Swap Contracts

Overview

Swaps are OTC derivative contracts where two parties exchange cash flows based on a notional amount. They are the largest derivative market globally, with notional values exceeding $500 trillion.

Difficulty advanced

Interest Rate Swaps

Plain Vanilla Swap

Party A: Pays fixed rate, receives floating (LIBOR/SOFR)
Party B: Pays floating, receives fixed

Notional: $100 million
Fixed Rate: 4.00%
Floating: SOFR + 0.10%
Tenor: 5 years
Payment Frequency: Quarterly

Pricing

Fixed Rate = Par Swap Rate

The fixed rate that makes the swap have zero value at inception:

Σ (Fixed Rate × Δt_i × DF_i) = Σ (Forward Rate_i × Δt_i × DF_i)

Where:
Δt_i = Day count fraction for period i
DF_i = Discount factor for period i
Forward Rate_i = Implied forward rate for period i

where: Δt_i accrual fraction (e.g. 30/360, ACT/360) · DF_i discount factor from the OIS/SOFR curve · Forward Rate_i implied forward computed from the same curve. does: sets the par swap rate so PV(fixed leg) = PV(floating leg) at trade date — a weighted average of forwards, weighted by discounted accruals. Bootstrapped from observed par swap quotes to build the curve itself; every subsequent valuation discounts off this curve.

Swap Valuation

Value to Fixed Payer = PV(Floating Leg) - PV(Fixed Leg)

PV(Fixed Leg) = Notional × Fixed Rate × Σ(Δt_i × DF_i)
PV(Floating Leg) = Notional × (1 - DF_n)  # Par value at reset

where: Notional swap notional · Fixed Rate contractual fixed coupon · Δt_i · DF_i per-period discounted accrual (its sum = the swap's PV01 / annuity factor) · DF_n discount factor to final payment. does: marks an IRS to market — fixed leg values as an annuity of fixed coupons; floating leg telescopes to 1 − DF_n because floating coupons reset to par each period. A fixed-receiver is long duration: PV rises when rates fall. The annuity factor Σ(Δt_i · DF_i) is the swap's DV01 — the dollar P&L per 1bp rate move.

Credit Default Swaps (CDS)

Structure

Protection Buyer: Pays periodic premium (CDS spread)
Protection Seller: Pays par - recovery if credit event occurs

Notional: $10 million
CDS Spread: 150 bps (1.50% per year)
Tenor: 5 years
Reference Entity: XYZ Corp

CDS Spread

CDS Spread ≈ (1 - Recovery Rate) × Hazard Rate

Where:
Hazard Rate = Probability of default per period
Recovery Rate = Expected recovery after default (typically 40%)

Approximate 5-year CDS:
CDS Spread ≈ PD_5yr × (1 - R) / 5

where: Hazard Rate instantaneous default intensity (λ) under risk-neutral measure · Recovery Rate (R) fraction of par recovered on default, conventionally 40% for senior unsecured · PD_5yr cumulative 5-year default probability. does: decomposes the CDS premium into the expected loss per unit time. Practitioners use this to back out a market-implied default probability from a quoted spread (PD ≈ Spread / (1 − R)). Spread widens when credit deteriorates → protection buyer mark-to-market gain. Foundation for capital structure arb and basis trades vs cash bonds.

Protection Value

Upfront Payment ≈ (CDS Spread - Running Spread) × Duration

If CDS spread widens (credit deteriorates):
- Protection increases in value
- Protection buyer profits

where: CDS Spread current market spread · Running Spread standardized fixed coupon (100 bps for IG, 500 bps for HY post-Big Bang) · Duration risky annuity (≈ 4-5 for 5y). does: translates a quoted par spread into the upfront cash payment under the standardized SNAC convention. P&L for an existing CDS position equals ΔSpread × Duration × Notional — same DV01 logic as bonds, applied to credit. Drives jump-to-default and credit curve trades.

Total Return Swaps

TRS Receiver: Receives total return of asset (price + dividends)
TRS Payer: Receives LIBOR/SOFR + spread

Used for:
- Synthetic exposure without ownership
- Regulatory capital optimization
- Short exposure
- Emerging market access

Cross-Currency Swaps

Party A: Pays USD interest, receives EUR interest
Party B: Pays EUR interest, receives USD interest

Includes:
- Initial exchange of principal (at spot rate)
- Periodic interest payments in each currency
- Final re-exchange of principal at same rate

Major Swap Markets

Market	Notional Outstanding	Key Benchmark
Interest Rate	$400+ trillion	SOFR, EURIBOR
CDS	$10+ trillion	CDX, iTraxx
FX Swaps	$70+ trillion	Cross-currency basis
TRS	$10+ trillion	Various indices

Counterparty Risk

Swaps are OTC instruments with counterparty risk:

Credit Valuation Adjustment (CVA):
CVA = (1 - Recovery) × E[Exposure at Default] × PD

Debit Valuation Adjustment (DVA):
DVA = CVA from the counterparty's perspective

Collateral:
- CSA (Credit Support Annex) governs collateral
- Daily margin calls reduce counterparty exposure

where: Recovery recovery rate on counterparty default (typically 40%) · E[Exposure at Default] expected positive MTM exposure at default time (computed by Monte Carlo over the netting set) · PD counterparty default probability over the horizon. does: prices counterparty credit risk into an OTC derivative — it's the discount you'd take to sell a trade to a risk-free counterparty. CVA desks hedge it with single-name CDS or CDS indices; XVA changes flow through P&L when either counterparty's credit moves. DVA is the mirror image (your own default reduces your liability), controversial because banks book P&L when their own credit worsens.

Regulatory Framework

1. Dodd-Frank Act (US): Mandatory clearing, trade reporting
2. EMIR (EU): Clearing obligation, risk mitigation
3. Basel III: Capital requirements for swap exposures
4. IBOR Transition: LIBOR → SOFR/SONIA/EURIBOR

Risk Considerations

1. Counterparty Risk: Default of swap partner
2. Basis Risk: Floating rate may not match exposure
3. Liquidity Risk: Hard to unwind large positions
4. Legal Risk: Documentation disputes
5. Operational Risk: Settlement failures

Checklist

• [ ] ISDA Master Agreement in place
• [ ] Credit Support Annex (collateral terms) defined
• [ ] Counterparty credit limits established
• [ ] Daily P&L and MTM process
• [ ] Legal documentation complete
• [ ] Regulatory reporting obligations met
• [ ] Termination events and breach conditions understood

References

1. Hull, J.C. (2022). Options, Futures, and Other Derivatives (11th ed.). Pearson.
2. ISDA. (2023). "Swaps Information and Statistics." International Swaps and Derivatives Association.
3. Brigo, D. & Mercurio, F. (2006). Interest Rate Models: Theory and Practice (2nd ed.). Springer.