Zero rates, discounting and implied forwards
A zero rate discounts a single cash flow at maturity: DF(T) = e^(−z(T)·T) in continuous terms. Market quotes come as par swaps or deposits, not zero rates; bootstrapping converts them one by one, holding earlier DFs fixed: D = (1 − K·A_known)/(1 + K·αₙ) for the next node. Implied forwards fall out of the curve: f(t₁,t₂) = [z₂t₂ − z₁t₁]/(t₂−t₁) — today's price of a rate that starts in the future.
Interpolation: the choice that builds your forwards
Between knots what you 'see' depends on the method: linear on discount factors is simple and stable; log-linear on DF gives flat implied forwards between knots (more natural forward behavior); cubic splines on zero rates are smooth but global — adding one knot can move the whole curve. No method is universally right; the platform's rule is to make the choice explicit and test round-trip consistency.
Worked example: bootstrapping three nodes
- 6m deposit at 4% (simple): DF(0.5) = 1/(1 + 0.04×0.5) = 0.98039.
- 1y par swap at 4.2%, annual pay: only one payment → DF(1) = 1/1.042 = 0.95969.
- 2y par swap at 4.5%: A_known = 0.042 × 0.95969 = 0.04031 → DF(2) = (1 − 0.04031)/1.045 = 0.91571.
- Round-trip: K*(2y) = (1 − 0.91571)/(0.95969 + 0.91571) = 4.495% ≈ 4.5% ✓.
- Continuous zeros: z(1) = −ln(0.95969) = 4.12%; z(2) = −ln(0.91571)/2 = 4.40% — a mildly upward-sloping curve.
نتیجه: Bootstrapping is a chain of small solves; every node's error propagates forward. That is why par-rate round-trip is part of the platform's publication standard.