Interest Rate Swap - Valuation and Pricing

Valuation and Pricing

The valuation of vanilla swaps was often done using the so-called textbook formulas using a unique curve in each currency. Some early literature described some incoherence introduced by that approach and multiple banks were using different techniques to reduce them. It became even more apparent with the 2007–2012 global financial crisis that the approach was not appropriate. The now standard pricing framework is the multi-curves framework.

The present value of a plain vanilla (i.e. fixed rate for floating rate) swap can be computed by determining the present value (PV) of the fixed leg and the floating leg.

The value of the fixed leg is given by the present value of the fixed coupon payments known at the start of the swap, i.e.

where C is the swap rate, n is the number of fixed payments, N is the notional amount, is the accrual factor according to the day count convention for the fixed rate period and is the discount factor for the payment time .

The value of the floating leg is given by the present value of the floating coupon payments determined at the agreed dates of each payment. However, at the start of the swap, only the actual payment rates of the fixed leg are known in the future, whereas the forward rates are unknown. The forward rate for each floating payment date is calculated using the forward curves. The forward rate for the period with accrual factor is given by

where I is the index and is the discount factor associated to the relevant forward curve. The value of the floating leg is given by the following:

where m is the number of floating payments, is the accrual factor according to the floating leg day count convention.

The fixed rate offered in the swap is the rate which values the fixed rates payments at the same PV as the variable rate payments using today's forward rates, i.e.:

Therefore, at the time the contract is entered into, there is no advantage to either party, i.e.,

Thus, the swap requires no upfront payment from either party.

During the life of the swap, the same valuation technique is used, but since, over time, the both the discounting factors and the forward rates change, the PV of the swap will deviate from its initial value. Therefore, the swap will be an asset to one party and a liability to the other. The way these changes in value are reported is the subject of IAS 39 for jurisdictions following IFRS, and FAS 133 for U.S. GAAP. Swaps are marked to market by debt security traders to visualize their inventory at a certain time.