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

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.