# Replicating the price of a (Bermudan) callable bond using a Bermudan swaption and bond

I have heard that the price of a (Bermudan) callable bond can be replicated (at least approximately) by a Bermudan swaption and ordinary bond (assume the callable bond pays a fixed coupon).

I was wondering if it is possible, how can one construct the Berm swaption, bond, and also combine them?

• My own thoughts: I know the Berm. swaption is meant to capture the optionality, but entering (or exiting) a swap introduces cash flows (like a floating leg) that a callable bond does not produce. Including a vanilla bond in the replication along with the Berm swaption does not seem to help with this issue. – d_797 Dec 16 '20 at 22:28

I think the strategy is meant to be an overhedge. Consider a portfolio $$\Pi$$ consisting on long positions on both a fixed-rate bond $$B$$ and a Bermudan payer $$V$$. Then on all scenarios the payoff of $$\Pi$$ is at least equal to that of the callable bond $$C$$:
• If $$C$$ is not called, then the swaption $$V$$ can be relinquished and the fixed cash flows from $$B$$ and $$C$$ offset each other.
• If $$C$$ is called, then we exercise the swaption $$V$$ and receive a net variable cash flow of LIBOR plus spread from the swap.
Hence $$\Pi$$ must be more expensive than $$C$$. When trading this kind of bond a dealer might charge a fee to compensate for the overhedge cost.