How Negative Convexity Affects a Yield-Curve Steepener

Summary

The answer explains why a curve steepener can carry negative convexity when it sells a longer-maturity rate product and buys a shorter-maturity product in matched delta. Longer-maturity bonds and swaps generally have greater convexity, and convexity tends to grow faster with maturity than delta. The resulting position can therefore be net short gamma even if it is designed to benefit from a steeper curve.

Whether that exposure offsets the steepening gain depends on how the overall level of rates moves and on market volatility. The answer contrasts a steepening while average rates stay low with a steepening accompanied by a substantial rise in rates; the latter can create a large adverse convexity contribution. It gives an illustrative calculation using a fixed gamma approximation, while noting that actual deltas and gammas decline as rates move, so the example exaggerates the cost. The discussion is qualitative and does not provide a general pricing model or trade sizing guidance.

Key ideas

• A matched-delta steepener that sells longer maturity and buys shorter maturity may be net short convexity.
• Longer-maturity rate instruments generally have greater convexity than shorter-maturity instruments.
• The convexity impact on profit and loss depends on the rate level and market volatility.
• A fixed-gamma example can overstate losses because instrument sensitivities change as rates move.

Tags

# Curve steepner and convexity


# 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?

## Answer by Attack68 (score 3)

https://quant.stackexchange.com/a/41588

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)

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