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Fabien Le Floc'h

Publications and source records attributed to Fabien Le Floc'h.

At least 19 recordsLinked to original sources
arXiv2026

Explicit Rational Formulae for Bachelier (Normal) Implied Volatility

We present two explicit rational formulae for Bachelier, or normal, implied volatility. The formulae take the option price, forward, strike, and expiry as inputs and return the implied normal volatility without iteration. They follow the branch structure of LFK-4, but use the simpler near-the-money variable given by the absolute forward-strike difference divided by the tail time value, avoiding a logarithm and a small-argument Taylor branch in that region. LFK-2026 is the accuracy-oriented formula and approximates reciprocal absolute standardized moneyness directly in the far tail. LFK-2026C keeps the same shifted out-of-the-money rational tail approximation, but splits the near-the-money branch two low degree rationals. In double precision tests both remain close to machine accuracy, while LFK-2026C is the faster scalar implementation on the current benchmark mix.

q-fin.CP↗
arXiv2026

A Practical Guide to Strip Caplet Volatilities

We study caplet stripping, the problem of recovering a caplet volatility term structure consistent with quoted cap volatilities. Many academic papers on the Libor market model assume caplet volatilities are readily available, whereas practitioners know they are not and extracting them is a complex task. This paper presents a practical workflow, structuring the presentation around a constructive algorithm. We start with criteria on the input data based on cap time-value monotonicity. If time values fail this check, we show how to correct the quotes using robust outlier detection based on the modified Z-score. The time-value proposition naturally leads to a direct non-bootstrap stripping approach by interpolating cap time values, which yields arbitrage-free caplet volatilities by construction. We then revisit the classic sequential bootstrap approach. We introduce compact-kernel transition interpolants (flat-linear and $C^1$ flat-smooth) that preserve bootstrap equivalence. Finally, for a richer, smoother curve, we introduce global search methods using midpoint node placement with positivity-preserving calibration. Pathological cases and detailed analyses of oscillations are provided in the appendix.

q-fin.CP↗
arXiv2026

Faster Monotone Implied Volatility Solver

We present ThiopheneIV, a Black-Scholes implied-volatility solver with a monotone core and explicit production guards. The solver starts from the simple Choi-Huh-Su L3 lower-bound seed and applies three Euler-Chebyshev steps on a lower branch and three Halley steps on the remaining upper branch. We prove that, in exact arithmetic, the seed lies below the root and both maps increase monotonically without overshooting. We also detail the practical challenges encountered for a double-precision implementation: parity normalisation, microscopic Bachelier-limit handling, saturated price treatment, and an optional Jäckel-Newton polish. Across standard grids, market-like data, high-volatility cases, and adversarial corners, ThiopheneIV agrees closely with multiprecision Black reference prices at low latency. We provide detailed comparisons with recent solvers, including Jäckel's Let's Be Rational. The broader lesson is that a convergence proof gives a clean core, but robust production inversion still depends on boundary handling and on the pricing objective one chooses to match.

q-fin.CP↗
arXiv2026

Implying Volatility: How Fast Can We Go?

FlashIV is a low-latency Black--Scholes implied-volatility solver for production use. It normalises each input to an out-of-the-money price and solves a tail-stable erfcx/log-price residual. The hot path combines a cheap Li/asymptotic seed with a fixed, branch-light Householder refinement and guarded boundary handling. Across regular and stressed benchmarks, FlashIV stays close to the expanded Jäckel reference price while running materially faster than a normalised Java port of Jäckel's \emph{Let's Be Rational}. FlashIV+ adds an optional Jäckel--Newton correction for applications that need tighter agreement with that reference price, trading latency for reference-price alignment.

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arXiv2025

How to choose my stochastic volatility parameters? A review

Based on the existing literature, this article presents the different ways of choosing the parameters of stochastic volatility models in general, in the context of pricing financial derivative contracts. This includes the use of stochastic volatility inside stochastic local volatility models.

q-fin.PR↗
arXiv2025

Asian Basket Spread Options: A New Approximation Based on Stochastic Taylor Expansions

We present closed analytical approximations for the pricing of Asian basket spread options under the Black-Scholes model. The formulae are obtained by using a stochastic Taylor expansion around a log-normal proxy model and are found to be highly accurate for Asian and spread options in practice. Unlike other approaches, they do not require any numerical integration or root solving.

q-fin.PR↗
arXiv2025

The Quadratic Local Variance Gamma Model: an arbitrage-free interpolation of class C3 for option prices

This paper generalizes the local variance gamma model of Carr and Nadtochiy, to a piecewise quadratic local variance function. The formulation encompasses the piecewise linear Bachelier and piecewise linear Black local variance gamma models. The quadratic local variance function results in an arbitrage-free interpolation of class C3. The increased smoothness over the piecewise-constant and piecewise-linear representation allows to reduce the number of knots when interpolating raw market quotes, thus providing an interesting alternative to regularization while reducing the computational cost.

q-fin.CP↗
arXiv2024

Stochastic Expansion for the Pricing of Asian and Basket Options

We present closed analytical approximations for the pricing of basket options, also applicable to Asian options with discrete averaging under the Black-Scholes model with time-dependent parameters. The formulae are obtained by using a stochastic Taylor expansion around a log-normal proxy model and are found to be highly accurate for Asian options in practice as well as for vanilla options with discrete dividends.

q-fin.PR↗
arXiv2023

Instabilities of explicit finite difference schemes with ghost points on the diffusion equation

Ghost, or fictitious points allow to capture boundary conditions that are not located on the finite difference grid discretization. We explore in this paper the impact of ghost points on the stability of the explicit Euler finite difference scheme in the context of the diffusion equation. In particular, we consider the case of a one-touch option under the Black-Scholes model. The observations and results are however valid for a much wider range of financial contracts and models.

math.NA↗
arXiv2023

Maximum Implied Variance Slope -- Practical Aspects

In the Black-Scholes model, the absence of arbitrages imposes necessary constraints on the slope of the implied variance in terms of log-moneyness, asymptotically for large log-moneyness. The constraints are used for example in the SVI implied volatility parameterization to ensure the resulting smile has no arbitrages. This note shows that those no-arbitrage contraints are very mild, and that arbitrage is almost always guaranteed in a large range of slopes where the contraints are enforced.

q-fin.PR↗
arXiv2022

On the Bachelier implied volatility at extreme strikes

What kind of implied volatility extrapolation is appropriate? Roger Lee proved that the Black-Scholes implied variance can not grow faster than linearly in log-moneyness. This paper investigates what happens in the Bachelier (or Normal) implied volatility world, making sure to cover the various aspects of vanilla option arbitrages.

q-fin.MF↗
arXiv2022

Roughness of the Implied Volatility

The measures of roughness of the volatility in the litterature are based on the realized volatility of high frequency data. Some authors show that this leads to a biased estimate, and does not necessarily indicate roughness of the underlying volatility process. Here, we attempt to measure the roughness of the implied volatility of short term options, as well as of the VIX index, and evaluate whether they may be more appropriate proxies of the underlying instant volatility.

q-fin.MF↗
arXiv2022

Double sweep LU decomposition for American options under negative rates

The classic Brennan-Schwartz algorithm to solve the linear complementary problem, which arises from the finite difference discretization of the partial differential equation related to American option pricing does not lead to the exact solution under negative interest rates. This is due to the two exercise boundaries which may appear under negative interest rate, while the algorithm was proven to lead to the exact solution in the case of a single exercise boundary only. This paper explains that two sweeps of the Brennan-Schwartz algorithm in two directions is enough to recover the exact solution.

q-fin.CP↗
arXiv2021

Positive Stochastic Collocation for the Collocated Local Volatility Model

This paper presents how to apply the stochastic collocation technique to assets that can not move below a boundary. It shows that the polynomial collocation towards a lognormal distribution does not work well. Then, the potentials issues of the related collocated local volatility model (CLV) are explored. Finally, a simple analytical expression for the Dupire local volatility derived from the option prices modelled by stochastic collocation is given.

q-fin.PR↗
arXiv2021

Pricing American options with the Runge-Kutta-Legendre finite difference scheme

This paper presents the Runge-Kutta-Legendre finite difference scheme, allowing for an additional shift in its polynomial representation. A short presentation of the stability region, comparatively to the Runge-Kutta-Chebyshev scheme follows. We then explore the problem of pricing American options with the Runge-Kutta-Legendre scheme under the one factor Black-Scholes and the two factor Heston stochastic volatility models, as well as the pricing of butterfly spread and digital options under the uncertain volatility model, where a Hamilton-Jacobi-Bellman partial differential equation needs to be solved. We explore the order of convergence in these problems, as well as the option greeks stability, compared to the literature and popular schemes such as Crank-Nicolson, with Rannacher time-stepping.

q-fin.CP↗