Interest Rate Swap

An interest rate swap is a deal to exchange two kinds of interest on the same notional amount: one side pays a fixed rate agreed today, the other pays a floating rate that resets with the market. Only the interest difference changes hands — the notional never moves.

Why would anyone? Because rate risk is everywhere. A company with floating-rate debt fears rate rises: it enters a swap to pay fixed and receive floating — its floating loan cost is now offset, and it effectively pays a known fixed rate. A pension fund needing long-term fixed income does the reverse.

Swaps are the largest derivatives market on the planet (notionals in the hundreds of trillions) because they are the cheapest tool ever devised for moving interest-rate risk from those who fear it to those paid to hold it.

Anatomy of a standard swap

• Notional: reference amount, e.g. $100M (never exchanged).
• Fixed leg: the "swap rate", paid annually/semi-annually — this is what's quoted.
• Floating leg: today, compounded overnight rates (SOFR, €STR) paid periodically — the post-LIBOR standard.
• Clearing: most swaps clear at CCPs (LCH, CME) with daily margin; the credit risk of old bilateral swaps is largely gone.

The swap curve

Swap rates quoted across maturities form the swap curve — effectively the market's fixed-income backbone, often more liquid than government bonds. A swap's fixed rate equals the average of expected floating rates over its life (plus tiny adjustments), so the curve is a readout of expected central-bank policy.

Marking to market

Rates move, and your swap gains or loses value. Pay fixed at 3% and the 10-year swap rate rises to 4%? Your position is a winner: you locked cheap fixed payments. Receivers profit when rates fall. A 10y swap's sensitivity: roughly $80k per basis point per $100M notional.

Pricing: two legs, one curve (now)

Value each leg on the OIS discount curve \(P(0,t)\). The floating leg (paying the same overnight index used for discounting) is worth par-like telescoping sums; the fair swap rate makes the legs equal:

— the swap rate is a discount-weighted average of forward rates \(f_i\). The denominator \(A = \sum \delta_i P(0,T_i)\) is the annuity; a seasoned swap's value is \(V = (S_{now} - S_{traded})\times A\times N\) for the receiver of \(S_{traded}\)… with sign per side.

Risk language

• DV01: dollar value of a basis point, ≈ \(A \times N \times 10^{-4}\).
• Curve risk: bucketed key-rate DV01s; traders trade 2s10s steepeners, butterflies, etc., as spread packages.
• Convexity/gamma: linear at first order; second-order effects matter at size and in spread trades vs. futures.

The multi-curve legacy

Pre-2008, one LIBOR curve did projection and discounting. The crisis broke that: OIS discounting, tenor bases, and cross-currency bases created the multi-curve framework. RFR reform re-simplified domestic swaps (projection = discounting = OIS) but bases persist across currencies and legacy tenors.

Swap spreads

Swap rate minus government yield at the same maturity — driven by balance-sheet costs, issuance flows and collateral scarcity; famously negative at long US maturities since 2015, a standing puzzle priced as the cost of dealer intermediation.

The formulas above are standard textbook formulations, simplified for teaching. They explain the mechanism — they are not a valuation tool, and they will not reproduce a dealer’s price.