Estimating Convexity in a 20-Year/30-Year Swap Spread

Summary

The document explains how to estimate the change in delta as rates move for a curve trade that offsets the DV01 of 20-year and 30-year swaps. It gives a rule of thumb that relates convexity to swap tenor and PV01, then offers a shape-sensitive approximation using the analytic deltas of successive one-year forward swaps. When those forward deltas are equal, the more detailed expression reduces to the rule of thumb.

The answer attributes the approximations to a practical guide to swap trading and notes that more accurate calculations can separate discounting and forecasting cross-gamma effects. It characterizes the simpler estimate as splitting the effect roughly evenly between those two components. These are approximations, not a full valuation model; the shape-sensitive version depends on the forward-trade deltas used. The document does not provide a worked curve scenario or market data to validate the estimates, and it does not give a specific numerical result for the net convexity of the hedged spread.

Key ideas

• A swap spread hedged to flat DV01 can regain rate exposure as yields move because the legs have different convexities.
• A tenor-based rule of thumb estimates PV01 convexity using tenor and the position's PV01.
• A shape-sensitive estimate weights the analytic deltas of successive one-year forward swaps.
• The shape-sensitive formula reduces to the simpler estimate when the forward deltas are equal.
• More detailed methods can distinguish discounting, forecasting, and cross-gamma risks.

Tags

# 20s30s curve convexity


# 20s30s curve convexity












Let’s assume I trade a 20s30s spread on the curve and i’m flat delta (-100k on 20Y swap, 100k on 30y swap dv01). If the market moves, i’m not flat delta anymore. Is there a simple way to estimate the convexity in this trade i.e gamma (forecast/forecast delta)/cross gamma (discount basis delta/forecast delta) ? Am i right in saying that this convexity comes from the dynamics of the 20Y vs 20y10y annuity?

Thanks

## Answer by Attack68 (score 2, accepted)

https://quant.stackexchange.com/a/41083

Yes.

#### Simple Approximation - Rule of Thumb

Use the formula:

$$ \gamma \text{(pv01/bp)} = -\frac{1+tenor}{10,000 (bps)} pv01 $$

So for the 20Y and 30Y tenors respectively this formula gives 210 and -310 respectively. Of which half is produced from PnL component (discount risk) and half is produced from forecasting risk.

#### Approximation Accounting for Shape of Curve

Use the formula:

$$ \gamma \text{(pv01/bp)} = -\frac{pv01}{10,000 (bps)} * \frac{\sum_{j=1}^{N}2jA_j}{\sum_{j=1}^NA_j} $$

where $A_j$ is the analytic delta of a 1Y forward trade, so for a 3Y swap ($N=3$) you would use the analytic delta of a 0y1y, 1y1y, 2y1y. Note this reduces to the approximation above if $A_j=1$.

#### Further Detail

These formulae are derived in Pricing and Trading Interest Rate Derivatives: A Practical Guide to Swaps by Darbyshire. The bibliography includes code that has even more accurate formulae calculating the specific cross-gamma risks, and methods of converting between par and forward representations.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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