Formulating a Swap Curve Calibration Objective with Cubic Splines

Summary

The document sets up a global curve calibration problem using a piecewise cubic spline to represent rates across maturities. It proposes choosing the spline coefficients so that prices or rates implied by the resulting discount curve match observed market instrument quotes. Discount factors are constructed from the interpolated rates, and the example expresses a squared-error objective for fitting par swap rates.

The question focuses on whether the proposed residual is correctly formulated, with a simplifying assumption that the floating leg is worth par and OIS discounting is omitted. It does not include an answer, calibration results, or a treatment of practical details such as payment schedules, day-count conventions, instrument-specific cash flows, weighting, or spline boundary conditions. Accordingly, it is best read as an introduction to the structure of curve fitting rather than a complete calibration recipe.

Key ideas

• A cubic spline can represent a rate curve piecewise between instrument maturities.
• Calibration adjusts spline parameters to align model-implied quotes with observed instrument quotes.
• The proposed objective sums squared differences between implied and observed par swap rates.
• The example simplifies floating-leg valuation and omits OIS discounting.
• Accurate implementation also depends on instrument cash flows and market conventions.

Tags

# Bootstrapping and Curve Calibration Objective Function


# Bootstrapping and Curve Calibration Objective Function












I'm confused about the form of the objective function for some global curve calibration. It seems simple enough: minimized the squared loss of the price of the input instruments and the price stripped off the built curve.

Suppose I want to interpolate rates using cubic splines, then I can write

$$r(a,b,c,d; t) = r(t) = a_i + b_i(t-t_i) + c_i(t-t_i)^2 + d_i(t-t_i)^3$$

where $t_i \leq t < t_{i+1}$ and the maturity of the input instruments are $t_1, t_2, \ldots, t_n.$ We want to find the parameters $a,b,c,d$ such that the the instrument prices stripped from this curve are consistent with the input prices. Write $Z(0,t) = e^{-r(t)\tau(0,t)}$ for the discount factor associated with the rate $r.$

In the case of a par swap (forget OIS discounting for now so that the floating leg is priced at 1) the swap rate $R_j$ is known so we would like to minimize

$$error = \sum_j\left(\frac{1 - Z(0,t_j)}{\sum_{k=1} Z(0,t_k)} - R_j\right)^2$$

which is the squared difference between the swap rate stripped off the interpolated curve and the input rate $R_j$ for each instrument $j$.

Is the correct? Is the error in the correct form? Please feel free to fill in any details I'm missing or clear up confusions I am having.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.