How would you price an interest rate swap?

Pricing an interest rate swap involves determining the value of the fixed and floating payments for the particular structure of the swap. An interest rate swap is a contract between two parties where one party agrees to pay a fixed interest rate on a notional principal amount, while the other party pays a floating interest rate on the same notional principal amount. The notional amount is never actually exchanged, but it serves as the basis for calculating the respective interest payments.

Here is a step-by-step process for pricing an interest rate swap:

1. Determine the cash flows: To price an interest rate swap, we need to determine the cash flows that will be exchanged between the parties. For this, we need to know the notional principal amount, fixed interest rate, floating rate index (e.g., LIBOR or EURIBOR), payment frequency, and swap tenor.

2. Calculate the Floating Rate Payments: The floating rate payments are calculated using the forward rates derived from the interest rate yield curve (i.e., zero-coupon yield curve) at each payment date. These forward rates represent the market’s expectation of future interest rates.

Assuming we have forward rates f₁, f₂, …, f_N and the notional principal amount is N, the floating payments at each payment date are calculated as:
PF_i = N × f_i × Δt_i

where Δt_i is the year fraction between consecutive payment dates.

3. Discount the Cash Flows: The value of the cash flows has to be converted to present value terms using the discount factors obtained from the zero-coupon yield curve.

Suppose the discount factors corresponding to the payment dates are d₁, d₂, …, d_N. Then, the present value of the floating payments is:

$$PV_{floating} = \sum_{i=1}^{N} PF_i \times d_i$$

4. Calculate the Fixed Rate Payments: The fixed rate payments are calculated using the fixed interest rate, notional principal amount, and payment frequency.

Assuming the fixed interest rate is r and the notional amount is N, the fixed payments at each payment date are calculated as:
P_i = N × r × Δt_i

5. Discount the fixed Cash Flows: Similar to the floating cash flows, the fixed cash flows also need to be discounted. The present value of the fixed payments is:

$$PV_{fixed} = \sum_{i=1}^{N} P_i \times d_i$$

6. Determine the Swap Value: The value of the interest rate swap to one party is the present value of the cash flows they would receive minus the present value of the cash flows they would pay.

For the party paying the fixed rate and receiving the floating rate, the swap value is:
SwapValue = PV_floating − PV_fixed

For the party paying the floating rate and receiving the fixed rate, the swap value is:
SwapValue = PV_fixed − PV_floating

If the swap is being entered into at inception, the swap should be priced such that the value is zero for both parties, which is done by equating the present value of the fixed and floating legs:
PV_fixed = PV_floating

This helps determine the fixed swap rate that would make the swap fairly priced for both parties at the start of the agreement.