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Jean-Philippe Aguilar

Publications and source records attributed to Jean-Philippe Aguilar.

16 recordsLinked to original sources
arXiv2026

Fast and explicit European option pricing under tempered stable processes

We provide series expansions for the tempered stable densities and for the price of European-style contracts in the exponential Lévy model driven by the tempered stable process. These formulas recover several popular option pricing models, and become particularly simple in some specific cases such as bilateral Gamma process and one-sided TS process. When compared to traditional Fourier pricing, our method has the advantage of not involving delicate choices of hyperparameters, but only a truncation order for the resulting analytical pricing formulas. We also provide a detailed numerical analysis and show that our technique is competitive with state-of-the-art pricing methods.

q-fin.CP↗
arXiv2026

The bilateral generalized inverse Gaussian process with applications to financial modeling

We introduce and document a class of probability distributions, called bilateral generalized inverse Gaussian (BGIG) distributions, that are obtained by convolution of two generalized inverse Gaussian distributions supported by the positive and negative semi-axis. We prove several results regarding their analyticity, shapes and asymptotics, and we introduce the associated Lévy processes as well as their main properties. We study the behaviour of these processes under change of measure, their simulations and the structure of their sample paths, and we introduce a stock market model constructed by means of exponential BGIG processes. Based on real market data, we show that this model is easy to calibrate thanks notably to idiosyncratic properties of BGIG distributions, and that it is well suited to Monte Carlo and Fourier option pricing.

math.PR↗
arXiv2025

Calibration and Option Pricing with Stochastic Volatility and Double Exponential Jumps

This work examines a stochastic volatility model with double-exponential jumps in the context of option pricing. The model has been considered in previous research articles, but no thorough analysis has been conducted to study its quality of calibration and pricing capabilities thus far. We provide evidence that this model outperforms challenger models possessing similar features (stochastic volatility and jumps), especially in the fit of the short term implied volatility smile, and that it is particularly tractable for the pricing of exotic options from different generations. The article utilizes Fourier pricing techniques (the PROJ method and its refinements) for different types of claims and several generations of exotics (Asian options, cliquets, barrier options, and options on realized variance), and all source codes are made publicly available to facilitate adoption and future research. The results indicate that this model is highly promising, thanks to the asymmetry of the jumps distribution allowing it to capture richer dynamics than a normal jump size distribution. The parameters all have meaningful econometrics interpretations that are important for adoption by risk-managers.

q-fin.PR↗
arXiv2021

A structural approach to default modelling with pure jump processes

We present a general framework for the estimation of corporate default based on a firm's capital structure, when its assets are assumed to follow a pure jump Lévy processes; this setup provides a natural extension to usual default metrics defined in diffusion (log-normal) models, and allows to capture extreme market events such as sudden drops in asset prices, which are closely linked to default occurrence. Within this framework, we introduce several pure jump processes featuring negative jumps only and derive practical closed formulas for equity prices, which enable us to use a moment-based algorithm to calibrate the parameters from real market data and to estimate the associated default metrics. A notable feature of these models is the redistribution of credit risk towards shorter maturity: this constitutes an interesting improvement to diffusion models, which are known to underestimate short term default probabilities. We also provide extensions to a model featuring both positive and negative jumps and discuss qualitative and quantitative features of the results. For readers convenience, practical tools for model implementation and GitHub links are also included.

q-fin.PR↗
arXiv2021

The value of power-related options under spectrally negative Lévy processes

We provide analytical tools for pricing power options with exotic features (capped or log payoffs, gap options ...) in the framework of exponential Lévy models driven by one-sided stable or tempered stable processes. Pricing formulas take the form of fast converging series of powers of the log-forward moneyness and of the time-to-maturity; these series are obtained via a factorized integral representation in the Mellin space evaluated by means of residues in $\mathbb{C}$ or $\mathbb{C}^2$. Comparisons with numerical methods and efficiency tests are also discussed.

q-fin.PR↗
arXiv2020

Explicit option valuation in the exponential NIG model

We provide closed-form pricing formulas for a wide variety of path-independent options, in the exponential Lévy model driven by the Normal inverse Gaussian process. The results are obtained in both the symmetric and asymmetric model, and take the form of simple and quickly convergent series, under some condition involving the log-forward moneyness and the maturity of instruments. Proofs are based on a factorized representation in the Mellin space for the price of an arbitrary path-independent payoff, and on tools from complex analysis. The validity of the results is assessed thanks to several comparisons with standard numerical methods (Fourier and Fast Fourier transforms, Monte-Carlo simulations) for realistic sets of parameters. Precise bounds for the convergence speed and the truncation error are also provided.

q-fin.PR↗
arXiv2020

Some pricing tools for the Variance Gamma model

We establish several closed pricing formula for various path-independent payoffs, under an exponential Lévy model driven by the Variance Gamma process. These formulas take the form of quickly convergent series and are obtained via tools from Mellin transform theory as well as from multidimensional complex analysis. Particular focus is made on the symmetric process, but extension to the asymmetric process is also provided. Speed of convergence and comparison with numerical methods (Fourier transform, quadrature approximations, Monte Carlo simulations) are also discussed; notable feature is the accelerated convergence of the series for short term options, which constitutes an interesting improvement of numerical Fourier inversion techniques.

q-fin.PR↗
arXiv2019

On expansions for the Black-Scholes prices and hedge parameters

We derive new formulas for the price of the European call and put options in the Black-Scholes model, under the form of uniformly convergent series generalizing previously known approximations. We also provide precise boundaries for the convergence speed and apply the results to the calculation of hedge parameters (Greeks).

q-fin.PR↗
arXiv2018

Series representation of the pricing formula for the European option driven by space-time fractional diffusion

In this paper, we show that the price of an European call option, whose underlying asset price is driven by the space-time fractional diffusion, can be expressed in terms of rapidly convergent double-series. The series formula can be obtained from the Mellin-Barnes representation of the option price with help of residue summation in $\mathbb{C}^2$. We also derive the series representation for the associated risk-neutral factors, obtained by Esscher transform of the space-time fractional Green functions.

q-fin.MF↗
arXiv2018

Option Pricing Models Driven by the Space-Time Fractional Diffusion: Series Representation and Applications

In this paper, we focus on option pricing models based on space-time fractional diffusion. We briefly revise recent results which show that the option price can be represented in the terms of rapidly converging double-series and apply these results to the data from real markets. We focus on estimation of model parameters from the market data and estimation of implied volatility within the space-time fractional option pricing models.

q-fin.MF↗
arXiv2017

Non-Gaussian analytic option pricing: a closed formula for the Lévy-stable model

We establish an explicit pricing formula for the class of Lévy-stable models with maximal negative asymmetry (Log-Lévy model with finite moments and stability parameter $1<α\leq 2$) in the form of rapidly converging series. The series is obtained with help of Mellin transform and the residue theory in $\mathbb{C}^2$. The resulting formula enables the straightforward evaluation of an European option with arbitrary accuracy without the use of numerical techniques. The formula can be used by any practitioner, even if not familiar with the underlying mathematical techniques. We test the efficiency of the formula, and compare it with numerical methods.

q-fin.PR↗
arXiv2016

Distributional Mellin calculus in $\mathbb{C}^n$, with applications to option pricing

We discuss several aspects of Mellin transform, including distributional Mellin transform and inversion of multiple Mellin-Barnes integrals in $\mathbb{C}^n$ and its connection to residue expansion or evaluation of Laplace integrals. These mathematical concepts are demonstrated on several option-pricing models. This includes European option models such as Black-Scholes or fractional-diffusion models, as well as evaluation of quantities related to the optimal exercise price of American options.

q-fin.PR↗
arXiv2016

Regularization and analytic option pricing under $α$-stable distribution of arbitrary asymmetry

We consider a non-Gaussian option pricing model, into which the underlying log-price is assumed to be driven by an $α$-stable distribution. We remove the a priori divergence of the model by introducing a Mellin regularization for the Lévy propagator. Using distributional and $\mathbb{C}^n$ tools, we derive an analytic closed formula for the option price, valid for any stability $α\in]1,2]$ and any asymmetry. This formula is very efficient and recovers previous cases (Black-Scholes, Carr-Wu); we calibrate the formula on market datas, make numerical tests, and discuss its many interesting properties.

q-fin.PR↗
arXiv2008

The effect of classical noise on a quantum two-level system

We consider a quantum two-level system perturbed by classical noise. The noise is implemented as a stationary diffusion process in the off-diagonal matrix elements of the Hamiltonian, representing a transverse magnetic field. We determine the invariant measure of the system and prove its uniqueness. In the case of Ornstein-Uhlenbeck noise, we determine the speed of convergence to the invariant measure. Finally, we determine an approximate one-dimensional diffusion equation for the transition probabilities. The proofs use both spectral-theoretic and probabilistic methods.

math.PR↗
arXiv2008

Muon Anomaly from Lepton Vacuum Polarization and The Mellin--Barnes Representation

We evaluate, analytically, a specific class of eighth--order and tenth--order QED contributions to the anomalous magnetic moment of the muon. They are generated by Feynman diagrams involving lowest order vacuum polarization insertions of leptons $l=e,μ$, and $τ$. The results are given in the form of analytic expansions in terms of the mass ratios $m_e/m_μ$ and $m_μ/m_τ$. We compute as many terms as required by the error induced by the present experimental uncertainty on the lepton masses. We show how the Mellin--Barnes integral representation of Feynman parametric integrals allows for an easy analytic evaluation of as many terms as wanted in these expansions and how its underlying algebraic structure generalizes the standard renormalization group properties. We also discuss the generalization of this technique to the case where two independent mass ratios appear. Comparison with previous numerical and analytic evaluations made in the literature, whenever pertinent, are also made.

hep-ph↗