# Interest Rate Swap Pricing

**Definition of an Interest Rate Swap**

An interest rate swap is an agreement in which one party agrees to pay / receive a fixed interest rate, in exchange for receiving / paying a floating interest rate over a set period of time for a set notional amount.

**The traditional approach**

The discount curve takes centre stage in both the traditional and modern approaches. In the traditional approach, the discount curve is derived from the par curve following these steps.

- First we compose the par curve from deposit rates and (IRS) swap rates.
- Next, the bootstrap process converts these par rates into discount factors.
- This discount curve is used for several purposes. The first and most straightforward is to calculate the present value of a future cash flow. Another purpose is to derive the forward curve.
- The floating interest rates are calculated from the discount curve (in simple words: a 1 year interest rate starting in 1 years’ time can synthetically be created by borrowing the notional for 2 years, and simultaneously depositing the notional for 1 years’ time). The calculated forward interest rates are used to calculate the forward floating coupons.
- Next, all coupons are present valued via the discount function to calculate the net present value (NPV) of all coupons (fixed and floating), which would give the fair value of the swap.

**The multiple curve approach**

The very first thing we did in the traditional approach was composing the par curve from deposit rates and swap rates, but they don’t fit together smoothly at all.

They don’t generate the same cash flows. Their rates can only be used as if they were par rates if they are somehow similar. That’s the case if the risk associated to the deposits and swaps are very low. But they aren’t, not anymore! That’s why this approach was valid before the crisis but is obsolete now.

The risk associated with deposits and swaps is no longer negligible. Moreover, they don’t carry the same risk. A deposit with a longer tenor is more risky. As for the swap, the risk depends on the period of the floating leg.

Nowadays there is also a substantial difference between instruments with different compounding periods. This difference is called the basis swap spread and it is no longer negligible.

Long term rates are the rates for instruments with maturities longer than one year. Swap rates are still suited for this purpose. The longer the floating leg period of a swap, the higher the risk associated with it. So to get the instrument with the lowest risk, we can take the swap with the shortest floating leg period. The overnight index swap, the OIS, is a swap between a fixed leg and a floating leg based on the overnight index. The fixed leg can vary from 1 week up to 50 years. This means the floating leg period of an OIS is one day.

So the first change to the traditional approach is to use the OIS rates to build the par curve and to derive the discount function from this curve. This curve is used to calculate the present value of all sorts of cash flows. It’s still a very important curve but it’s not the only one in the game anymore.

The blue line is the par curve composed out of deposit rates and IRS rates. The orange line below is a par curve composed out of deposit rates for the overnight and tom/next point only and OIS rates for all other maturities.

Forward rates get their own curves, one for each commonly used tenor. So there is a one month forward curve, a three month forward one and so on. If you want a floating rate of a certain tenor, or a quantity derived from it, you take the forward curve with that tenor. For example, the six month forward curve can be used for six month forward rate agreements, for swaps against against six month floating leg or more generally, for any financial instrument derived from six month floating rates.

If you want something with one month floating rates, you’ll need the one month floating curve. And so on.

These forward curves can be derived in two different and complimentary ways.

The first starts from IRS rates. The floating leg of the IRS determines the tenor of the forward rates. So an IRS with six month floating leg is used to derive a six month forward curve, and so on. For each IRS rate, list the cash flows of the IRS and calculate the net present value of these flows. Require this net present value to be zero, which gives rise to an equation. This equation features the IRS rate, present value factors and unknown forward rates. So the derived forward rates depend on the IRS rates and the previously derived discount curve.

Another way of deriving forward curves starts from basis swap spreads. One of the floating legs of the basis swap has to correspond to the tenor of the forward curve which will be derived. The other has to be equal to the tenor of a forward curve which is already known. List the cash flows of a basis swap and require their net present value to be zero. We get an equation depending on the basis swap spread, the OIS discount curve and the pseudo discount curve of the other floating leg. There is an equation for every maturity at which a basis swap spread is quoted.

**Let's recapitulate**

We want five curves in the multiple curve approach. The first one is the one and only discount curve, derived from OIS rates.

After that, use IRS rates to derive pseudo discount curves to calculate the forward curves. The discount curve, the real one, is used in this process. For example, you can use IRS against six month floating leg to derive the six month pseudo discount curve.

Then, use basis swap spreads to derive the other pseudo discount curves. For this process you need the real discount curve and the pseudo discount curve for the other tenor. For example, you can use a three month versus six month basis swap to derive the three month pseudo discount curve from the six month one. We could follow up with the one month versus three month basis swap and get the one month pseudo discount curve from the three month one. We could also derive a one year pseudo discount curve from the six month one if we use the six month versus one year basis swaps.

In such a way we get the five curves: one discount curve to discount cash flows and four pseudo discount curves to derive forward rates.

It is possible to derive the pseudo discount curves in a different order, as long as the inputs are known. You can’t derive a one month pseudo discount curve from the one month versus three month basis swaps if you didn’t derive the three month pseudo discount curve first.

Also, if other interest rate swaps and basis swaps are quoted, another order of derivation is required. For example, the standard interest rate swap for the dollar is against a three month floating leg, so the first pseudo curve you can derive is the three month one.