# A Practical Guide to Swaps: Pricing and Trading Interest Rate Derivatives

## Pricing and Trading Interest Rate Derivatives: A Practical Guide

Interest rate derivatives are financial contracts that derive their value from the underlying interest rates. They are widely used by banks, corporations, investors, and governments to hedge against interest rate risk, speculate on future interest rate movements, or enhance portfolio returns. In this article, we will provide a practical guide on how to price and trade interest rate derivatives, covering the main concepts, factors, models, instruments, and strategies.

## PricingandTradingInterestRateDerivativesAPracticalGu

## What are interest rate derivatives?

Interest rate derivatives are financial contracts that pay or receive cash flows based on the level or change of interest rates. For example, an interest rate swap is a contract where two parties agree to exchange fixed-rate and floating-rate payments over a period of time. An interest rate future is a contract where the buyer agrees to pay a fixed price for a certain amount of money at a future date, based on the prevailing interest rate at that time. An interest rate option is a contract that gives the buyer the right, but not the obligation, to enter into an interest rate derivative contract at a specified price within a specified time.

## Why are interest rate derivatives important?

Interest rate derivatives are important because they allow market participants to manage their exposure to interest rate risk, which is the risk of losing money due to changes in interest rates. Interest rate risk can arise from various sources, such as borrowing or lending money, issuing or investing in bonds, or holding cash or other assets that are sensitive to interest rates. By using interest rate derivatives, market participants can hedge their existing positions, lock in favorable interest rates, or reduce their borrowing costs.

Interest rate derivatives are also important because they provide opportunities for market participants to profit from their views on future interest rate movements. By using interest rate derivatives, market participants can speculate on the direction or volatility of interest rates, or exploit arbitrage opportunities between different markets or instruments. Interest rate derivatives can also enhance portfolio returns by creating synthetic positions that have different risk-return profiles than the underlying assets.

However, interest rate derivatives also entail risks that need to be carefully managed. These include market risk, which is the risk of losing money due to adverse movements in interest rates; credit risk, which is the risk of losing money due to default or downgrade of the counterparty; liquidity risk, which is the risk of not being able to buy or sell the contract at a fair price; operational risk, which is the risk of losing money due to errors or failures in processing or settlement; and legal risk, which is the risk of losing money due to disputes or changes in regulations.

## How are interest rate derivatives priced?

The pricing of interest rate derivatives depends on various factors, such as the type and features of the contract, the current and expected level and shape of the yield curve, the volatility and correlation of interest rates, and the creditworthiness and preferences of the counterparties. To price interest rate derivatives, market participants use various models that capture these factors and generate fair values and risk measures for the contracts.

### Discount factors and zero-coupon rates

A discount factor is a number that represents the present value of one unit of currency to be received at a future date. A zero-coupon rate is the annualized interest rate that equates the present value of one unit of currency to be received at a future date with the current price of a zero-coupon bond that pays one unit of currency at that date. The discount factor and the zero-coupon rate are inversely related, such that:

DF(t) = 1 / (1 + r(t) * t)

where DF(t) is the discount factor for time t, r(t) is the zero-coupon rate for time t, and t is the time in years.

The discount factor and the zero-coupon rate are the basic building blocks for pricing interest rate derivatives, as they allow us to calculate the present value of any future cash flow. For example, the present value of a fixed-rate payment of F to be received at time T is:

PV(F,T) = F * DF(T)

where PV(F,T) is the present value of the payment, F is the payment amount, and DF(T) is the discount factor for time T.

### Forward rates and swap rates

A forward rate is an interest rate that applies to a loan or deposit that starts at a future date and ends at another future date. A swap rate is an interest rate that applies to a fixed-rate payment in an interest rate swap that starts at a present date and ends at a future date. The forward rate and the swap rate are related, such that:

f(t1,t2) = (DF(t1) / DF(t2) - 1) / (t2 - t1)

s(t) = (DF(0) - DF(t)) / sum(DF(ti))

where f(t1,t2) is the forward rate between time t1 and t2, DF(t1) and DF(t2) are the discount factors for time t1 and t2, s(t) is the swap rate for time t, DF(0) is the discount factor for time 0, and sum(DF(ti)) is the sum of the discount factors for all the payment dates ti in the swap.

The forward rate and the swap rate are useful for pricing interest rate derivatives, as they allow us to compare different interest rates across different time periods. For example, the present value of a floating-rate payment of L * (f(t1,t2) + s) to be received at time t2, where L is the notional amount and s is the spread, is:

PV(L * (f(t1,t2) + s),t2) = L * (f(t1,t2) + s) * DF(t2)

### Yield curves and term structure models

A yield curve is a graphical representation of the relationship between interest rates and maturities. A term structure model is a mathematical model that describes how the yield curve evolves over time. The yield curve and the term structure model are essential for pricing interest rate derivatives, as they reflect the market expectations and dynamics of interest rates.

There are different types of yield curves, such as spot yield curve, forward yield curve, par yield curve, and swap yield curve. Each type of yield curve has its own interpretation and application. For example, the spot yield curve shows the current interest rates for different maturities; the forward yield curve shows the expected future interest rates for different maturities; the par yield curve shows the interest rates for coupon-bearing bonds with different maturities; and the swap yield curve shows the interest rates for fixed-rate payments in interest rate swaps with different maturities.

There are also different types of term structure models, such as equilibrium models, no-arbitrage models, stochastic models, and deterministic models. Each type of term structure model has its own assumptions and implications. For example, equilibrium models assume that interest rates are determined by macroeconomic factors; no-arbitrage models assume that there are no arbitrage opportunities in the market; stochastic models assume that interest rates follow random processes; and deterministic models assume that interest rates follow predetermined functions.

### Option pricing models

## How are interest rate derivatives traded?

Interest rate derivatives are traded in various markets and platforms, such as over-the-counter (OTC) markets, exchange-traded markets, electronic trading systems, and interdealer brokers. The choice of market and platform depends on factors such as the type and size of the contract, the availability and cost of liquidity, the transparency and efficiency of pricing, and the regulatory and operational requirements.

There are many types of interest rate derivatives that are traded in the market, each with its own characteristics and applications. In this section, we will focus on three main types of interest rate derivatives: interest rate swaps, interest rate futures, and interest rate options.

### Interest rate swaps

An interest rate swap is a contract where two parties agree to exchange fixed-rate and floating-rate payments over a period of time, based on a notional amount. The fixed-rate payer pays a predetermined fixed interest rate on the notional amount to the floating-rate payer, while the floating-rate payer pays a variable interest rate on the notional amount to the fixed-rate payer. The variable interest rate is usually based on a reference rate, such as LIBOR or SOFR, plus or minus a spread.

Interest rate swaps are used for various purposes, such as hedging against interest rate risk, converting between fixed-rate and floating-rate liabilities or assets, enhancing portfolio returns, or exploiting arbitrage opportunities. For example, a borrower who has a floating-rate loan can use an interest rate swap to hedge against rising interest rates by paying a fixed rate and receiving a floating rate. Alternatively, an investor who has a fixed-rate bond can use an interest rate swap to enhance returns by receiving a fixed rate and paying a floating rate.

There are two main types of interest rate swaps: plain vanilla swaps and basis swaps.

#### Plain vanilla swaps

A plain vanilla swap is the simplest and most common type of interest rate swap. It involves exchanging fixed-rate and floating-rate payments based on the same currency and the same notional amount. The fixed rate is also known as the swap rate, which is determined by the market such that the present value of the fixed-rate payments equals the present value of the floating-rate payments at the inception of the swap.

The valuation of a plain vanilla swap is straightforward, as it involves calculating the present value of the future cash flows using the discount factors and the forward rates. The value of a plain vanilla swap at time t is:

V(t) = sum(Fi * DF(ti) - Li * (fi(t1i,t2i) + s) * DF(t2i))

where V(t) is the value of the swap at time t, Fi is the fixed-rate payment at time ti, Li is the notional amount at time ti, fi(t1i,t2i) is the forward rate between time t1i and t2i, s is the spread, and DF(ti) and DF(t2i) are the discount factors for time ti and t2i.

#### Basis swaps

A basis swap is a type of interest rate swap that involves exchanging floating-rate payments based on different reference rates or different currencies. For example, a LIBOR-SOFR basis swap is a contract where one party pays LIBOR plus or minus a spread and receives SOFR plus or minus a spread. A cross-currency basis swap is a contract where one party pays a floating rate in one currency and receives a floating rate in another currency.

Basis swaps are used for various purposes, such as hedging against basis risk, which is the risk of divergence between different reference rates; converting between different currencies; or exploiting arbitrage opportunities. For example, a bank that has liabilities based on LIBOR and assets based on SOFR can use a LIBOR-SOFR basis swap to hedge against basis risk by paying LIBOR and receiving SOFR.

The valuation of a basis swap is similar to that of a plain vanilla swap, except that it involves using different discount factors and forward rates for different reference rates or currencies. The value of a basis swap at time t is:

V(t) = sum(L1i * (f1(t1i,t2i) + s1) * DF1(t2i) - L2i * (f2(t1i,t2i) + s2) * DF2(t2i))

where V(t) is the value of the swap at time t, L1i and L2i are the notional amounts in currency 1 and currency 2 at time ti, f1(t1i,t2i) and f2(t1i,t2i) are the forward rates in currency 1 and currency 2 between time t1i and t2i, s1 and s2 are the spreads in currency 1 and currency 2, and DF1(t2i) and DF2(t2i) are the discount factors in currency 1 and currency 2 for time t2i.

### Interest rate futures

An interest rate future is a contract where the buyer agrees to pay a fixed price for a certain amount of money at a future date, based on the prevailing interest rate at that time. The seller agrees to deliver the money at the future date and receive the fixed price. The fixed price is also known as the futures price, which is determined by the market such that the present value of the futures contract equals zero at the inception of the contract.

Interest rate futures are used for various purposes, such as hedging against interest rate risk, speculating on future interest rate movements, or exploiting arbitrage opportunities. For example, a borrower who expects interest rates to rise can use an interest rate future to lock in a lower interest rate by buying a futures contract and paying a fixed price. Alternatively, an investor who expects interest rates to fall can use an interest rate future to profit from lower interest rates by selling a futures contract and receiving a fixed price.

There are many types of interest rate futures that are traded in the market, each with its own characteristics and applications. In this section, we will focus on two main types of interest rate futures: Eurodollar futures and Treasury futures.

#### Eurodollar futures

A Eurodollar future is a contract that represents the implied three-month LIBOR for a certain date in the future. The contract size is $1 million, and the futures price is quoted as 100 minus the implied LIBOR. For example, if the futures price for a Eurodollar future expiring in December is 98.50, it means that the implied LIBOR for December is 1.50%.

Eurodollar futures are used for various purposes, such as hedging against LIBOR risk, speculating on LIBOR movements, or exploiting arbitrage opportunities. For example, a bank that has liabilities based on LIBOR can use Eurodollar futures to hedge against LIBOR risk by buying futures contracts and paying a fixed price. Alternatively, an investor who expects LIBOR to rise can use Eurodollar futures to profit from higher LIBOR by selling futures contracts and receiving a fixed price.

The valuation of a Eurodollar future is straightforward, as it involves calculating the present value of the difference between the futures price and the implied LIBOR using the discount factor. The value of a Eurodollar future at time t is:

V(t) = L * (F - r) * DF(T)

where V(t) is the value of the future at time t, L is the contract size ($1 million), F is the futures price (100 minus the implied LIBOR), r is the actual LIBOR at time t, T is the expiration date of the future, and DF(T) is the discount factor for time T.

#### Treasury futures

the price and delivery of a U.S. Treasury security with a certain maturity and coupon rate. The contract size is $100,000, and the futures price is quoted as a percentage of the face value of the security. For example, if the futures price for a Treasury future expiring in December is 99.00, it means that the buyer agrees to pay 99% of $100,000, or $99,000, for the security at delivery.

Treasury futures are used for various purposes, such as hedging against interest rate risk, speculating on interest rate movements, or exploiting arbitrage opportunities. For example, a borrower who expects interest rates to rise can use Treasury futures to lock in a higher interest rate by selling futures contracts and receiving a fixed price. Alternatively, an investor who expects interest rates to fall can use Treasury futures to profit from lower interest rates by buying futures contracts and paying a fixed price.

The valuation of a Treasury future is complicated, as it involves accounting for the conversion factor and the cheapest-to-deliver option. The conversion factor is a number that adjusts the futures price for the differences in coupon rates and maturities among the eligible securities for delivery. The cheapest-to-deliver option is the right of the seller to choose the security that minimizes the cost of delivery. The value of a Treasury future at time t is:

V(t) = L * (F / CF - P) * DF(T)

where V(t) is the value of the future at time t, L is the contract size ($100,000), F is the futures price, CF is the conversion factor for the cheapest-to-deliver security, P is the price of the cheapest-to-deliver security at time t, T is the delivery date of the future, and DF(T) is the discount factor for time T.

### Interest rate options

An interest rate option is a contract that gives the buyer the right, but not the obligation, to enter into an interest rate derivative contract at a specified price within a specified time. The seller receives a premium from the buyer for granting this right. The specified price is also known as the strike price or exercise price, and the specified time is also known as the expiration date or maturity date. The buyer can exercise the option at any time before or on the expiration date, depending on whether the option is American or European.

Interest rate options are used for various purposes, such as hedging against interest rate risk, speculating on interest rate movements or volatility, or enhancing portfolio returns. For example, a borrower who expects interest rates to rise can use an interest rate option to hedge against interest rate risk by buying a cap option and paying a premium. A cap option gives the buyer the right to receive a payment if the reference rate exceeds the strike rate. Alternatively, an investor who expects interest rates to fall can use an interest rate option to profit from lower interest rates by buying a floor option and paying a premium. A floor option gives the buyer the right to receive a payment if the reference rate falls below the strike rate.

There are many types of interest rate options that are traded in the market, each with its own characteristics and applications. In this section, we will focus on two main types of interest rate options: caps and floors and swaptions.

#### Caps and floors

the right to receive a payment if the reference rate exceeds the strike rate at each payment date. The payment amount is calculated as the difference between the reference rate and the strike rate multiplied by the notional amount and the day count fraction. A floor option is a contract that gives the buyer the right to receive a payment if the reference rate falls below the strike rate at each payment date. The payment amount is calculated as the difference between the strike rate and the reference rate multiplied by the notional amount and the day count fraction.

Caps and floors are used for various purposes, such as hedging against interest rate risk, speculating on interest rate movements or volatility, or enhancing portfolio returns. For example, a borrower who has a floating-rate loan can use a cap option to hedge against rising interest rates by buying a cap option and paying a premium. The cap option will pay the borrower if the reference rate exceeds the strike rate, offsetting the higher interest payments on the loan. Alternatively, an investor who has a floating-rate bond can use a floor option to enhance returns by buying a floor option and paying a premium. The floor option will pay the investor if the reference rate falls below the strike rate, supplementing the lower interest payments on the bond.

The valuation of a cap or floor option is complex, as it involves estimating the expected value of the future payments using an option pricing model. The value of a cap or floor option at time t is:

V(t) = sum(E(max(r(ti) - K,0) * L * DCF(ti)) * DF(ti))

V(t) = sum(E(max(K - r(ti),0) * L *