Hedging a Forward Swap with Zero-Coupon Swaps and a Libor Deposit

Summary

The document examines whether a forward-starting swap can be statically replicated using spot-starting zero-coupon swaps. The proposed hedge combines paying fixed on a one-year-forward, one-year swap, paying fixed on a spot-starting one-year zero-coupon swap, and receiving fixed on a spot-starting two-year zero-coupon swap. The apparent mismatch is that the two-year swap has no cash flow at the one-year point.

The response adds a fourth transaction: invest or borrow the one-year cash flow at Libor until year two. Under the assumption that Libor funding is freely available, this makes the cash flows cancel and leaves no convexity adjustment. The hedge therefore requires an intervening funding transaction and is not fully static. The answer also cautions that the assumption is imperfect under modern Fed funds discounting, though it describes the approximation as close.

Key ideas

• The one-year and two-year zero-coupon swaps alone do not offset all intermediate cash flows.
• Investing or borrowing the one-year cash flow at Libor bridges the timing gap to year two.
• The replication implies zero convexity adjustment under the stated free Libor funding assumption.
• The hedge is not fully static, and the funding assumption is only approximate with Fed funds discounting.

Tags

# Static hedge forward swap using zero coupon swaps


# Static hedge forward swap using zero coupon swaps












I'm trying to create a static hedge for a forward swap using two spot starting zero coupon swaps (to prove that there is no convexity adjustment needed). Here are the instruments -

- Paying fixed in 1y1y forward swap

- Paying fixed in spot starting 1y zero coupon swap

- Receiving fixed in spot starting 2y zero coupon swap

Problem is #3 does not have cash flows at t=1y, so there is no way to cancel out that exposure from #2. Could someone please point out my mistake.

## Answer by dm63 (score 2)

https://quant.stackexchange.com/a/42302

There needs to be a 4th transaction: the cashflow at t=1y needs to be invested at Libor until t=2y. You will then find that all the cashflows cancel. This means that the hedge is not quite static, but the convexity adjustment is still zero, because we assume money can be invested (or borrowed) at Libor for free at any time. (Note: this is not quite correct in the current era of Fed funds discounting, but it's pretty close).

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