Why Equal-DV01 Swap Positions Still Have Convexity Risk
Summary
The document explains why a 10-year receiver and 20-year payer swap with equal and opposite DV01s may not offset under a parallel rate move. Equal DV01s cancel the positions’ first-order sensitivity at the starting curve, but they do not ensure equal convexity. Convexity describes how each swap’s DV01 changes as rates move, analogous to gamma for an option.
Because the two swaps can have different convexities, a rate shift changes their DV01s by different amounts. The combined position can then develop a net sensitivity, producing mark-to-market gains or losses. The example illustrates the distinction between initial DV01 neutrality and neutrality after rates move; it does not quantify the swaps’ convexities or provide a full valuation calculation.
Key ideas
- Equal and opposite DV01s offset only the positions’ first-order rate sensitivity at the starting point.
- Convexity measures how a swap’s DV01 changes as its underlying rate changes.
- Swaps with different tenors can have different convexities even when their DV01 magnitudes match.
- A parallel rate move can therefore leave the combined position with residual rate exposure.
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# If 10s20s steepener have equal DV01 weighting on each swap then why does convexity play a role in MtM # If 10s20s steepener have equal DV01 weighting on each swap then why does convexity play a role in MtM Receiver Swap 10yrs Notional: 1,000,000 DV01: +1,300 Tenor: 10yrs Rate: 4% Payer Swap 20yrs Notional: 500,000 DV01: -1,300 Tenor: 20yrs Rate: 5% Looking at this fictitious example, I want to understand why this position will generate negative mark-to-market whenever the yield curve is shifted (parallel) up or down? The DV01s of both swaps are of equal magnitude (i.e. notionals have been adjusted in such a way to render both swaps having same DV01 magnitude, correct?), so therfore a parallel move in any direction of the yields will mean that the MtM of the swaps offset each other? Surely, convexity does not play a role because the DV01s are of equal magnitude? I am confused, and any help is greatly appreciated. ## Answer by thetableed (score 4) https://quant.stackexchange.com/a/47565 Even though your two swaps have offsetting DV01s, in general they do not have the same convexity. Swap (or bond) convexity is analogous to option gamma - it's the change in the delta (in this case, DV01) of the derivative when the underlying value changes, which for your swaps is the par rate. For a truly parallel shift in rates, say 20bps for example, the DV01 of each swap will change due to its own convexity, and they will not change equally - thus, your DV01s will cease to be neutral and the position will have a delta lean.
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