# Supernatural Libor Coupons

Natural Libor coupons in swaps are fixed at (or usually some few days before) the start of their accrual period while the payment is made at the end of that period. Sometimes the fixing is delayed until the end of the period, so that the payment is done only a few days after the fixing. Such coupons are said to be fixed in arrears. What seems to be a slight modification of the terms distinguishes a simple case of model independent reading-the-price-off-the-forward curve from a highly exotic and model dependent endeavor in pricing and hedging.

The exact criterion for being a natural coupon boils down to the equality of the index estimation end date and the payment date. Everything else (accrual start and end date, fixing date) does not matter. Note that all these dates are in general different. In particular the index estimation end date does not need to coincide with the accrual end date (the latter being usually identical to the payment date). This is due to the swap schedule construction where an intermediate accrual start date may be moved a few days forward due to non-business days adjustments while the corresponding accrual end date falls into the regular schedule, which then causes the index estimation end date lying a few days after the accrual end date.

While in principle all cases with index estimation end date not equal to the payment date require a so called timing adjustment for pricing, it is only significant in cases where there is a bigger lag between the two dates.

Either case is relevant: While there are swaps with in arrears fixings, i.e. a payment long before the index end date, or average rate swaps (corresponding to a basket of coupons fixed somewhere between in advance and in arrears) there are also delayed payments occurring in some adjustable rate mortgages loans (I seem to remember that it is perfectly usual for Spanish mortgages, but I am not totally sure about that at the moment …).

Pricing models range from simple adjustment formulas coming from simple Black style models over replication approaches taking into account the whole relevant caplet smile using some terminal rate model (similar to the well known replication models for CMS coupons) up to full term structure model implied adjustments.

Where does QuantLib stand. Currently quite at the baseline, we only have a simple Black76 adjustment for the case of in arrears fixings, which is implemented in BlackIborCouponPricer::adjustedFixing(). Nothing in place for delayed payments or for fixings somewhere between in advance and in arrears.

Some time ago I wrote down the formula which generalizes the simple Black76 in arrears adjustment to the case of an arbitrary configuration of index end and payment date. You can find the paper here. I am not going into the derivation here, it’s better to read that for oneself in a lonely hour. Instead I will visualize an example.

The implementation is QuantLib of the formulas is straight forward, you can see the relevant code here.

The following graph shows the adjustments for an Euribor 6m coupon on evaluation date 24-08-2015 with fixing on 24-08-2025, $2\%$ forward rate and associated caplet volatility of $40\%$ (lognormal, no shift). The payment date slides between 26-08-2025 ($t \approx 10$) and 26-08-2026 ($t \approx 11$) and for each we compute the corresponding timing adjustment. The index estimation period end date is 26-02-2026 ($t \approx 10.5$). Remember that this is also the natural payment time with zero timing adjustment.

For payments before the natural payment time 26-02-2026 the adjustment is positive, for delayed payments negative, ranging between plus/mins three basis points approximately. We can introduce a correlation parameter $\rho$ (see the paper for the model behind). Normally the parameter will be close to one, but for very large delays it may also be sensible to choose a correlation below one. However for consistency is must be ensured that $\rho \rightarrow 1$ if the payment time approaches the natural payment time from the left (again see the paper for the rationale behind). Therefore we set

$\rho(\tau) = \rho_0 + (1-\rho_0) e^{-10\tau}$

with $\tau$ defined to be the year fraction between the payment time and the natural payment time. For payment times after the natural payment time we just assume a constant $\rho = \rho_0$.