# Convexity in a DV01 neutral trade

I have got a question about DV01 neutral trades. Generally speaking: if you perform a 2s10s steepener on a generic govt yield curve, would convexity be a risk? If so, in what measures?

Technically, as we are DV01 neutral, I imagine that convexity is also somewhat mitigated, yet I fail to understand where this convexity risk might arise from. Might you be provide me with an example?

• "would convexity be a risk? If so, in what measures?" - bond convexity is higher on 10s than 2s when '01 neutral. That's where the risk comes from. E.g. on CT2 and CT10 Govt, BBG has DV01s as 1.99 and 9.12 but convexity as 0.05 and 0.955. Mar 1 at 15:44

Let’s say you do a 2s-10s steepener, dv01 neutral. What does this mean ? It means you are using the current dv01s of the 2s and 10s, which are approximately 1.99 and 9.12, to weight the relative principal amounts of the bonds. Now, the key point is, when the market moves, these dv01s move and you no longer have a balanced trade. That is the convexity risk. For example , 10yr rates go up, so the dv01 of 10s goes down, so you have to make your position larger if you want to restore the balance. If you are long the 10s this effect will be a benefit (long convexity ) and if you are short it will be a cost (short convexity ).

• In this example if we have a 2s10s steepener, long 2's and short 10's, and yields move up in a parallel fashion you're duration will get longer and you'll need to sell more 10s. Vice versa in a rally you'll need to buy more 10's back as you're position is too long. Notice that you're always buying and selling at the worse times. You're getting shorter into a rally and longer into a selloff, which is the opposite of a long option position with positive convexity. The real value of the convexity of the position is based on a interest rate path distribution assumption. Mar 2 at 21:43

An intuitive explanation of why we have IR gamma: when rates are 10 bps, then 1 bp change is a somewhat bigger deal then when rates are 1,000 bps. (Maybe if instead of dv01 being a sensitivity to a 1bp move, we used some bump size dependent on rate level, we'd have less gamma.)

If you have an IR swap then:

• you can calculate a matrix of cross-gammas between all the tenors of your swap curve. Most of its entries will be close enough to 0 to be indistinguishable from numeric noise. Not at all useful for a swap, in my opinion, but may be useful for non-linear products.

• more useful, you can calculate a single number for the gamma, risk-weighted by the tenors where you have the IR delta. This is helpful not for risk management, but if you are trying to achieve P&L explain with little unexplained residual, especially in longer maturities than 10 years. For risk management, put the risk amount on the risk report, see how much P&L it causes, so everyone can see that it's not worth hedging.

Edit: as @dm63 points out, you can also see how much your IR deltas by tenor bucket have changed because of the IR gamma $$\times$$ rate change.