Fed Funds Average Swap Convexity

Per Multi-Curve Construction, the Fed Funds curve itself is a dependent node discounted off the SOFR curve, not off itself (see that document's Fed Funds/SOFR discounting note). Within the Fed Funds curve's own construction, three instrument families are stitched together tenor by tenor:

1. Short end — FOMC-dated instruments: flat-forward step functions anchored to discrete Federal Open Market Committee meeting dates (the same FOMC-dated short-end technique documented for SOFR in FOMC Dates).
2. Belly — Fed Funds OIS (compounded): the floating leg compounds daily realised Fed Funds fixings geometrically: \(\prod_i (1 + r_i \cdot \Delta t_i)\).
3. Long end — Fed Funds Average Swaps (arithmetic): the floating leg is the simple arithmetic mean of daily realised fixings: \(\frac{1}{N}\sum_i r_i\).

Because the belly and the long end use two different floating-leg paradigms — geometric compounding vs. arithmetic averaging — forcing a single continuous curve through both introduces a bias. For a fluctuating rate process, Jensen's inequality guarantees the expectation of the arithmetic average diverges from the expectation of the geometric compound: \(E[\text{avg}(r)] > \text{avg}(E[r])\) when the rate has positive variance. Plotted against a "base curve" built purely from OIS/compounded instruments, the FFAS-implied curve bows upward — a convexity premium that grows with rate volatility and increases with tenor.

Naively bootstrapping an FFAS pillar as though it were a compounded OIS pillar ignores this bias entirely. The concrete failure modes are:

• Distorted forward rates at the long end.
• Mispriced long-end Fed Funds derivatives.
• Flawed bucketed DV01 sensitivities for any portfolio with long-dated Fed Funds exposure.

A correct bootstrap injects a dynamic convexity adjustment directly into the curve's root solver, translating the arithmetic FFAS cash flows onto the same footing as the compounded instruments before solving for discount factors. Per the source material, addressing this properly is not a one-line deterministic fix:

• A purely deterministic model (discount forwards with no convexity term) fails outright — it cannot capture the bias at all.
• Capturing the distribution of future rate paths (needed because the bias scales with realised rate volatility, not just the rate level) requires a stochastic short-rate model — Hull-White is the model named in the source material.
• Calibration is two-dimensional: the model must fit both the OIS base level and the FFAS "bow" (the convexity spread) simultaneously.

This document records the existence and shape of the gap, not a chosen remedy for ORE Studio; no FFAS generation or convexity-adjustment implementation exists in ORE Studio today. Whether and how to model this is open should Fed Funds Average Swaps ever need to be primary pillar instruments here (see Synthetic market data generation: approach, which does not currently generate Fed Funds curves or their long end).