Curve Construction for Bond Pricing

Core Architecture

Modern curve construction uses the multi-curve framework:

┌─────────────────────────────────────────────────────────────┐
│                    CURVE HIERARCHY                          │
├─────────────────────────────────────────────────────────────┤
│  DISCOUNT CURVES (OIS-based)                                │
│  ├── USD: SOFR OIS                                          │
│  ├── EUR: €STR OIS                                          │
│  ├── GBP: SONIA OIS                                         │
│  └── Used for: Present value discounting                    │
├─────────────────────────────────────────────────────────────┤
│  PROJECTION CURVES (tenor-specific)                         │
│  ├── SOFR Term (1M, 3M)                                     │
│  ├── EURIBOR (1M, 3M, 6M)                                   │
│  └── Used for: Forward rate estimation                      │
├─────────────────────────────────────────────────────────────┤
│  GOVERNMENT YIELD CURVES                                    │
│  ├── Treasury (USD), Gilts (GBP), Bunds (EUR)              │
│  ├── Par, Zero, Forward representations                     │
│  └── Used for: Spread benchmarks, risk-free rates           │
├─────────────────────────────────────────────────────────────┤
│  SPREAD/CREDIT CURVES                                       │
│  ├── Z-spread curves by rating/sector                       │
│  ├── CDS hazard rate curves                                 │
│  └── Used for: Corporate bond pricing, credit analysis      │
├─────────────────────────────────────────────────────────────┤
│  INFLATION CURVES                                           │
│  ├── Real yield curves (TIPS, linkers)                      │
│  ├── Breakeven inflation                                    │
│  └── Used for: Inflation-linked bond pricing                │
├─────────────────────────────────────────────────────────────┤
│  CROSS-CURRENCY CURVES                                      │
│  ├── EUR collateralized in USD                              │
│  ├── GBP collateralized in USD                              │
│  └── Used for: Multi-currency asset swaps, XCCY pricing     │
├─────────────────────────────────────────────────────────────┤
│  REPO/FINANCING CURVES                                      │
│  ├── GC repo term structure                                 │
│  ├── Special repo rates by bond                             │
│  └── Used for: Carry, rolldown, funding analysis            │
└─────────────────────────────────────────────────────────────┘

Build Order (Critical)

Always construct curves in this sequence:

1. OIS discount curves - SOFR, SONIA, €STR (anchor currency first, typically USD)
2. Projection curves - Using OIS for discounting
3. Cross-currency curves - Using domestic OIS + FX forwards + XCCY basis

Quick Reference: Key Equations

OIS Bootstrapping

D(0,Tᵢ) = [1 - sᵢ × Σⱼ₌₁ⁱ⁻¹ D(0,Tⱼ) × τⱼ] / [1 + sᵢ × τᵢ]

Multi-Curve Forward Rate

Fₘ(t; T₁,T₂) = (1/τₘ) × [Pₘ(t,T₁)/Pₘ(t,T₂) - 1]

Cross-Currency Basis

D_for^USD-coll(T) = D_USD(T) × S / F(T)

Present Value (Multi-Curve)

PV = Σᵢ D_discount(Tᵢ) × τₘ × F_projection(Tᵢ₋₁, Tᵢ)

Z-Spread

P = Σᵢ CFᵢ × exp(-(r(tᵢ) + z) × tᵢ)

Breakeven Inflation

BEI(T) = y_nominal(T) - y_real(T)

Nelson-Siegel Zero Rate

r(τ) = β₀ + β₁×[(1-e^(-τ/λ))/(τ/λ)] + β₂×[(1-e^(-τ/λ))/(τ/λ) - e^(-τ/λ)]

CDS Hazard Rate (Approximation)

λ ≈ spread / (1 - Recovery)

Carry

Carry = Coupon_Accrual - (Dirty_Price × Repo_Rate × Days/360)

Implementation Workflow

Step 1: OIS Curve Bootstrap

fn bootstrap_ois(instruments: &[OISSwap]) -> Curve {
    let mut nodes: Vec<(f64, f64)> = vec![(0.0, 1.0)]; // D(0,0) = 1
    
    for inst in instruments.iter().sorted_by_key(|i| i.maturity) {
        let df = solve_discount_factor(inst, &nodes);
        nodes.push((inst.maturity, df));
    }
    
    Curve::new(nodes, Interpolation::LogLinear)
}

fn solve_discount_factor(inst: &OISSwap, prior: &[(f64, f64)]) -> f64 {
    // Direct solve if no intermediate dates needed
    // Otherwise use Brent solver with 1e-12 tolerance
}

See references/ois-curves.md for overnight compounding, meeting-date construction, and turn-of-year handling.

Step 2: Projection Curve Bootstrap

fn bootstrap_projection(
    instruments: &[SwapInstrument],
    discount_curve: &Curve,  // OIS curve
) -> Curve {
    // Use OIS curve for discounting
    // Solve for projection curve nodes
}

See references/multi-curve.md for tenor basis, convexity adjustments, and dual-curve stripping.

Step 3: Cross-Currency Curve Bootstrap

fn bootstrap_xccy(
    fx_forwards: &[FXForward],      // Short end
    xccy_swaps: &[XCCYBasisSwap],   // Long end  
    domestic_ois: &Curve,
    spot_fx: f64,
) -> Curve {
    // Phase 1: FX forward implied (< 1Y)
    // Phase 2: XCCY basis swap bootstrap (> 1Y)
}

See references/cross-currency.md for basis swap mechanics, collateral switching, and simultaneous calibration.

Interpolation Selection

See references/interpolation.md for algorithms and positivity constraints.

Sensitivity Calculation

Bucketed PV01 requires the calibration Jacobian:

fn compute_key_rate_durations(
    curve: &Curve,
    portfolio: &Portfolio,
) -> Vec<f64> {
    // Use algorithmic differentiation (5x faster than bump)
    // Or finite difference with bump = 1e-4 to 1e-6
}

See references/numerical.md for AD implementation, solver selection, and performance optimization.

Collateral Currency Rules

Performance Targets

Common Pitfalls

Curve Construction

1. Wrong build order - Always OIS first, then projection, then XCCY
2. Missing convexity adjustment - Futures ≠ forwards (adjust 1-2bp short end)
3. Ignoring turn-of-year - Can be 5-500bp jump at year-end
4. Single-curve pricing - Post-2008 spreads can reach 200bp
5. Wrong collateral curve - 10%+ error on long-dated swaps possible

Bond Spread Calculations

6. Z-spread compounding mismatch - Use semi-annual for USD, match convention
7. Dirty vs clean price confusion - Always use dirty for PV calculations
8. Benchmark maturity mismatch - G-spread uses wrong point; prefer Z-spread
9. Ignoring accrued interest - Critical for short-dated bonds

Inflation-Linked Bonds

10. Missing indexation lag - TIPS use 3-month CPI lag
11. Ignoring seasonality - CPI seasonal patterns affect short-dated breakevens
12. Forgetting deflation floor - TIPS principal cannot go below par at maturity

Credit/CDS

13. Recovery rate assumption - Typically 40% for senior unsecured, but varies
14. Ignoring accrued premium - CDS premium accrues to default date
15. Hazard rate vs spread confusion - λ ≈ spread/(1-R), not equal

Repo/Financing

16. Using wrong repo rate - GC vs special can differ 50bp+
17. Day count convention - Repo uses ACT/360, bonds vary by market
18. Ignoring haircuts - Affects actual funding amount