arxiv: 2605.05140 · v2 · submitted 2026-05-06 · 💱 q-fin.CP

Recognition: unknown

What Can Go Wrong During Caplet Stripping ?

Fabien Le Floc'h

Pith reviewed 2026-05-08 15:03 UTC · model grok-4.3

classification 💱 q-fin.CP

keywords caplet strippingvolatility interpolationcap quotesbootstrap equivalencepositive volatility curvesinterpolation kernelsnode placementfinancial derivatives

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The pith

Interpolation choices and node placement are the main causes of oscillations and negative values when stripping caplets from market quotes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates why extracting caplet volatilities from cap quotes frequently leads to unstable results like large swings or negative numbers. It identifies the interpolation method between data points and the location of those points as the chief sources of the problem, with occasional bad market quotes making things worse. The authors demonstrate that switching to continuous flat-linear or C1 flat-smooth kernels, placing nodes at midpoints, solving globally, and enforcing positivity through exponential reparametrization or Hyman splines produces stable, positive caplet curves. These changes keep the extracted volatilities exactly consistent with the original cap prices while cutting oscillations sharply. The result is a practical workflow that works reliably on real market data without heavy manual fixes.

Core claim

Instability during exact caplet stripping arises primarily from the interpolation scheme and node placement, which can be worsened by isolated poor quotes. Continuous flat-linear and C1 flat-smooth kernels that preserve bootstrap equivalence, combined with midpoint node placement and a global solver, plus positivity enforcement via exponential reparametrization or Hyman non-negative C1 splines, deliver substantially reduced oscillations, robust positive caplet curves, and negligible repricing error.

What carries the argument

Continuous flat-linear and C1 flat-smooth interpolation kernels with midpoint node placement and global solver, plus exponential reparametrization or Hyman splines for positivity.

If this is right

The workflow produces production-ready positive caplet curves without manual smoothing or quote filtering.
Repricing error on the input caps remains negligible, preserving exact market consistency.
Data quality checks become simpler because the method tolerates isolated bad quotes better.
The same kernels and placement can be reused across different tenors without retuning.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar kernel and node choices might reduce instability in other single-curve bootstraps such as swaption volatilities.
In live risk systems the positivity enforcement could be run continuously to flag when input data quality drops.
The approach opens the door to fully automated daily caplet surface construction with minimal human oversight.

Load-bearing premise

Market cap quotes are sufficiently clean and the proposed kernels preserve exact bootstrap equivalence without adding material bias under typical conditions.

What would settle it

Applying the flat-linear kernel, midpoint placement, and exponential reparametrization to a standard set of market cap quotes and still obtaining negative or highly oscillatory caplet volatilities would falsify the stability claim.

Figures

Figures reproduced from arXiv: 2605.05140 by Fabien Le Floc'h.

**Figure 1.** Figure 1: Caplet volatilities obtained with bootstrapping and piecewise-constant interpolation. The cap vols appear shifted by 1M because a cap maturity of 3M actually has its last fixing at 2M for a forward looking rate with 1M tenor. Maturity (months) 6 9 12 15 18 21 24 27 30 N orm al v ol (b p) 0 50 100 Zoom 6–30M view at source ↗

**Figure 2.** Figure 2: Caplet volatilities after bootstraping, zoom on the 6M-30M caplets. How would a child fill in the gaps between the dots? A child would simply join the dots with a line. This leads to the flat-linear interpolation: at each node boundary, instead of an instantaneous jump the vol transitions linearly over a short ramp of width 𝛽∆. Crucially, if 0 < 𝛽 < 1, all caplet maturities see exactly the same vol as the … view at source ↗

**Figure 3.** Figure 3: shows the impact of the ramp width 𝛽 on the caplet volatilities. Maturity (months) 0 3 6 9 12 15 18 21 24 N orm al v ol (b p) 0 50 100 Piecewise Constant vs Flat-Linear — Zoom 0–24M Flat (piecewise constant) Flat-linear β=0.25 Flat-linear β=0.5 Flat-linear β=0.8 Cap vol quotes Caplet vols (flat) view at source ↗

**Figure 4.** Figure 4: shows the flat-smooth interpolation with 𝛽 = 1. Maturity (months) 0 3 6 9 12 15 18 21 24 N orm al v ol (b p) 0 50 100 Piecewise Constant vs Flat-Smooth β=1 (Hermite) — Zoom 0–24M Flat (piecewise constant) Flat-smooth β=1 Cap vol quotes Caplet vols (flat) view at source ↗

**Figure 5.** Figure 5: shows that both approaches work. With the exponential change of variable, the linear interpolation at midpoints does not reach zero anymore. The C2 cubic spline presents a small oscillation, while the C1 cubic spline with non-negative filter does not present any oscillation and is smoother than the linear interpolation with exponential transform. Maturity (months) 0 50 100 150 N orm al v ol (b p) 0 50 100 … view at source ↗

**Figure 6.** Figure 6: Caplet volatilities obtained with linear interpolation at maturity nodes exhibit strong oscillations. 4.1. The Reason for the Oscillations With linear interpolation and at-maturity nodes 𝜏𝑘 = 𝑇𝑘 , the vol at caplet time 𝑡𝑖 ∈ (𝑇𝑘−1, 𝑇𝑘 ] is 𝜎(𝑡𝑖 ) = 𝑣𝑘−1 𝑇𝑘 − 𝑡𝑖 𝑇𝑘 − 𝑇𝑘−1 + 𝑣𝑘 𝑡𝑖 − 𝑇𝑘−1 𝑇𝑘 − 𝑇𝑘−1 . The sequential bootstrap solves for 𝑣𝑘 given (𝑣1 , … , 𝑣𝑘−1) already fixed. The incremental constraint (cap 𝑘 … view at source ↗

**Figure 7.** Figure 7: Caplet volatilities obtained with outliers removed. with the quotes, and that we should investigate further by looking at the outliers with the modified Z-score method. 6. Conclusion When using a simple bootstrap algorithm to strip the caplet volatilities, there is no good reason to use a pure piecewise-constant interpolation. A flat-linear interpolation is always preferable as it produces a continuous vol… view at source ↗

read the original abstract

We study exact and near exact extraction of caplet volatilities from market cap quotes and identify why some common choices produce extreme oscillations or negative vols. Interpolation scheme and node placement are shown to be the primary drivers of instability, which can be amplified by isolated bad quotes. We propose practical, production ready remedies: continuous flat-linear and C1 flat-smooth kernels that preserve bootstrap equivalence, midpoint node placement with a global solver, positivity enforcement via an exponential reparametrization or Hyman non-negative C1 splines. We also introduce simple data quality checks. Numerical experiments demonstrate substantially reduced oscillations, robust positive caplet curves, and negligible repricing error, delivering a fast and stable caplet stripping workflow suitable for real-world use.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows that interpolation scheme and node placement drive most caplet stripping instability and gives concrete, tested fixes that keep vols positive with tiny repricing error.

read the letter

The main takeaway is that standard caplet stripping often blows up because of how people interpolate and place nodes, and the authors give practical changes that fix it in their tests without breaking the bootstrap math. They introduce continuous flat-linear and C1 flat-smooth kernels, switch to midpoint nodes solved globally, and enforce positivity with either exponential reparametrization or Hyman non-negative C1 splines, plus a couple of simple data checks. These are presented as production-ready tweaks rather than a new theory. The numerical experiments show clear drops in oscillation and near-zero repricing error, which is the part that actually matters for daily use. The evidence is all numerical, so there is no formal guarantee it holds in every regime, and the work assumes market quotes are mostly clean rather than heavily contaminated. That is a real but limited limitation for a methods paper. The citation pattern looks standard for this corner of fixed-income quant work. This is for people who actually implement and maintain caplet curves in trading systems. A desk quant or risk modeler would find the before-and-after plots and the explicit kernel definitions useful. It is narrow but the problem is real and the fixes look implementable, so it deserves a serious referee. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript studies exact and near-exact extraction of caplet volatilities from market cap quotes. It identifies interpolation scheme and node placement as the primary drivers of instability (oscillations or negative vols), which can be amplified by isolated bad quotes. Practical remedies are proposed: continuous flat-linear and C1 flat-smooth kernels that preserve bootstrap equivalence, midpoint node placement with a global solver, positivity enforcement via exponential reparametrization or Hyman non-negative C1 splines, and simple data quality checks. Numerical experiments are reported to demonstrate substantially reduced oscillations, robust positive caplet curves, and negligible repricing error.

Significance. If the numerical evidence holds, the work supplies directly usable, production-ready guidance for a common task in interest-rate volatility modeling. The emphasis on preserving bootstrap equivalence while enforcing positivity and stability addresses a practical pain point; the combination of targeted kernel choices, node placement, and reparametrization offers a concrete workflow that can be implemented without altering the economic meaning of the stripped curve.

major comments (2)

[kernel section] Section describing the kernels (around the flat-linear and C1 flat-smooth proposals): the claim that these kernels 'preserve bootstrap equivalence' is central to the practical value of the remedies. The manuscript should supply either an explicit algebraic verification that the interpolated forward vols reproduce the input cap prices exactly, or a side-by-side numerical check (e.g., repricing error before and after the change) on the same market data set used for the oscillation experiments.
[numerical experiments] Numerical experiments section: while reduced oscillations and positive curves are reported, the manuscript does not state the precise quantitative metrics employed (maximum oscillation amplitude, integrated negativity measure, or L2 deviation from a reference curve) nor the number and diversity of market regimes tested. Without these, it is difficult to judge whether the improvement is robust or specific to the chosen data snapshots.

minor comments (2)

[figures] Figure captions should explicitly list the market data date, the tenor structure, and the solver tolerance used for the global optimization so that readers can reproduce the oscillation plots.
[data quality] The data-quality checks are introduced as 'simple'; a short pseudocode or explicit threshold values (e.g., quote deviation in basis points) would make them immediately usable by practitioners.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the helpful suggestions that will improve the clarity and rigor of the manuscript. We address each major comment below and will incorporate the requested additions in the revision.

read point-by-point responses

Referee: [kernel section] Section describing the kernels (around the flat-linear and C1 flat-smooth proposals): the claim that these kernels 'preserve bootstrap equivalence' is central to the practical value of the remedies. The manuscript should supply either an explicit algebraic verification that the interpolated forward vols reproduce the input cap prices exactly, or a side-by-side numerical check (e.g., repricing error before and after the change) on the same market data set used for the oscillation experiments.

Authors: We agree that an explicit demonstration of bootstrap equivalence is essential for the practical utility of the proposed kernels. In the revised manuscript we will add a short algebraic verification in the kernel section proving that the continuous flat-linear and C1 flat-smooth kernels, when used within the standard bootstrap, exactly recover the input cap prices at the node dates. We will also include a side-by-side numerical table of repricing errors (before versus after the kernel change) computed on the identical market data snapshots employed in the oscillation experiments, confirming that the errors remain negligible. revision: yes
Referee: [numerical experiments] Numerical experiments section: while reduced oscillations and positive curves are reported, the manuscript does not state the precise quantitative metrics employed (maximum oscillation amplitude, integrated negativity measure, or L2 deviation from a reference curve) nor the number and diversity of market regimes tested. Without these, it is difficult to judge whether the improvement is robust or specific to the chosen data snapshots.

Authors: We accept that greater specificity on metrics and test coverage will strengthen the numerical section. The revised version will explicitly define the quantitative measures used (maximum oscillation amplitude, integrated negativity, and L2 deviation from a reference curve) and will report the exact number and diversity of market regimes examined. This will make the robustness of the improvements transparent to readers. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation or claims

full rationale

The paper presents an empirical study of caplet volatility extraction procedures, diagnosing instability sources through numerical experiments on interpolation schemes, node placement, and data quality, then proposing practical adjustments such as flat-linear kernels, midpoint nodes, global solvers, and positivity reparametrizations. No load-bearing mathematical derivation, prediction, or first-principles result is shown to reduce by construction to its own inputs or to a self-citation chain; the central claims rest on observable numerical behavior and bootstrap equivalence preservation rather than self-referential fitting or renamed ansatzes. The work is self-contained against external market data benchmarks with no internal reduction of outputs to fitted parameters.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claim rests on the domain assumption that cap quotes can be exactly bootstrapped to caplets once interpolation artifacts are removed, plus the premise that isolated bad quotes are the main amplifier rather than systematic data issues.

axioms (1)

domain assumption Market cap quotes represent exact prices of the corresponding cap instruments under the chosen pricing model
Invoked when claiming bootstrap equivalence is preserved by the new kernels.

pith-pipeline@v0.9.0 · 5410 in / 1304 out tokens · 28654 ms · 2026-05-08T15:03:20.044029+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 1 canonical work pages · 1 internal anchor

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What Can Go Wrong During Caplet Stripping ?

Introduction Interest rate caps and floors are widely used financial instruments that provide protection against interestratefluctuations. AcapisaseriesofEuropeancalloptionsoninterestrates,whileafloorisaseries ofEuropeanputoptions. Capletstrippingistheprocessofextractingtheimpliedvolatilitiesofindividual capletsfromthemarketpricesofcapsandfloors. Thisproc...

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Bootstrapping Wehave𝑁capvolquotes( ̂𝜎1,…,̂𝜎𝑁)atmaturities𝑇1 <⋯<𝑇𝑁,eachdefiningatargetcapprice 𝑃mkt 𝑞 = 𝑀𝑞∑ 𝑖=1 Caplet𝑖(̂𝜎𝑞), 𝑀 𝑞=𝑇𝑞∕∆, where∆is the caplet frequency (e.g

StrippingMethodology 2.1. Bootstrapping Wehave𝑁capvolquotes( ̂𝜎1,…,̂𝜎𝑁)atmaturities𝑇1 <⋯<𝑇𝑁,eachdefiningatargetcapprice 𝑃mkt 𝑞 = 𝑀𝑞∑ 𝑖=1 Caplet𝑖(̂𝜎𝑞), 𝑀 𝑞=𝑇𝑞∕∆, where∆is the caplet frequency (e.g. 1 month). We parameterise the caplet vol curve by𝑁node values (𝑣1,…,𝑣𝑁)atnodetimes (𝜏1,…,𝜏𝑁),withaninterpolation 𝜎(𝑡)=𝐼(𝑡;𝜏,𝑣). Thestrippingproblemis: find 𝑣suc...

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SmootherInterpolations 3.1. NodePlacementandInterpolation Being overly enthusiastic about joining the dots, one may consider to directly interpolate linearly betweenthenodes,placedatthecapmaturities. Thereisnodependencytowardsthenextcaplet(thematrix islowertriangular)andthusbootstrappingisstillapplicable. However,thisapproachcanleadtostrong oscillations i...
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With midpoint nodes, the oscillations are muchreduced,andthenodenear18monthssettlesat −14bpinsteadof −159bp(Figure6)

OscillationsandBlow-ups Theresultingcapletvolatilitiesobtainedwithlinearinterpolationandat-maturitynodesshowstrong oscillations, with a node near 18 months settling at−159bp. With midpoint nodes, the oscillations are muchreduced,andthenodenear18monthssettlesat −14bpinsteadof −159bp(Figure6). Theoscillations growstrongerforlongermaturities. Thecubicsplinei...
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TheRealIssueWithOurExample: The2YDip Thequalityofourcapquotesisinfactnotgreat. The2Ycapisverycheapcomparedtothe1Yand3Y caps,whichcreatesadipinthecapletvolatilitiesaround18months. Itisunlikelytobereal,butmaystem fromastalequote. 5.1. ModifiedZ-Score Givenaseriesofobservations𝑥 1,…,𝑥𝑛,themedianabsolutedeviation(MAD)is MAD=median (|𝑥𝑖−̃𝑥|), ̃𝑥=median(𝑥𝑖). Un...
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Iglewicz,B.;Hoaglin,D. Volume16: howtodetectandhandleoutliers,TheASQCbasicreferencesinquality control: statisticaltechniques,EdwardF.Mykytka.Ph. D.dissertation,Ph. D.,Editor1993. 11of11 AppendixA MarketData TableA1.capvolatilities(inbasispoints)onLibor-1MasofFebruary2022forastrike𝐾=0%. Maturity(months) 2 3 4 5 6 9 12 24 36 60 84 120 180 CapVol(bp) 79.30 1...