arXiv subjects

Ahmed Kebaier

Publications and source records attributed to Ahmed Kebaier.

At least 19 recordsLinked to original sources

arXiv2026

Weak error approximation for rough and Gaussian mean-reverting stochastic volatility models

For a class of stochastic models with Gaussian and rough mean-reverting volatility that embeds the genuine rough Stein-Stein model, we study the weak approximation rate when using a Euler type scheme with integrated kernels. Our first result is a weak convergence rate for the discretised rough Ornstein-Uhlenbeck process, that is essentially in $\min(3α-1,1)$, where $\frac{t^{α-1}}{Γ(α)} $ is the fractional convolution kernel with $α\in (1/2,1)$. Then, our main result is to obtain the same convergence rate for the corresponding stochastic rough volatility model with polynomial test functions.

math.PR↗

arXiv2026

How can the dual martingale help solving the primal optimal stopping problem?

Motivated by recent results on the dual formulation of optimal stopping problems, we investigate in this short paper how the knowledge of an approximating dual martingale can improve the efficiency of primal methods. In particular, we show on numerical examples that accurate approximations of a dual martingale efficiently reduce the variance for the primal optimal stopping problem.

q-fin.CP↗

arXiv2026

Credit Spreads' Term Structure: Stochastic Modeling with CIR++ Intensity

This paper introduces a novel stochastic model for credit spreads. The stochastic approach leverages the diffusion of default intensities via a CIR++ model and is formulated within a risk-neutral probability space. Our research primarily addresses two gaps in the literature. The first is the lack of credit spread models founded on a stochastic basis that enables continuous modeling, as many existing models rely on factorial assumptions. The second is the limited availability of models that directly yield a term structure of credit spreads. An intermediate result of our model is the provision of a term structure for the prices of defaultable bonds. We present the model alongside an innovative, practical, and conservative calibration approach that minimizes the error between historical and theoretical volatilities of default intensities. We demonstrate the robustness of both the model and its calibration process by comparing its behavior to historical credit spread values. Our findings indicate that the model not only produces realistic credit spread term structure curves but also exhibits consistent diffusion over time. Additionally, the model accurately fits the initial term structure of implied survival probabilities and provides an analytical expression for the credit spread of any given maturity at any future time.

q-fin.RM↗

arXiv2025

Local asymptotic properties for the growth rate of a jump-type CIR process

In this paper, we consider a one-dimensional jump-type Cox-Ingersoll-Ross process driven by a Brownian motion and a subordinator, whose growth rate is an unknown parameter. Considering the process observed continuously or discretely at high frequency, we derive the local asymptotic properties for the growth rate in both ergodic and non-ergodic cases. Local asymptotic normality (LAN) is proved in the subcritical case, local asymptotic quadraticity (LAQ) is derived in the critical case, and local asymptotic mixed normality (LAMN) is shown in the supercritical case. To obtain these results, techniques of Malliavin calculus and a subtle analysis on the jump structure of the subordinator involving the amplitude of jumps and number of jumps are essentially used.

math.PR↗

arXiv2024

A pure dual approach for hedging Bermudan options

This paper develops a new dual approach to compute the hedging portfolio of a Bermudan option and its initial value. It gives a "purely dual" algorithm following the spirit of Rogers (2010) in the sense that it only relies on the dual pricing formula. The key is to rewrite the dual formula as an excess reward representation and to combine it with a strict convexification technique. The hedging strategy is then obtained by using a Monte Carlo method, solving backward a sequence of least square problems. We show convergence results for our algorithm and test it on many different Bermudan options. Beyond giving directly the hedging portfolio, the strength of the algorithm is to assess both the relevance of including financial instruments in the hedging portfolio and the effect of the rebalancing frequency.

q-fin.MF↗

arXiv2024

Deep Calibration of Interest Rates Model

For any financial institution, it is essential to understand the behavior of interest rates. Despite the growing use of Deep Learning, for many reasons (expertise, ease of use, etc.), classic rate models such as CIR and the Gaussian family are still widely used. In this paper, we propose to calibrate the five parameters of the G2++ model using Neural Networks. Our first model is a Fully Connected Neural Network and is trained on covariances and correlations of Zero-Coupon and Forward rates. We show that covariances are more suited to the problem than correlations due to the effects of the unfeasible backpropagation phenomenon, which we analyze in this paper. The second model is a Convolutional Neural Network trained on Zero-Coupon rates with no further transformation. Our numerical tests show that our calibration based on deep learning outperforms the classic calibration method used as a benchmark. Additionally, our Deep Calibration approach is designed to be systematic. To illustrate this feature, we applied it to calibrate the popular CIR intensity model.

q-fin.ST↗

arXiv2024

Financial Stochastic Models Diffusion: From Risk-Neutral to Real-World Measure

This research presents a comprehensive framework for transitioning financial diffusion models from the risk-neutral (RN) measure to the real-world (RW) measure, leveraging results from probability theory, specifically Girsanov's theorem. The RN measure, fundamental in derivative pricing, is contrasted with the RW measure, which incorporates risk premiums and better reflects actual market behavior and investor preferences, making it crucial for risk management. We address the challenges of incorporating real-world dynamics into financial models, such as accounting for market premiums, producing realistic term structures of market indicators, and fitting any arbitrarily given market curve. Our framework is designed to be general, applicable to a variety of diffusion models, including those with non-additive noise such as the CIR++ model. Through case studies involving Goldman Sachs' 2024 global credit outlook forecasts and the European Banking Authority (EBA) 2023 stress tests, we validate the robustness, practical relevance and applicability of our methodology. This work contributes to the literature by providing a versatile tool for better risk measures and enhancing the realism of financial models under the RW measure. Our model's versatility extends to stress testing and scenario analysis, providing practitioners with a powerful tool to evaluate various what-if scenarios and make well-informed decisions, particularly in pricing and risk management strategies.

q-fin.MF↗

arXiv2024

Interpolated Drift Implicit Euler MLMC Method for Barrier Option Pricing and application to CIR and CEV Models

Recently, Giles et al. [14] proved that the efficiency of the Multilevel Monte Carlo (MLMC) method for evaluating Down-and-Out barrier options for a diffusion process $(X_t)_{t\in[0,T]}$ with globally Lipschitz coefficients, can be improved by combining a Brownian bridge technique and a conditional Monte Carlo method provided that the running minimum $\inf_{t\in[0,T]}X_t$ has a bounded density in the vicinity of the barrier. In the present work, thanks to the Lamperti transformation technique and using a Brownian interpolation of the drift implicit Euler scheme of Alfonsi [2], we show that the efficiency of the MLMC can be also improved for the evaluation of barrier options for models with non-Lipschitz diffusion coefficients under certain moment constraints. We study two example models: the Cox-Ingersoll-Ross (CIR) and the Constant of Elasticity of Variance (CEV) processes for which we show that the conditions of our theoretical framework are satisfied under certain restrictions on the models parameters. In particular, we develop semi-explicit formulas for the densities of the running minimum and running maximum of both CIR and CEV processes which are of independent interest. Finally, numerical tests are processed to illustrate our results.

math.PR↗

arXiv2024

Data-driven uncertainty quantification for constrained stochastic differential equations and application to solar photovoltaic power forecast data

In this work, we extend the data-driven Itô stochastic differential equation (SDE) framework for the pathwise assessment of short-term forecast errors to account for the time-dependent upper bound that naturally constrains the observable historical data and forecast. We propose a new nonlinear and time-inhomogeneous SDE model with a Jacobi-type diffusion term for the phenomenon of interest, simultaneously driven by the forecast and the constraining upper bound. We rigorously demonstrate the existence and uniqueness of a strong solution to the SDE model by imposing a condition for the time-varying mean-reversion parameter appearing in the drift term. The normalized forecast function is thresholded to keep such mean-reversion parameters bounded. The SDE model parameter calibration is applied to user-selected approximations of the likelihood function. Another novel contribution is estimating the unknown transition density of the forecast error process with a tailored kernel smoothing technique with the control variate method, coupling an adequate SDE to the original one. We provide a theoretical study about how to choose the optimal bandwidth. We fit the model to the 2019 photovoltaic (PV) solar power daily production and forecast data in Uruguay, computing the daily maximum solar PV production estimation. Two statistical versions of the constrained SDE model are fit, with the beta and truncated normal distributions as proxies for the transition density. Empirical results include simulations of the normalized solar PV power production and pathwise confidence bands generated through an indirect inference method. An objective comparison of optimal parametric points associated with the two selected statistical approximations is provided by applying our innovative kernel smoothing estimation technique of the transition function of the forecast error process.

stat.ME↗

arXiv2022

Approximation of Stochastic Volterra Equations with kernels of completely monotone type

In this work, we develop a multifactor approximation for $d$-dimensional Stochastic Volterra Equations (SVE) with Lipschitz coefficients and kernels of completely monotone type that may be singular. First, we prove an $L^2$-estimation between two SVEs with different kernels, which provides a quantification of the error between the SVE and any multifactor Stochastic Differential Equation (SDE) approximation. For the particular rough kernel case with Hurst parameter lying in $(0,1/2)$, we propose various approximating multifactor kernels, state their rates of convergence and illustrate their efficiency for the rough Bergomi model. Second, we study a Euler discretization of the multifactor SDE and establish a convergence result towards the SVE that is uniform with respect to the approximating multifactor kernels. These obtained results lead us to build a new multifactor Euler scheme that reduces significantly the computational cost in an asymptotic way compared to the Euler scheme for SVEs. Finally, we show that our multifactor Euler scheme outperforms the Euler scheme for SVEs for option pricing in the rough Heston model.

math.PR↗

arXiv2021

Asymptotic behavior of the multilevel type error for SDEs driven by a pure jump Lévy process

Motivated by the multilevel Monte Carlo method introduced by Giles [5], we study the asymptotic behavior of the normalized error process $u_{n,m}(X^n-X^{nm})$ where $X^n$ and $X^{nm}$ are respectively Euler approximations with time steps $1/n$ and $1/nm$ of a given stochastic differential equation $X$ driven by a pure jump Lévy process. In this paper, we prove that this normalized multilevel error converges to different non-trivial limiting processes with various sharp rates $u_{n,m}$ depending on the behavior of the Lévy measure around zero. Our results are consistent with those of Jacod [9] obtained for the normalized error $u_n(X^n-X)$, as when letting $m$ tends to infinity, we recover the same limiting processes. For the multilevel error, the proofs of the current paper are challenging since unlike [9] we need to deal with $m$ dependent triangular arrays instead of one.

math.PR↗

arXiv2021

Quantifying Uncertainty with a Derivative Tracking SDE Model and Application to Wind Power Forecast Data

We develop a data-driven methodology based on parametric Itô's Stochastic Differential Equations (SDEs) to capture the real asymmetric dynamics of forecast errors. Our SDE framework features time-derivative tracking of the forecast, time-varying mean-reversion parameter, and an improved state-dependent diffusion term. Proofs of the existence, strong uniqueness, and boundedness of the SDE solutions are shown under a principled condition for the time-varying mean-reversion parameter. Inference based on approximate likelihood, constructed through the moment-matching technique both in the original forecast error space and in the Lamperti space, is performed through numerical optimization procedures. We propose another contribution based on the fixed-point likelihood optimization approach in the Lamperti space. All the procedures are agnostic of the forecasting technology, and they enable comparisons between different forecast providers. We apply our SDE framework to model historical Uruguayan normalized wind power production and forecast data between April and December 2019. Sharp empirical confidence bands of future wind power production are obtained for the best-selected model.

stat.ME↗

arXiv2020

Local asymptotic properties for Cox-Ingersoll-Ross process with discrete observations

In this paper, we consider a one-dimensional Cox-Ingersoll-Ross (CIR) process whose drift coefficient depends on unknown parameters. Considering the process discretely observed at high frequency, we prove the local asymptotic normality property in the subcritical case, the local asymptotic quadraticity in the critical case, and the local asymptotic mixed normality property in the supercritical case. To obtain these results, we use the Malliavin calculus techniques developed recently for CIR process by Alòs et {\it al.} \cite{AE08} and Altmayer et {\it al.} \cite{AN14} together with the $L^p$-norm estimation for positive and negative moments of the CIR process obtained by Bossy et {\it al.} \cite{BD07} and Ben Alaya et {\it al.} \cite{BK12,BK13}. In this study, we require the same conditions of high frequency $Δ_n\rightarrow 0$ and infinite horizon $nΔ_n\rightarrow\infty$ as in the case of ergodic diffusions with globally Lipschitz coefficients studied earlier by Gobet \cite{G02}. However, in the non-ergodic cases, additional assumptions on the decreasing rate of $Δ_n$ are required due to the fact that the square root diffusion coefficient of the CIR process is not regular enough. Indeed, we assume $nΔ_n^{3}\to 0$ for the critical case and $Δ_n^{2}e^{-b_0nΔ_n}\to 0$ for the supercritical case.

math.ST↗

arXiv2020

Central Limit Theorem for the $σ$-antithetic multilevel Monte Carlo method

In this paper, we introduce the $σ$-antithetic multilevel Monte Carlo (MLMC) estimator for a multi-dimensional diffusion which is an extended version of the original antithetic MLMC one introduced by Giles and Szpruch \cite{a}. Our aim is to study the asymptotic behavior of the weak errors involved in this new algorithm. Among the obtained results, we prove that the error between on the one hand the average of the Milstein scheme without Lévy area and its $σ$-antithetic version build on the finer grid and on the other hand the coarse approximation stably converges in distribution with a rate of order 1. We also prove that the error between the Milstein scheme without Lévy area and its $σ$-antithetic version stably converges in distribution with a rate of order $1/2$. More precisely, we have a functional limit theorem on the asymptotic behavior of the joined distribution of these errors based on a triangular array approach (see e.g. Jacod \cite{c}). Thanks to this result, we establish a central limit theorem of Lindeberg-Feller type for the $σ$-antithetic MLMC estimator. The time complexity of the algorithm is carried out.

math.PR↗

arXiv2019

Asymptotic properties of maximum likelihood estimator for the growth rate of a stable CIR process based on continuous time observations

We consider a stable Cox--Ingersoll--Ross process driven by a standard Wiener process and a spectrally positive strictly stable Lévy process, and we study asymptotic properties of the maximum likelihood estimator (MLE) for its growth rate based on continuous time observations. We distinguish three cases: subcritical, critical and supercritical. In all cases we prove strong consistency of the MLE in question, in the subcritical case asymptotic normality, and in the supercritical case asymptotic mixed normality are shown as well. In the critical case the description of the asymptotic behavior of the MLE in question remains open.

math.ST↗

arXiv2018

Non-asymptotic error bounds for The Multilevel Monte Carlo Euler method applied to SDEs with constant diffusion coefficient

In this paper, we are interested in deriving non-asymptotic error bounds for the multilevel Monte Carlo method. As a first step, we deal with the explicit Euler discretization of stochastic differential equations with a constant diffusion coefficient. We prove that, as long as the deviation is below an explicit threshold, a Gaussian-type concentration inequality optimal in terms of the variance holds for the multilevel estimator. To do so, we use the Clark-Ocone representation formula and derive bounds for the moment generating functions of the squared difference between a crude Euler scheme and a finer one and of the squared difference of their Malliavin derivatives.

math.PR↗

arXiv2018

Asymptotic behavior of maximum likelihood estimators for a jump-type Heston model

We study asymptotic properties of maximum likelihood estimators of drift parameters for a jump-type Heston model based on continuous time observations, where the jump process can be any purely non-Gaussian Lévy process of not necessarily bounded variation with a Lévy measure concentrated on $(-1,\infty)$. We prove strong consistency and asymptotic normality for all admissible parameter values except one, where we show only weak consistency and mixed normal (but non-normal) asymptotic behavior. It turns out that the volatility of the price process is a measurable function of the price process. We also present some numerical illustrations to confirm our results.

math.ST↗

arXiv2017

Improved adaptive Multilevel Monte Carlo and applications to finance

This paper focuses on the study of an original combination of the Multilevel Monte Carlo method introduced by Giles [10] and the popular importance sampling technique. To compute the optimal choice of the parameter involved in the importance sampling method, we rely on Robbins-Monro type stochastic algorithms. On the one hand, we extend our previous work [2] to the Multilevel Monte Carlo setting. On the other hand, we improve [2] by providing a new adaptive algorithm avoiding the discretization of any additional process. Furthermore, from a technical point of view, the use of the same stochastic algorithms as in [2] appears to be problematic. To overcome this issue, we employ an alternative version of stochastic algorithms with projection (see e.g. Laruelle, Lehalle and Pagès [20]). In this setting, we show innovative limit theorems for a doubly indexed stochastic algorithm which appear to be crucial to study the asymptotic behavior of the new adaptive Multilevel Monte Carlo estimator. Finally, we illustrate the efficiency of our method through applications from quantitative finance.

math.PR↗