Convexity, Hedging, and Dealer Pricing in Zero-Coupon Swaps

# Why is it desirable to receive fixed on a zero coupon swap, and undesirable to pay fixed on a zero coupon swap?


# Why is it desirable to receive fixed on a zero coupon swap, and undesirable to pay fixed on a zero coupon swap?












In most established rates markets, swaps are discounted using risk-free reference rates, such as Sonia in the GBP market and Eonia in the EUR market, as opposed to Libor.

Because of the way zero-coupon Libor swaps are valued (i.e. forward cash flows compounded up to the maturity date of the swap vs Libor, and then discounted back vs Sonia), this creates a convexity adjustment that needs to be accounted for when pricing them.

In other words, a dealer will need to charge a client more if the client wants receive fixed on the zero-coupon swap, and does not need to charge anything more if the client wants to pay fixed on the zero-coupon swap. So not accounting for any extra 'convexity' charges, it is desirable to receive fixed on a zero-coupon swap, but undesirable to pay fixed.

The below was explained to me by someone, but I'm not entirely sure I understand exactly why/how this works from a mathematical point of view. Would it be the other way around if the Libor curve happened to lie below the Sonia curve?

I'm essentially looking for a more concise/clear explanation of this phenomenon.

## Answer by Attack68 (score 5, accepted)

https://quant.stackexchange.com/a/50307

I notice you mention GBP. This effect is particularly apparent there since a large number of insurance, pension and asset management companies like to trade ZCS. They do this because the forward risk profile of a ZCS more accurately reflects the increasing notional of their portfolio and avoids them having to deal with interim coupon payments. They almost exclusively receive fixed.

That being said those companies execute a trade and hold to maturity generally, whilst dealers mtm and manage risk exposures on a daily basis.

Dealers also have to hedge with IRS, since there is no ZCS interdealer market.

If, as a dealer, you execute perfect delta hedges in IRSs for a ZCS you will not gain or lose any money if interest rates rise or fall but what you will acquire are cross gamma risks. Suppose you are a dealer and paid fixed on a 20Y ZCS and received IRSs to perfectly delta hedge. If interest rates rise you will gain a large positive cashflow at (and only at) the 20y maturity on the ZCS and some offsetting cashflows on the IRSs any many different times between 0-20Y. The NPV of these is zero (you have no made or lost money), but you now have a cashflow profile.

This cashflow profile is discounted at OIS so whilst you have not acquired any outright delta risk you will need to execute LIBOR/OIS basis to hedge.

If the correlation between rates and the LIBOR/OIS basis is zero then theoretically the expected monetary effect of this is zero since you will gain as much as you lose from the effect. However, the PnL volatility is increased which is unappealing from a dealers point of view so would typically charge a convexity adjustment in either direction.

The fact that the market is so one-way makes it much more preferable for dealers to receive fixed to offset their existing large positions.

Also the correlation of OIS/LIBOR basis with rates was not a zero correlation since the large size of the dealers positions meant it was self defeating (as they hedged and pension funds didnt) and so the correlation was a cost.