Zero coupon swap

In finance, a zero coupon swap (ZCS) is an interest rate derivative (IRD). In particular it is a linear IRD, that in its specification is very similar to the much more widely traded interest rate swap (IRS).

A Zero coupon swap (ZCS) is a derivative contract made between two parties with terms defining two 'legs' upon which each party either makes or receives payments. One leg is the traditional fixed leg, whose cashflows are determined at the outset, usually defined by an agreed fixed rate of interest. A second leg is the traditional floating leg, whose payments at the outset are forecast but subject to change and dependent upon future publication of the interest rate index upon which the leg is benchmarked. This is same description as with the more common interest rate swap (IRS). A ZCS differs from an IRS in one major respect; timings of scheduled payments. A ZCS takes its name from a zero coupon bond which has no interim coupon payments and only a single payment at maturity. A ZCS, unlike an IRS, also only has a single payment date on each leg at the maturity of the trade. The calculation methodology for determing payments is, as a result, slightly more complicated than for IRSs.

Zero coupon swaps (ZCSs) hedged by the more commonly traded interest rate swaps (IRSs) introduce cross-gamma into an IRD portfolio. As such, and due to correlation between different instruments, ZCSs are required to have a pricing adjustment, to equate their value to IRSs under a no arbitrage principle. Otherwise this is considered rational pricing. This adjustment is referred to in literature as the zero coupon swap convexity adjustment (ZCA).

ZCSs are bespoke financial products whose customisation can include changes to payment dates, accrual period adjustment and calculation convention changes (such as a day count convention of 30/360E to ACT/360 or ACT/365).

A vanilla ZCS is the term used for standardised ZCSs. Typically these will have none of the above customisations, and instead exhibit constant notional throughout, implied payment and accrual dates and benchmark calculation conventions by currency. A vanilla ZCS is also characterised by one leg being 'fixed' and the second leg 'floating' often referencing an -IBOR index. The net present value (PV) of a vanilla ZCS can be computed by determining the PV of each fixed leg and floating leg separately and summing. For pricing a mid-market ZCS the underlying principle is that the two legs must have the same value initially; see further under Rational pricing.

Calculating the fixed leg requires discounting the known, single cashflow by an appropriate discount factor:

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