Pricing and Valuation of Interest Rates and Other Swaps | CFA Level I Derivatives

Understanding Swaps and Their Connection to Forwards

Welcome back to our discussion on the pricing and valuation of swaps. Today, we’ll dive into the general principles of swap pricing and take a closer look at interest rate swaps.

Swaps share some similarities with forwards, such as: Typically requiring no payment by either party at initiation. Being custom instruments with non-standard contracts. Largely less regulated. Exposing both parties to counterparty risk.

Swaps can also be viewed as a series of forwards. For example, two parties can enter into a single swap agreement where they exchange payments at multiple settlement dates instead of initiating separate forward contracts.

Swaps as a Series of Forwards

To understand the principles behind the pricing and valuation of swap contracts, it’s helpful to view swaps as a series of forwards.

For example, one party agrees to buy an underlying asset at a fixed price at the end of every year for the next 3 years. The other party agrees to sell this underlying asset at the end of every year for this period. Instead of a single contract, the two parties can enter into 3 separate forward contracts, each with the same settlement price, expiring in 1, 2, and 3 years. The price of each forward is equal to the swap fixed rate.

However, the principle of no arbitrage states that the forward price at initiation should be a function of the length of the forward contract. For no arbitrage to hold, all these 3 contracts should have different pricing.

But for a swap, all the fixed payments are equal. So, how can we equate a swap to a series of forward contracts? It turns out that we can, but we have to violate an important principle about forward pricing.

Off-Market Forwards and Swap Pricing

A forward contract that starts with a nonzero value is called an off-market forward. A swap, with a fixed swap price for every settlement, can be decomposed into a series of off-market forwards initiated at time 0 with the same settlement price, equal to the swap price.

Such an arrangement means that some of the forward contracts would have positive values, and some would have negative values, at initiation. However, to adhere to the principles of no-arbitrage, their combined values would equal zero. This also means that the value of the swap is 0 to both parties at initiation.

Pricing Interest Rate Swaps

For the Level 1 curriculum, we’ll only focus on interest rate swaps. The pricing and valuation of other types of swaps are covered in Level 2.

In essence, pricing a swap is to determine the equilibrium price where the values of all the forwards sum to zero. We’ll use the concept of replication to help guide us to this price.

Under the principle of no-arbitrage, a position in a derivative can be replicated by buying the underlying asset and having a short position in a risk-free asset.

A plain vanilla interest rate swap can generally be replicated in the same manner. The long promises to pay a fixed rate and receives the floating rate from the short.

The cash flows can be replicated by borrowing at a fixed interest rate and using the proceeds to buy a bond that provides the floating rate payments. We denote the payment on the fixed-rate loan as F, and the interest received on the floating rate bond is MRR (Market Reference Rate). These are the forward rates that correspond to the preceding period of each cash flow.

Implied forward rates can be calculated based on the current spot rates. So, let’s say we have a 3-year swap with annual settlements and these are the current 1-year, 2-year, and 3-year spot rates. By now, you should be familiar with calculating implied forward rates based on spot rates using a specific equation.

For the swap to be of zero value at initiation, the present value (PV) of all fixed-rate cash flows must equal the PV of all floating-rate cash flows. When we solve for F, we get 3.97%, which is the no-arbitrage fixed rate, also known as the par swap rate.

It’s unlikely you’ll be asked to make calculations like this in the exam. Instead, understanding the concept of interest rate swap pricing should suffice for the Level 1 curriculum.

Conclusion

And that concludes our lesson on the pricing and valuation of swaps. In the next lesson, we’ll move on to options.