Interest Rate Derivatives

Swaps, swaptions, caps/floors, curve construction, OIS discounting, and multi-curve framework.

When to Activate

• User pricing interest rate swaps or computing swap rates
• Constructing yield curves from market instruments (bootstrapping)
• Pricing swaptions, caps, or floors
• Implementing multi-curve framework with OIS discounting
• Understanding convexity adjustments for non-standard rate products
• Analyzing interest rate risk and DV01/duration for derivatives portfolios

Core Concepts

Interest Rate Swap (IRS) Mechanics

Plain Vanilla IRS

• Exchange fixed rate for floating rate on a notional principal
• Fixed leg: pays C * delta_i * N at each period (C = swap rate, N = notional)
• Floating leg: pays L_i * delta_i * N at each period (L = LIBOR/SOFR, set in advance or arrears)
• No exchange of notional (hence "notional" principal)
• At inception: swap rate is set so PV(fixed leg) = PV(floating leg) — zero NPV

Swap Rate Calculation

• Par swap rate: C = [1 - DF(T_N)] / sum(delta_i * DF(T_i))
• DF(T_i) = discount factor to time T_i
• Numerator: difference between initial and final discount factors
• Denominator: sum of discounted accrual factors (the annuity or PV01)

DV01 and Duration

• DV01 = change in swap value for 1bp parallel shift in the curve
• For a swap: DV01 approximately = notional * PV01 * 0.0001
• PV01 (annuity) = sum of delta_i * DF(T_i)
• A 10Y swap on $100M has DV01 approximately $85K-$95K depending on the rate environment

Yield Curve Construction (Bootstrapping)

Instruments Used

• Short end (0-2Y): deposit rates, FRAs, or overnight rate futures (SOFR futures)
• Medium term (2-5Y): interest rate swaps (or futures for liquid tenors)
• Long end (5-30Y+): interest rate swaps
• Convexity adjustments needed for futures-based curves

Bootstrapping Process

1. Start with the shortest maturity deposit rate: DF(T_1) = 1 / (1 + r_1 * T_1)
2. Use successive swap rates to solve for each discount factor:
   • C_n * sum(delta_i * DF(T_i)) + DF(T_n) = 1
   • DF(T_n) = (1 - C_n * sum(delta_i * DF(T_i) for i<n)) / (1 + C_n * delta_n)
3. Interpolate between pillar points: log-linear on discount factors or monotone convex on zero rates
4. Result: a complete discount factor curve DF(T) for any maturity T

Interpolation Methods

• Linear on zero rates: simple but can produce non-smooth forwards
• Log-linear on discount factors: ensures positive forwards
• Cubic spline on zero rates: smooth but can oscillate
• Monotone convex: smooth and produces well-behaved forward rates — preferred in practice
• Tension splines: control oscillation with a tension parameter

Multi-Curve Framework (Post-2008)

Before 2008, LIBOR was used for both discounting and projection. Post-crisis:

OIS Discounting

• Collateralized derivatives earn the overnight rate (SOFR/ESTR/SONIA) on posted collateral
• Therefore, discount using the OIS curve, not the LIBOR/swap curve
• OIS-LIBOR basis: LIBOR > OIS due to bank credit risk (was ~10bp pre-crisis, spiked to 350bp in 2008)

Projection Curves

• Each tenor (1M, 3M, 6M LIBOR or SOFR) has its own forward curve
• Forward 3M rate is read from the 3M projection curve
• Discount factor for present value comes from the OIS curve
• Basis swaps (e.g., 3M vs 6M) calibrate the relationship between projection curves

LIBOR to SOFR Transition

• LIBOR ceased publication (USD: June 2023 for most tenors)
• SOFR (Secured Overnight Financing Rate) replaced USD LIBOR
• SOFR is an overnight rate; term SOFR derived from futures
• Compounded SOFR in arrears replaces forward-looking LIBOR
• Legacy LIBOR contracts: fallback spread adjustments (ISDA protocol)

Methodology

Swaption Pricing

Black's Model for Swaptions

• Swaption gives the right to enter a swap at a predetermined fixed rate
• Payer swaption: right to pay fixed (benefits from rising rates)
• Receiver swaption: right to receive fixed (benefits from falling rates)
• Under Black's model: swaption price = A * Black76(F, K, sigma_N * sqrt(T), T)
  • A = annuity factor (PV01 of the underlying swap)
  • F = forward swap rate
  • K = strike swap rate
  • sigma_N = normal (Bachelier) vol or sigma_LN = lognormal vol
  • T = option expiry

Normal vs. Lognormal Quoting

• Normal (Bachelier) vol: quoted in bp; payoff proportional to (F - K)
• Lognormal (Black) vol: quoted in %; payoff proportional to F * (F/K - 1)
• Normal vol is now market standard for rates (handles negative rates naturally)
• Conversion: sigma_N approximately = sigma_LN * F for ATM

Swaption Cube

• Three dimensions: option expiry, underlying swap tenor, strike (or moneyness)
• ATM swaption matrix: expiry (1M to 30Y) x tenor (1Y to 30Y)
• Smile: parameterized by SABR at each expiry-tenor point
• The swaption cube drives pricing for all rate-exotic products

Caps and Floors

Cap: portfolio of caplets, each paying max(L_i - K, 0) * delta_i * N at time T_{i+1} Floor: portfolio of floorlets, each paying max(K - L_i, 0) * delta_i * N at time T_{i+1} Cap-Floor Parity: Cap - Floor = Swap (payer)

Caplet Pricing

• Each caplet is a European call on the forward rate
• Black's formula: Caplet = DF(T_{i+1}) * delta_i * N * [F_i * N(d1) - K * N(d2)]
• Or normal model: Caplet = DF(T_{i+1}) * delta_i * N * [(F_i - K)N(d) + sigmasqrt(T)*phi(d)]

Flat Vol vs. Spot Vol

• Flat vol: a single vol that prices the entire cap correctly (not individual caplets)
• Spot vol: individual caplet vols — the term structure of caplet volatilities
• Stripping spot vols from flat vols: bootstrap forward from the shortest cap
• Spot vol term structure typically humps around 1-3 years then declines

Convexity Adjustments

Futures vs. Forward Rate

• Eurodollar/SOFR futures prices imply rates that differ from forward rates
• Convexity adjustment: Forward = Futures_rate - 0.5 * sigma^2 * T_1 * T_2
• Adjustment is positive: forward rate < futures-implied rate
• Larger for longer-dated futures (can be 10-20bp for 5Y+ futures)

CMS (Constant Maturity Swap) Convexity

• CMS rate = swap rate observed at a future date, paid with a delay
• CMS rate expectation differs from the forward swap rate due to convexity
• CMS convexity adjustment: depends on the swaption smile (particularly the curvature/vol of vol)
• Replication approach: express CMS payoff as an integral over swaptions across all strikes
• CMS adjustment can be 5-30bp depending on tenor and market conditions

Interest Rate Risk Measures

DV01 (Dollar Value of 01)

• Change in PV for a 1bp parallel shift in the zero curve
• DV01 = -dV/dy * 0.0001

Key Rate Duration (KRD)

• Sensitivity to shifts at specific tenor points (2Y, 5Y, 10Y, 30Y)
• Sum of key rate durations approximately = effective duration
• Reveals curve exposure: a portfolio can be duration-neutral but have significant curve risk

Gamma (Convexity in Rates)

• Second-order sensitivity: d^2V/dy^2
• Positive convexity: bonds, receiver swaptions (benefit from both rate rises and falls)
• Negative convexity: callable bonds, mortgage-backed securities (lose from big moves)

Examples

Swap Rate Bootstrapping

Market instruments:
  6M deposit: 4.50%
  1Y swap: 4.60%
  2Y swap: 4.70%
  5Y swap: 4.80%

DF(0.5) = 1/(1 + 0.045 * 0.5) = 0.97799

1Y swap (semi-annual): 0.046/2 * DF(0.5) + (1 + 0.046/2) * DF(1.0) = 1
  0.023 * 0.97799 + 1.023 * DF(1.0) = 1
  DF(1.0) = (1 - 0.02249) / 1.023 = 0.95553

2Y swap: 0.023 * [DF(0.5) + DF(1.0) + DF(1.5) + DF(2.0)] + DF(2.0) = 1
  Interpolate DF(1.5), solve for DF(2.0) = 0.91098

Zero rate at 2Y: -ln(0.91098)/2 = 4.66% (continuously compounded)

Swaption Pricing (Normal Model)

1Y expiry into 5Y swap (1Y5Y swaption)
Forward swap rate: 4.50%
Strike: 4.50% (ATM)
Normal vol: 85bp (0.0085)
Annuity factor (PV01): 4.35 per 100 notional
Notional: $50M

ATM normal model: Payer swaption = A * sigma * sqrt(T) * phi(0) * N
  = 4.35 * 0.0085 * 1.0 * 0.3989 * $50M
  = $737,700

Receiver swaption = same price (ATM put-call parity for normal model)

Cap Stripping

1Y cap (2 caplets, semi-annual): flat vol = 80bp, price = $42,000 on $10M
2Y cap (4 caplets): flat vol = 82bp, price = $98,000

First 2 caplets are known from 1Y cap.
Remaining 2 caplets: $98,000 - $42,000 = $56,000
Spot vol for 1.5Y and 2.0Y caplets: solve for vol that prices $56,000 using 2 caplets.
Spot vol (1.5Y-2.0Y): 84bp (higher than flat vol — term structure is upward sloping).

Quality Gate

• Bootstrapped curve must reprice all input instruments to within 0.1bp
• Discount factors must be monotonically decreasing; forward rates must be non-negative (post-negative rate era: floors at reasonable levels)
• OIS discounting must be used for all collateralized derivatives — single-curve pricing is incorrect post-2008
• Swaption prices must satisfy put-call parity: payer - receiver = forward swap value
• Cap-floor parity must hold: cap - floor = swap at the same strike
• SABR calibration for swaptions must fit ATM vol, risk reversal, and butterfly within market bid-ask
• Convexity adjustments must be applied for futures-based curve construction and CMS products
• Interpolation method must produce smooth and well-behaved forward curves (check visually)
• Key rate durations must sum to approximately the effective duration (within 5%)
• Multi-curve framework must use consistent OIS curve for discounting across all tenor projection curves