Piecewise Quadratics, Locality, and Curve Hedging (Chapter 21, Part 2)
Book: Fixed Income Securities: Tools for Today’s Markets | Author: Bruce Tuckman & Angel Serrat | ISBN: 978-0-470-89169-8
Previous: Fitting Discount Curves with Flat Forwards (Chapter 21, Part 1) | Next: Fixed Income Securities Series Wrap-Up
Part 1 gave you flat forwards: ugly steps, clean hedges. Part 2 asks what happens when you smooth the steps away.
Why Smooth at All?
Flat forward curves jump at segment boundaries. For many jobs that is fine. For reporting, client charts, or models that differentiate forwards, jumps annoy people.
Pure smoothers like cubic splines fit beautifully and lie badly. Forwards oscillate unless you regularize heavily. Worse, smoothing couples the whole curve. Shift the 25y par rate, refit, and your 5y forward moves because the spline tied them together. That violates locality.
Piecewise quadratics on top of flat forwards are the compromise Tuckman recommends.
How Piecewise Quadratics Work
Take the flat forward curve from Part 1. Put midpoints between segment boundaries. Between midpoints, let the instantaneous forward be a quadratic g_i(t) = a_i + b_i t + c_i t^2.
Constraints:
- Each quadratic starts at one flat forward and ends at the next (value match at midpoints).
- Average forward over each original flat segment equals that segment’s flat rate (so integrated forwards still reprice benchmarks).
Condition (2) is the clever part. Benchmark securities were priced off flat forwards. If the smooth curve preserves segment-average forwards, it still prices every benchmark exactly. Smoothing is cosmetic on top of a fitted skeleton.
Solve sequentially: first quadratic from t_0 to first midpoint (special boundary), then bootstrap quadratics using flat forwards and prior segments.
Result on May 28, 2010 USD curve: smooth line through midpoint knots, same discount factors at benchmark dates as the flat curve.
The Cost: Dependency Across the Curve
Change flat forward f_3 only. Quadratic g_2 must adjust to keep segment averages. That forces g_3 to move to preserve the next segment average, then g_4, and so on. A local market move propagates down the term structure.
Smooth forwards are continuous but not C^1 smooth: first derivative jumps at midpoints. Ask for equal slopes at joins and you add more linkage. More elegance, less independence.
Locality and Hedging: Table 21.4
Portfolio: pay 2.61% on $100mm 5.5y swap, pay 3.853% on $100mm 17y swap. Partial DV01: bump one benchmark 1 bp, refit curve, measure P&L.
Flat forwards: 5y bump adds $23,517. Hedge with ~$49.5mm 5y receive-fixed swap (DV01 $475 per $100mm). Exposures sit near neighboring benchmarks (5y, 6y for the 5.5y swap; 15y, 20y for the 17y). Local.
Smooth quadratics: Same 5y bump still matters, but you also pick up sensitivity to 4y, 7y, 12y, 25y when you refit. Hedge notionals spread across distant benchmarks. The 25y bump moves 17y swap value because smoothing tied them together.
Which hedge is “right”? If your curve method is flat forwards, hedge locally. If you use smooth forwards, you inherit the model’s long-range correlations, whether or not they exist in the market.
This is not academic. Wrong partial DV01 means your “hedged” book bleeds when unrelated points on the curve move.
Cubic Splines and Other Methods
The book mentions cubics as a cautionary tale: C2 smooth, famous for ringing. Practitioners either constrain heavily or avoid for production curves.
Flat forwards + quadratic overlay hits a sweet spot for LIBOR desks circa 2010: exact fit, reasonable shape, hedges that make sense to traders.
Fitting Imperfect Benchmarks
Footnote nuance: if benchmark prices have noise (on-the-run Treasury specials), you might minimize pricing errors instead of forcing zero error. With LIBOR swaps and ED futures quoting tightly, exact fit became standard.
Closing the Technical Arc
Chapter 21 completes the bridge from Chapters 1-2 bootstrap examples to production curve engines used in swaps (Ch 16), two-curve pricing (Ch 17), and option models (Ch 18). You now have:
- A forward representation
- A bootstrap that respects market instruments
- A smoothing layer with known tradeoffs
- A hedging story that depends on the layer you chose
The book’s last chapter is not the end of interest rate modeling (Part Three already went deeper), but it is the end of the desk-pricing thread: given liquid quotes, how do I discount arbitrary cash flows today?
Next post: series wrap-up and what stuck after 21 chapters of Tuckman and Serrat.