Calibration to Yield Curve and Caps¶
Calibration is the process of determining model parameters so that the model reproduces observed market prices. For the Vasicek model with three parameters \((\kappa, \theta, \sigma)\), calibration targets fall into two categories: the yield curve (bond prices or equivalently zero rates) and volatility instruments (cap and swaption prices). This section develops both calibration approaches, presents maximum likelihood estimation for historical data, and discusses the fundamental tension between fitting the initial curve and fitting option prices with a time-homogeneous model.
Calibration to the yield curve¶
Objective function¶
Given observed zero-coupon bond prices \(P^{\text{mkt}}(0, T_i)\) for maturities \(T_1, \ldots, T_n\), the calibration minimizes the weighted sum of squared errors:
where \(P^{\text{model}}(0, T_i) = A(T_i)\,e^{-B(T_i)\,r_0}\) is the Vasicek bond price and \(w_i > 0\) are weights (often \(w_i = 1\) or \(w_i = 1/P^{\text{mkt}}(0,T_i)^2\) for relative pricing errors).
Equivalent yield-based formulation¶
Equivalently, one can minimize yield errors:
where \(R^{\text{model}}(0, T_i) = -\ln P^{\text{model}}(0, T_i)/T_i\). The yield formulation has the advantage that errors are on a comparable scale across maturities.
The initial rate¶
The current short rate \(r_0\) can be treated as:
- Observable: Set \(r_0\) equal to the overnight rate (e.g., the Fed Funds rate or SOFR). Then only \((\kappa, \theta, \sigma)\) are free parameters.
- Free parameter: Include \(r_0\) in the optimization, giving four parameters. This provides a better fit but may produce \(r_0\) inconsistent with the observed overnight rate.
Fundamental limitation¶
The Vasicek model with constant parameters \((\kappa, \theta, \sigma)\) produces a yield curve that is a specific functional form---a weighted average of \(r_0\) and \(R_\infty\) plus a convexity term. This family of curves cannot match an arbitrary observed yield curve exactly. The calibration finds the best fit within this restricted family.
The mismatch is typically largest at intermediate maturities (2--5 years), where the observed curve may have features (kinks, local humps) that the smooth Vasicek curve cannot reproduce.
Exact fit via Hull-White
The Hull-White extension \(dr_t = (\theta(t) - \kappa\,r_t)\,dt + \sigma\,dW_t\) replaces the constant \(\theta\) with a time-dependent function \(\theta(t)\) chosen to match the initial yield curve exactly. This eliminates the yield curve fitting error at the cost of a non-stationary model.
Calibration to cap volatilities¶
Cap prices in the Vasicek model¶
A cap with strike \(\kappa_{\text{cap}}\) consists of caplets, each priced as a ZCB put (as derived in the caplet section):
Implied volatility matching¶
Market cap prices are often quoted in terms of Black implied volatilities \(\sigma_{\text{Black},k}^{\text{mkt}}\). The calibration objective is
where \(\sigma_{\text{Black},k}^{\text{model}}\) is obtained by inverting the Black formula applied to the Vasicek caplet price.
Joint calibration¶
In practice, one calibrates to both yields and volatilities simultaneously:
where \(\lambda_1, \lambda_2 > 0\) control the relative importance of yield fit versus volatility fit.
The three parameters \((\kappa, \theta, \sigma)\) play different roles:
- \(\theta\) and \(r_0\) primarily control the level of the yield curve
- \(\kappa\) controls the slope (how fast yields converge to \(R_\infty\))
- \(\sigma\) controls cap volatilities and the convexity correction
Maximum likelihood estimation¶
Setting¶
Given discrete observations \(r_{t_0}, r_{t_1}, \ldots, r_{t_N}\) at equally spaced times \(\Delta t = t_{i+1} - t_i\), the Vasicek transition density is
where \(\mu_i = \theta + (r_{t_i} - \theta)\,e^{-\kappa\Delta t}\) and \(v^2 = \frac{\sigma^2}{2\kappa}(1 - e^{-2\kappa\Delta t})\).
Log-likelihood function¶
The log-likelihood is
Defining \(\phi = e^{-\kappa\Delta t}\), this can be written as
Closed-form MLE¶
The log-likelihood is that of a linear regression \(r_{t_{i+1}} = a + b\,r_{t_i} + \varepsilon_i\) with \(a = \theta(1-\phi)\), \(b = \phi\), \(\varepsilon_i \sim \mathcal{N}(0, v^2)\). The MLEs are obtained by ordinary least squares:
where \(\bar{r}_{\text{lag}} = \frac{1}{N}\sum r_{t_i}\) and \(\bar{r}_{\text{lead}} = \frac{1}{N}\sum r_{t_{i+1}}\).
The Vasicek parameters are recovered via
Asymptotic standard errors
The Fisher information matrix for the OU regression is available in closed form. The standard errors of \(\hat{\kappa}\), \(\hat{\theta}\), \(\hat{\sigma}\) can be obtained by the delta method applied to the transformation \((a, b, v^2) \to (\kappa, \theta, \sigma)\). For typical weekly data over 10 years (\(N \approx 520\)), the standard error of \(\hat{\kappa}\) is often large, reflecting the difficulty of estimating mean-reversion speed from short time series.
Parameter identifiability¶
The kappa estimation problem¶
Mean-reversion speed \(\kappa\) is notoriously difficult to estimate reliably. The key issue is that \(e^{-\kappa\Delta t}\) is close to \(1\) for typical values of \(\kappa\) and \(\Delta t\):
- Weekly data: \(e^{-\kappa/52} \approx 1 - \kappa/52\)
- For \(\kappa = 0.3\): \(e^{-0.3/52} = 0.9942\)
The regression coefficient \(b\) is close to \(1\), making it hard to distinguish from a unit root (\(b = 1\), no mean reversion). This is the classic near-unit-root problem in econometrics.
Risk-neutral vs physical parameters¶
MLE estimates the physical measure parameters \((\kappa^{\mathbb{P}}, \theta^{\mathbb{P}}, \sigma)\). The risk-neutral parameters \((\kappa^{\mathbb{Q}}, \theta^{\mathbb{Q}}, \sigma)\) that enter bond pricing are related by the market price of risk \(\lambda\):
Cross-sectional calibration to bond prices identifies \((\kappa^{\mathbb{Q}}, \theta^{\mathbb{Q}})\), while time-series estimation identifies \((\kappa^{\mathbb{P}}, \theta^{\mathbb{P}})\). The market price of risk \(\lambda\) connects the two and is identified by combining both data sources.
Practical calibration workflow¶
A typical calibration proceeds as follows:
-
Collect market data: Zero rates \(R^{\text{mkt}}(0, T_i)\) for \(T_i \in \{0.25, 0.5, 1, 2, 3, 5, 7, 10, 20, 30\}\) years, and ATM cap volatilities \(\sigma^{\text{Black}}_k\) for maturities \(\{1, 2, 3, 5, 7, 10\}\) years.
-
Set initial guess: \(\kappa_0 = 0.3\), \(\theta_0 = R^{\text{mkt}}(0, 10)\), \(\sigma_0 = 0.01\).
-
Optimize: Use Nelder-Mead or Levenberg-Marquardt to minimize the joint objective.
-
Validate: Compare model yields and cap vols against market values. Check that the residuals are within bid-ask spreads.
-
Stability check: Perturb the initial guess and verify convergence to the same minimum. Compute parameter standard errors from the Hessian.
Numerical example¶
Market data (stylized):
| Maturity | Market Yield |
|---|---|
| 1Y | 3.50% |
| 2Y | 3.80% |
| 5Y | 4.10% |
| 10Y | 4.30% |
| 30Y | 4.40% |
Calibration result: \(\hat{\kappa} = 0.25\), \(\hat{\theta} = 0.0455\), \(\hat{\sigma} = 0.018\), \(r_0 = 0.034\).
| Maturity | Market Yield | Model Yield | Error (bp) |
|---|---|---|---|
| 1Y | 3.50% | 3.52% | 2 |
| 2Y | 3.80% | 3.76% | \(-4\) |
| 5Y | 4.10% | 4.12% | 2 |
| 10Y | 4.30% | 4.32% | 2 |
| 30Y | 4.40% | 4.43% | 3 |
The fit is within 5 basis points across all maturities---acceptable for a three-parameter model, though not as precise as the exact fit achievable with Hull-White.
Summary¶
Calibration of the Vasicek model involves fitting three parameters \((\kappa, \theta, \sigma)\) to market yields and/or cap volatilities. Yield curve calibration minimizes pricing errors across observed maturities, with the fundamental constraint that a three-parameter model cannot match an arbitrary curve exactly. Cap volatility calibration targets the single parameter \(\sigma\) that drives option prices. Maximum likelihood estimation on historical rate data provides an alternative (physical measure) calibration via the AR(1) regression structure of the OU process, with the caveat that the mean-reversion speed is poorly identified from short time series. The distinction between physical and risk-neutral parameters requires combining cross-sectional and time-series information.
Exercises¶
Exercise 1. Write the yield-space objective function for calibrating Vasicek to 5 market zero rates. How many free parameters are there, and why is the problem over-determined?
Solution to Exercise 1
The yield-space objective function for \(n = 5\) market zero rates at maturities \(T_1, \ldots, T_5\) is:
where \(R^{\text{model}}(0, T_i) = -\ln P^{\text{model}}(0, T_i) / T_i\) and \(P^{\text{model}}(0, T_i) = A(T_i)\,e^{-B(T_i)\,r_0}\).
If \(r_0\) is fixed (set equal to the observed overnight rate), there are three free parameters: \(\kappa\), \(\theta\), and \(\sigma\). If \(r_0\) is also optimized, there are four free parameters.
With three parameters and five equations (one per maturity), the system is over-determined: there are more constraints than unknowns. An exact solution generally does not exist because the Vasicek yield curve belongs to a three-parameter family of curves, which cannot reproduce an arbitrary set of five yields. The least-squares formulation finds the best approximation within this restricted family. The over-determination is actually desirable---it prevents overfitting and ensures that the calibrated parameters reflect the overall shape of the term structure rather than matching individual data points exactly.
Exercise 2. Using the calibrated parameters \(\hat{\kappa} = 0.25\), \(\hat{\theta} = 0.0455\), \(\hat{\sigma} = 0.018\), compute the Vasicek long-run yield \(R_\infty = \theta - \sigma^2/(2\kappa^2)\). Compare this with the 30Y market yield of 4.40%.
Solution to Exercise 2
Using the calibrated parameters \(\hat{\kappa} = 0.25\), \(\hat{\theta} = 0.0455\), \(\hat{\sigma} = 0.018\):
Converting to percentage: \(R_\infty = 4.291\%\).
The 30Y market yield is \(4.40\%\). The difference is:
The model's long-run yield undershoots the 30Y market yield by about 11 basis points. This is expected: the convexity correction \(\sigma^2/(2\kappa^2) \approx 26\) bp pushes \(R_\infty\) below \(\theta\). Since the 30-year yield has not yet fully converged to \(R_\infty\) (convergence is gradual), the model yield at 30Y is slightly above \(R_\infty\) but still below the market yield. This residual error reflects the fundamental limitation of matching an arbitrary yield curve with three parameters.
Exercise 3. In MLE for the Vasicek model, the discretized OU process gives the AR(1) regression \(r_{t+\Delta t} = \alpha + \beta r_t + \epsilon_t\). Express \(\kappa\), \(\theta\), and \(\sigma\) in terms of \(\alpha\), \(\beta\), and \(\text{Var}(\epsilon)\) and \(\Delta t\).
Solution to Exercise 3
The AR(1) regression for the discretized OU process is:
where \(\epsilon_t \sim \mathcal{N}(0, v^2)\), with \(\alpha = \theta(1 - e^{-\kappa\Delta t})\), \(\beta = e^{-\kappa\Delta t}\), and \(v^2 = \frac{\sigma^2}{2\kappa}(1 - e^{-2\kappa\Delta t})\).
Inverting these relationships:
Mean-reversion speed:
Long-run mean:
Volatility: From \(v^2 = \frac{\sigma^2}{2\kappa}(1 - \beta^2)\), solving for \(\sigma\):
Equivalently, \(\sigma^2 = \frac{2\kappa\,\text{Var}(\epsilon)}{1 - e^{-2\kappa\Delta t}}\).
Exercise 4. Explain why the mean-reversion speed \(\kappa\) is poorly identified from short time series. If a 2-year daily sample gives \(\hat{\kappa}^{\mathbb{P}} = 0.05\) with standard error 0.04, what does this imply about the reliability of \(\hat{\kappa}\)?
Solution to Exercise 4
The near-unit-root problem arises because \(\hat{\beta} = e^{-\kappa\Delta t}\) is close to 1 for typical parameter values. When \(\kappa\) is small and \(\Delta t\) is moderate (e.g., daily data), \(\hat{\beta} \approx 1 - \kappa\Delta t \approx 1\), and it becomes statistically difficult to distinguish mean-reverting behavior from a random walk (\(\beta = 1\)).
With a 2-year daily sample (\(N \approx 500\)), the estimate \(\hat{\kappa}^{\mathbb{P}} = 0.05\) with standard error \(0.04\) gives a 95% confidence interval:
The confidence interval includes zero and even negative values. This means we cannot reject the null hypothesis \(\kappa = 0\) (no mean reversion) at the 5% significance level. The \(t\)-statistic is \(0.05/0.04 = 1.25\), which is not significant.
This implies that the estimate of \(\hat{\kappa}\) is unreliable: the data do not contain enough information about the mean-reversion speed over a 2-year window. Longer time series (10--30 years) are needed to estimate \(\kappa\) with reasonable precision, because the process must be observed over multiple "mean-reversion cycles" of approximate duration \(1/\kappa\).
Exercise 5. A cap volatility calibration requires matching the Vasicek implied cap volatility \(\sigma_{\text{cap}}\) to the market Black volatility. Since the Vasicek cap price depends only on \(\sigma\) (with \(\kappa\) and \(\theta\) fixed from the yield curve), this is a one-dimensional root-finding problem. Describe the procedure using Newton's method and the cap vega.
Solution to Exercise 5
With \(\kappa\) and \(\theta\) fixed from the yield curve calibration, the cap price depends on \(\sigma\) alone. The goal is to find \(\sigma\) such that the model cap volatility matches the market Black volatility: \(\sigma_{\text{Black}}^{\text{model}}(\sigma) = \sigma_{\text{Black}}^{\text{mkt}}\).
Newton's method procedure:
- Start with an initial guess \(\sigma^{(0)}\) (e.g., 0.01).
- Compute the Vasicek caplet price \(C^{\text{model}}(\sigma^{(k)})\) using the ZCB put formula.
- Invert the Black formula to obtain \(\sigma_{\text{Black}}^{\text{model}}(\sigma^{(k)})\).
- Compute the error \(\epsilon^{(k)} = \sigma_{\text{Black}}^{\text{model}}(\sigma^{(k)}) - \sigma_{\text{Black}}^{\text{mkt}}\).
- Compute the cap vega \(\mathcal{V} = \partial C^{\text{model}} / \partial \sigma\) (sensitivity of the Vasicek cap price to \(\sigma\)). In the Vasicek model, this can be computed analytically since \(\sigma_P = B(\delta)\,\sigma\sqrt{(1-e^{-2\kappa T_0})/(2\kappa)}\) depends linearly on \(\sigma\).
- Update: \(\sigma^{(k+1)} = \sigma^{(k)} - \epsilon^{(k)} / (\partial \sigma_{\text{Black}}^{\text{model}} / \partial \sigma)\).
The derivative \(\partial \sigma_{\text{Black}}^{\text{model}} / \partial \sigma\) can be obtained via the chain rule through the Black formula inversion. Since the Vasicek cap price is smooth and monotonically increasing in \(\sigma\), Newton's method converges quickly (typically 3--5 iterations to machine precision).
Exercise 6. The calibrated model has residual yield errors of \(\{+2, -4, +2, +2, +3\}\) basis points across maturities \(\{1, 2, 5, 10, 30\}\)Y. Is this pattern random or systematic? What does a systematic pattern tell you about the model's yield curve shape limitations?
Solution to Exercise 6
The residual pattern \(\{+2, -4, +2, +2, +3\}\) bp across maturities \(\{1, 2, 5, 10, 30\}\)Y is systematic, not random. The pattern shows:
- The model overestimates the 1Y, 5Y, 10Y, and 30Y yields (positive errors)
- The model underestimates the 2Y yield (negative error)
If the errors were purely random (due to measurement noise or bid-ask spreads), we would expect roughly equal numbers of positive and negative errors with no discernible pattern. Instead, the negative error at 2Y surrounded by positive errors at adjacent maturities suggests the market yield curve has a local feature (a slight kink or extra curvature) around the 2-year maturity that the smooth Vasicek curve cannot capture.
The Vasicek yield curve is of the form \(R(\tau) = h(\tau)\,r_0 + (1-h(\tau))\,R_\infty + C(\tau)\), which is a smooth, monotone-approaching curve. It cannot reproduce local humps, kinks, or inflection points in the observed yield curve. The systematic residual pattern reveals these shape limitations: the three-parameter model is too rigid to accommodate the fine structure of the term structure, particularly at intermediate maturities where central bank policy expectations and term premia create local features.
Exercise 7. Distinguish physical measure parameters \((\kappa^{\mathbb{P}}, \theta^{\mathbb{P}}, \sigma)\) from risk-neutral parameters \((\kappa, \theta, \sigma)\). If historical estimation gives \(\kappa^{\mathbb{P}} = 0.10\), \(\theta^{\mathbb{P}} = 0.04\), and market calibration gives \(\kappa = 0.25\), \(\theta = 0.0455\), compute the market price of risk \(\lambda\) using \(\kappa = \kappa^{\mathbb{P}} + \lambda\).
Solution to Exercise 7
Physical measure parameters \((\kappa^{\mathbb{P}}, \theta^{\mathbb{P}}, \sigma)\) govern the actual (statistical) dynamics of the short rate:
These are estimated from historical time series data.
Risk-neutral parameters \((\kappa, \theta, \sigma)\) govern the pricing dynamics:
These are calibrated to cross-sectional market prices (bonds, caps).
The volatility \(\sigma\) is the same under both measures because the Girsanov theorem only changes the drift, not the diffusion coefficient.
Computing the market price of risk:
Verification using the relationship \(\theta = \kappa^{\mathbb{P}}\theta^{\mathbb{P}} / \kappa\):
However, the market calibration gives \(\theta = 0.0455 \neq 0.016\). This means the relationship \(\theta^{\mathbb{Q}} = \kappa^{\mathbb{P}}\theta^{\mathbb{P}} / (\kappa^{\mathbb{P}} + \lambda)\) requires a more general specification where \(\lambda\) may not be constant, or the risk-neutral drift takes the form \(\kappa(\theta - r_t) - \sigma\lambda\) with \(\lambda\) absorbing both the mean-reversion speed and level adjustments. Under the general Vasicek change of measure with constant \(\lambda\), the risk-neutral parameters are \(\kappa^{\mathbb{Q}} = \kappa^{\mathbb{P}} + \lambda\) and \(\theta^{\mathbb{Q}} = (\kappa^{\mathbb{P}}\theta^{\mathbb{P}} + \sigma\lambda') / \kappa^{\mathbb{Q}}\) where \(\lambda'\) may differ. The key takeaway is that \(\lambda = 0.15 > 0\), indicating a positive risk premium: investors require compensation for bearing interest rate risk, which increases the risk-neutral mean-reversion speed relative to the physical one.