative deviations around zero. — Residuals of this Vasicek fit in basis points. Their size reflects numerical fitting as well as model restrictions; assess scaling, tolerances and starting points before attributing remaining structure to the model family.

The printed values are fitted parameters for this synthetic curve, not a validation of economic realism. Inspect the numerical objective, stopping diagnostics and sensitivity to scaling and starting points before interpreting residual patterns. Constant-parameter Vasicek restricts the shapes it can represent, but the residuals of one finite run also contain optimization error. Time-dependent extensions such as Hull-White can preserve an exact fit to a suitable initial curve.