Fitting Smooth Front-End Swap Curves from Dense FRA Quotes

Summary

The document describes a curve-building problem in the front end of a EUR six-month swap curve. The market instruments include a six-month deposit followed by a dense sequence of forward rate agreements and short swaps. Using quoted maturities directly as knots in a cubic spline can produce jagged forward rates. A monotone-convex interpolation is reported to be smoother in general, although it can still jump on some days.

The author considers adding a synthetic deposit before the first quoted maturity, or fitting a shape-preserving forward curve by minimizing its length while repricing all market instruments. They ask whether a continuous, piecewise-linear forward curve is adequate for the densely quoted front end, and how banks typically handle the issue. No answer, implementation, comparison, or empirical results are included, so these are proposed approaches rather than validated recommendations. The central lesson is that smooth discount or swap curves do not guarantee smooth implied forwards, and interpolation choices must respect instrument repricing constraints.

Key ideas

• Naive cubic spline interpolation across dense front-end swap instruments can create jumpy forward rates.
• Monotone-convex interpolation may smooth the curve, but the author reports occasional remaining jumps.
• A proposed alternative is to fit a continuous, piecewise-linear forward curve while repricing the market instruments.
• The document poses these approaches as questions and does not provide a tested solution.

Tags

# Swap curve is unsmooth at front end with naive interpolation


# Swap curve is unsmooth at front end with naive interpolation












I am looking at swap curve building at front end and find it difficult to get a smooth forward curve with a fast generic algorithm. For example, EUR 6m curve has 6m deposit, and then a series of FRAs (1m by 7m, 2m by 8m,,,,etc) up to 2 year swap at front end. If I use a naive smooth interpolation method such as cubic spline and use 6m, 7m, 8m,..., as knot points on the spline, I get very unsmooth/jumpy forward curve. I also tried monotone-convex method in Hagan's paper, it gets much smoother, however, on a few days, the curve still gets very jumpy. One way to address is to add "synthetic" deposit before 6m. However, I want to apply a more generic algorithm to fit to this type of structure. I am thinking I can use some shape preserving algorithm: minimize the length of the forward curve with all the market instruments being repriced. However, with a polynomial spline, this optimization method will take some time to solve. Can I assume a piece-wise linear but continuous forward rate and set up this optimization problem just for the front end? (since they are densely spaced, I wonder if a high order polynomial is needed anyway). Did anyone try this approach before? Also does anyone know what is the approach taken at banks nowadays to address issue like this? Or is there any paper on this topic? Thank you.

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