# Curve steepner and convexity

Can someone please explain why a curve steepener trade has a negative convexity? And are the gains from the steepness of the curve offset by the negative convexity?

• Hi ababoua, by steepner trade do you mean a CMS spread swap ? Or another payoff ? – Jiem Sep 4 '18 at 22:31
• @Jiem no, I don’t mean CMS spread swap. I meant a curve steepener like trading a 5s10s and expecting this part of the curve to steepen. – ababoua Sep 5 '18 at 6:16

## 1 Answer

If you know what convexity is then you will know that products with longer maturity have higher convexity, e.g. swaps and bonds. Convexity (gamma) is also generally quadratic so increases faster than linear with maturity unlike delta which is broadly linear, so if you sell a longer product and buy a shorter product in the same delta you will likely have net sold convexity.

Take a look here 20s30s curve convexity, for a more detailed example.

Whether or not this impacts the PnL significantly depends upon the volatility of the market. Say you executed a curve steepender at mid-market price 25bps, with rates averaging around 1.5%. If the curve steepens to 35bp and rates still average around 1.5% then this will be far more profitable than if the curve steepens to 35bps but the market sells off and the average rates are 5.0%. In that case the gamma may have blown all profit on the steepening.

For example using the number in the 20s30s link for a 100k steepening position the gamma cost is approximately $\frac{1}{2}\times 350^2 \times -100=6.125mm$ (OK its not really as much as this since the gammas/deltas decline, but we have held -100, with increasing rates but for small moves this linear approximation is reasonable - here it is an exaggerated example)