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If we have a 30y swap with 5y early termination breaks exercisable by either party, at each termination period, who would exercise the early termination/choose to break the swap? Is there an optimal exercise behaviour? Also, what's the benefit of having such a break clause vs. just going back in 5y time and asking to unwind/terminate the trade?

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The break clause is contractual. Meaning that potential future exposure calculations can be made based on the swap terminating at most in 5y time. Without the break clause this would not be financially valid.

Usually break clauses are flagged and it is in both counterparties interests to resolve the trade before the break. Usually this involves rolling the swap to an on market swap and extending the break or agreeing some NPV to exchange to terminate the swap shortly before the break.

In the event the swap goes to official break, there is usually something written into the contract on how to execute this fairly at break time. The experience I have had is that interbank counterparties are requested to provide a quotation for the swap in question. Interbank counterparties usually receive the request and do not respond becuase it is not in their interest to do so and then the client is forced to rely on the market-makers valuation as a fallback, which is probably not in the clients best interest albeit will probably still be reasonably fair. On the off chance an interbank counterparty does respond it may be slightly favourable for one party or the other which is why the market-maker generally prefers to resolve this with certainity before the event.

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