# 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

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.

• thank you. Could you expand more on the simple approximation? Doesn’t look trivial tbh. The second one is basically balancing your risk over the analytic delta of each bucket, am I correct? – ababoua Aug 3 '18 at 7:11
• Well deriving the formulae is quite complicated, and the approximations come from having the precise versions and reducing them down. For example you can see that the simple approximation is the same as the one accounting for curve shape if you approximate $A_j=1$ for each annual forward. Do you want a clearer example of using the simple approximation or a more intuitive explanation of the terms? You can also look at this spreadsheet which compares the various approximations to the more precise calculations: www.tradinginterestrates.com/revised/ACG.xlsb – Attack68 Aug 3 '18 at 7:42
• Thank you.I was initially under the impression that the simple formula had some intuition behind it (for e.g why would it be tenor+1 and not tenor? But everything else is clear. Now if I want to get the convexity premium in bp terms, i probably need the whole cross gamma grid + var-covar matrix to do a proper estimation, is that correct? – ababoua Aug 3 '18 at 7:52
• Yes if you want to assess the value of the gamma you need volatilities and correlations of each instrument. However, again approximations, using just vol are often simpler and produce reasonable values. A further complication is trade holding period, which might be stochastically dependent on stop-profit levels. And don't forget skew (leptokurtosis of market movements). Personally I have never invested the time in developing complicated tools for considerations where basic approximations work reasonably well – Attack68 Aug 3 '18 at 9:12