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    Derivatives / rates

    Caplet bootstrap

    The market quotes caps with one flat volatility per maturity. A model needs one volatility per forward period. Because a cap is exactly the sum of its caplets, subtracting one cap price from the next isolates a single caplet and its volatility can be solved for — the first calibration step of any LIBOR market model.

    Periods6First caplet21.20%Curve shapeUpward
    Longest caplet vol
    29.70%
    period 6, 5–6 years
    Forward vol
    37.16%
    implied between the last two periods
    Cap price
    3.182e+6
    on 100m notional
    Repricing error
    0.0000
    should be at the solver tolerance
    Caplet strip
    Each cap price minus the one before it isolates a single caplet, whose volatility is then solved for.
    6 caplets
    Caplet strip
    PeriodForwardCaplet volForward volCaplet price
    0–1y4.081%20.00%20.00%7.790e+4
    1–2y4.081%21.20%21.20%3.537e+5
    2–3y4.081%23.29%25.20%5.055e+5
    3–4y4.081%25.39%29.14%6.344e+5
    4–5y4.081%27.52%33.11%7.511e+5
    5–6y4.081%29.70%37.16%8.592e+5

    The first caplet expires on the valuation date, so it carries no optionality and its volatility cannot be identified from the strip — it is set equal to the first cap volatility by convention, not by calibration.

    Inspect the model
    Choose a quick result view, a scaled comparison, or the complete diagnostic set.
    Caplet volatility curve
    The bootstrapped spot vols, which is what the model actually consumes.
    Caplet volatility curve0.0006.6513.3119.9626.6133.271y1y2y2y3y3y4y4y5y5y6y6yPercent
    Cap price against caplet sum
    The reconstruction should sit exactly on the market cap price.
    Cap price against caplet sum0.000750.901501.812252.713003.613754.5277.9077.901y1y431.59431.592y2y937.13937.133y3y4y4y5y5y6y6yMarketRebuilt
    Canonical equations
    The exact relationships implemented by this engine.

    Cap_n = Σ_{i≤n} Caplet_i · The identity the bootstrap rests on.

    N·τ·P(0,t_i)·[F_i N(d₁) − K N(d₂)] · Black on the forward rate.

    Caplet_n = Cap_n − Cap_{n−1} · Then invert for the volatility.

    σ_f² = (σ_n²t_n − σ_{n−1}²t_{n−1})/(t_n − t_{n−1}) · From additivity of variance.

    finance methodology

    A clear path from inputs to results.

    The market quotes caps with one flat volatility per maturity. A model needs one volatility per forward period. Because a cap is exactly the sum of its caplets, subtracting one cap price from the next isolates a single caplet and its volatility can be solved for.

    Step-by-step methodWorked exampleAssumptions explained

    How to use

    1. Enter the notional, the strike and a flat zero curve.
    2. Set the one-year and longest cap volatilities.
    3. Choose how many maturities the strip has.
    4. Read the bootstrapped caplet curve, the forward vols and the repricing error.

    The formula

    Capn=∑i=1nCapleti  ;  Capletn=Capn−Capn−1\mathrm{Cap}_n = \sum_{i=1}^{n} \mathrm{Caplet}_i \;;\; \mathrm{Caplet}_n = \mathrm{Cap}_n - \mathrm{Cap}_{n-1}

    Reference: Brace, Gatarek & Musiela (1997) for the LIBOR market model; the caplet identity is standard Black (1976) cap pricing.

    Worked example

    A six-year cap strip at 4%

    Inputs
    • 100m notional, 4% strike, 4% flat curve.
    • Cap vols rising from 20% to 26% across six maturities.
    Calculation path
    1. 1

      The one-year cap prices as a single caplet at 20% volatility.

    2. 2

      Each subsequent cap adds one more caplet, priced at that maturity's flat cap vol.

    3. 3

      Subtracting consecutive cap prices isolates each caplet and inverting Black gives its volatility.

    4. 4

      The second caplet bootstraps to 21.20% and the sixth to 29.70%.

    5. 5

      The forward volatility for the final period is 37.16%, far above the spot curve, because it is the second difference of total variance.

    6. 6

      The strip reprices to within 0.000 of the market caps.

    Result

    21.20% to 29.70% spot · 37.16% forward · repricing error 0.000.

    Interpreting the result

    The spot caplet curve is smooth and the forward curve is not: forward volatility is a small difference of large variances, so it amplifies every rounding in the quotes. That is why a trader quotes spot caplet vols and derives forwards, never the other way round.

    Assumptions & limitations

    • Annual periods on a flat continuously compounded curve.
    • Each cap is quoted with a single flat volatility across its caplets.
    • The first caplet expires on the valuation date, so its volatility is unidentifiable and set by convention.
    • No basis, no collateral adjustment, no smile.

    Frequently asked questions

    Why is the first caplet volatility unidentifiable?

    Because it expires today: there is no optionality left, so its price does not depend on volatility. The bootstrap sets it equal to the first cap volatility by convention.

    Why does the forward vol exceed the spot vols?

    Because the spot curve is rising steeply: the last period contributes more variance than its share of the average, so the isolated forward is higher than either endpoint.

    What is a forward volatility?

    The volatility implied for a single future period by two spot caplet vols, from the additivity of variance. It is what the model uses to price a forward-starting instrument.

    Does this calibrate an LMM?

    It is the first step. The LMM also needs a correlation matrix, which the cap strip alone cannot identify — swaptions are required for that.

    Methodology last reviewed 2026-10-22. Results are informational and not financial or engineering advice — see our disclaimer.