Zero-duration hedging strategy

I have seen a question that says: Describe a zero-duration hedging strategy using only the government bond portfolio and options on U.S Treasury bond futures contracts. No calculations required. Answer says: The correct strategy would be to short call options and go long put options. The short call position would create a negative cash flow if rates were to decline but the long put position would create a positive cash flow if rates were to increase. This fully hedges the portfolio. The call and put options should have the same exercise price and expiration date and the appropriate notional amounts.

My question: A short call gains when interest rates increase. A long put gains when interest rates increase. This doesn’t seem to be a hedge at all? Can someone please explain?

Well if you are long the bond portfolio, you are long the bonds which have a “positive” duration and therefore will increase in price if the interest rate goes down and fall in price if the interest rates go up.

A short call and a long put is analogous to a short position in the underlying, which moves in the opposite direction in relation to interest rate moves.

So you are long the underlying on the one hand, and have a synthetic short position on the other hand.

Short options never gain; they can only lose.

Long options never lose; they can only gain.

(nb: I’m assuming that the option holder isn’t an imbecile.)

Can someone please explain urther. I still don’t understand how a short call and long put hedges the portfolio here.

you are creating a collar - around the range of movement of the portfolio - along with the options.

Short a call option - you get the premium. and along with the underlying - now you have a Covered Call strategy.

Long the put - you are offsetting the call premium by paying the put premium.

when rates fall - your underlying rises in value. So the call options (sold) are in the money.

when rates rise - your underlying falls in value - put options get in the money.

so either way you can get out of your position if you felt like it.

The payoff on a short call looks like this:


The payoff on a long put looks like this:


When you combine them, the payoff looks like this:

¯\ + _ = \

The payoff on the underlying looks like this:


When you combine them, the payoff looks like this:

( ¯\ + _ ) + / = \ + / = –.

And that, my friend, is a hedge.

@S2000magician …wow that was some effort with the symbols to give an explanation!!! Isn’t this also called a risk reversal position? Secondly, the call and put should have the same delta rather than same strike right?

They need to be the same strike. Because otherwise the hedge wouldn’t work. That’s because you want to exactly offset the profile of the long position with the two options. If the strikes were different that’s not the case.

Just take into account that you want to have combine



_ \

in order to generate this: . This only works when both strikes match. Otherwise you would come up with something like:

_ \

I hope that helps?!

Yes I agree with the payoff diagram.My bad regarding the deltas… Both short call and long put are negative deltas.