arXiv subjects

José E. Figueroa-López

Publications and source records attributed to José E. Figueroa-López.

At least 19 recordsLinked to original sources

arXiv2026

An infinitesimal generator approach on weak convergence of regulated multi-class matching systems

We consider a regulated multi-class instantaneous matching system with reneging, in which each event requires $K \geq 2$ distinct impatient agents who wait in their respective queues. Each agent class is subject to a buffer capacity, allowing for the special case without buffers. Due to the instantaneous matching behavior, at any give time, at least one category has an empty queue. Under the Markovian assumption, the system dynamics are described by a Markov chain with innovative rate matrices that capture all possible queue configurations across all classes. To effectively circumvent the structural challenges introduced by instantaneous matching, we establish a non-trivial yet tractable diffusion approximation under heavy traffic conditions by leveraging the infinitesimal generator in conjunction with appropriate regulation and boundary conditions. This asymptotic analysis offers a direct explanation of the dynamics of the regulated coupled heavy-traffic limiting process. Furthermore, we demonstrate the connection between the diffusion-scaled limit derived from the generator approach and the one established in the literature. The latter is typically described by a regulated coupled stochastic integral equation.

math.PR↗

arXiv2026

Debiased Kernel Estimation of Spot Volatility in the Presence of Infinite Variation Jumps

Volatility estimation is a central problem in financial econometrics, but becomes particularly challenging when jump activity is high, a phenomenon observed empirically in highly traded financial securities. In this paper, we revisit the problem of spot volatility estimation for an Itô semimartingale with jumps of unbounded variation. We construct truncated kernel-based estimators and debiased variants that extend rate-optimal spot volatility estimation to a wider range of jump activity indices, from the previously available bound $Y<4/3$ to $Y<20/11$. Rate-suboptimal CLTs are also established for $Y>20/11$. Compared with earlier work, our approach achieves smaller asymptotic variances through the use of more general kernels and an optimal choice for the bandwidth convergence rate, and also has broader applicability under more flexible model assumptions. A comprehensive simulation study confirms that our procedures outperform competing methods in finite samples.

econ.EM↗

arXiv2025

Near-Maturity Asymptotics of Critical Prices of American Put Options under Exponential Lévy Models

In the present paper, we study the near-maturity ($t\rightarrow T^{-}$) convergence rate of the optimal early-exercise price $b(t)$ of an American put under an exponential Lévy model with a {\it nonzero} Brownian component. Two important settings, not previous covered in the literature, are considered. In the case that the optimal exercise price converges to the strike price ($b(T^{-})=K$), we contemplate models with negative jumps of unbounded variation (i.e., processes that exhibit high activity of negative jumps or sudden falls in asset prices). In the second case, when the optimal exercise price tend to a value lower than $K$, we consider infinite activity jumps (though still of bounded variations), extending existing results for models with finite jump activity (finitely many jumps in any finite interval). In both cases, we show that $b(T^{-})-b(t)$ is of order $\sqrt{T-t}$ with explicit constants proportionality. Furthermore, we also derive the second-order near-maturity expansion of the American put price around the critical price along a certain parabolic branch.

q-fin.MF↗

arXiv2025

Estimation of Integrated Volatility Functionals with Kernel Spot Volatility Estimators

For a multidimensional Itô semimartingale, we consider the problem of estimating integrated volatility functionals. Jacod and Rosenbaum (2013) studied a plug-in type of estimator based on a Riemann sum approximation of the integrated functional and a spot volatility estimator with a forward uniform kernel. Motivated by recent results that show that spot volatility estimators with general two-side kernels of unbounded support are more accurate, in this paper, an estimator using a general kernel spot volatility estimator as the plug-in is considered. A biased central limit theorem for estimating the integrated functional is established with an optimal convergence rate. Unbiased central limit theorems for estimators with proper de-biasing terms are also obtained both at the optimal convergence regime for the bandwidth and when applying undersmoothing. Our results show that one can significantly reduce the estimator's bias by adopting a general kernel instead of the standard uniform kernel. Our proposed bias-corrected estimators are found to maintain remarkable robustness against bandwidth selection in a variety of sampling frequencies and functions.

econ.EM↗

arXiv2024

Data-driven fixed-point tuning for truncated realized variations

Many methods for estimating integrated volatility and related functionals of semimartingales in the presence of jumps require specification of tuning parameters for their use in practice. In much of the available theory, tuning parameters are assumed to be deterministic and their values are specified only up to asymptotic constraints. However, in empirical work and in simulation studies, they are typically chosen to be random and data-dependent, with explicit choices often relying entirely on heuristics. In this paper, we consider novel data-driven tuning procedures for the truncated realized variations of a semimartingale with jumps based on a type of random fixed-point iteration. Being effectively automated, our approach alleviates the need for delicate decision-making regarding tuning parameters in practice and can be implemented using information regarding sampling frequency alone. We demonstrate our methods can lead to asymptotically efficient estimation of integrated volatility and exhibit superior finite-sample performance compared to popular alternatives in the literature.

math.ST↗

arXiv2024

Adaptive Optimal Market Making Strategies with Inventory Liquidation Cos

A novel high-frequency market-making approach in discrete time is proposed that admits closed-form solutions. By taking advantage of demand functions that are linear in the quoted bid and ask spreads with random coefficients, we model the variability of the partial filling of limit orders posted in a limit order book (LOB). As a result, we uncover new patterns as to how the demand's randomness affects the optimal placement strategy. We also allow the price process to follow general dynamics without any Brownian or martingale assumption as is commonly adopted in the literature. The most important feature of our optimal placement strategy is that it can react or adapt to the behavior of market orders online. Using LOB data, we train our model and reproduce the anticipated final profit and loss of the optimal strategy on a given testing date using the actual flow of orders in the LOB. Our adaptive optimal strategies outperform the non-adaptive strategy and those that quote limit orders at a fixed distance from the midprice.

q-fin.TR↗

arXiv2024

Efficient Integrated Volatility Estimation in the Presence of Infinite Variation Jumps via Debiased Truncated Realized Variations

Statistical inference for stochastic processes based on high-frequency observations has been an active research area for more than two decades. One of the most well-known and widely studied problems has been the estimation of the quadratic variation of the continuous component of an Itô semimartingale with jumps. Several rate- and variance-efficient estimators have been proposed in the literature when the jump component is of bounded variation. However, to date, very few methods can deal with jumps of unbounded variation. By developing new high-order expansions of the truncated moments of a locally stable Lévy process, we propose a new rate- and variance-efficient volatility estimator for a class of Itô semimartingales whose jumps behave locally like those of a stable Lévy process with Blumenthal-Getoor index $Y\in (1,8/5)$ (hence, of unbounded variation). The proposed method is based on a two-step debiasing procedure for the truncated realized quadratic variation of the process and can also cover the case $Y<1$. Our Monte Carlo experiments indicate that the method outperforms other efficient alternatives in the literature in the setting covered by our theoretical framework.

econ.EM↗

arXiv2022

Estimation of Tempered Stable Lévy Models of Infinite Variation

We propose a new method for the estimation of a semiparametric tempered stable Lévy model. The estimation procedure combines iteratively an approximate semiparametric method of moment estimator, Truncated Realized Quadratic Variations (TRQV), and a newly found small-time high-order approximation for the optimal threshold of the TRQV of tempered stable processes. The method is tested via simulations to estimate the volatility and the Blumenthal-Getoor index of the generalized CGMY model as well as the integrated volatility of a Heston-type model with CGMY jumps. The method outperforms other efficient alternatives proposed in the literature when working with a Lévy process (i.e., the volatility is constant), or when the index of jump intensity $Y$ is larger than $3/2$ in the presence of stochastic volatility.

econ.EM↗

arXiv2022

Kernel Estimation of Spot Volatility with Microstructure Noise Using Pre-Averaging

We first revisit the problem of estimating the spot volatility of an Itô semimartingale using a kernel estimator. We prove a Central Limit Theorem with optimal convergence rate for a general two-sided kernel. Next, we introduce a new pre-averaging/kernel estimator for spot volatility to handle the microstructure noise of ultra high-frequency observations. We prove a Central Limit Theorem for the estimation error with an optimal rate and study the optimal selection of the bandwidth and kernel functions. We show that the pre-averaging/kernel estimator's asymptotic variance is minimal for exponential kernels, hence, justifying the need of working with kernels of unbounded support as proposed in this work. We also develop a feasible implementation of the proposed estimators with optimal bandwidth. Monte Carlo experiments confirm the superior performance of the devised method.

econ.EM↗

arXiv2022

Efficient Volatility Estimation for Lévy Processes with Jumps of Unbounded Variation

Statistical inference for stochastic processes based on high-frequency observations has been an active research area for more than a decade. One of the most well-known and widely studied problems is that of estimation of the quadratic variation of the continuous component of an Itô semimartingale with jumps. Several rate- and variance-efficient estimators have been proposed in the literature when the jump component is of bounded variation. However, to date, very few methods can deal with jumps of unbounded variation. By developing new high-order expansions of the truncated moments of a Lévy process, we construct a new rate- and variance-efficient estimator for a class of Lévy processes of unbounded variation, whose small jumps behave like those of a stable Lévy process with Blumenthal-Getoor index less than $8/5$. The proposed method is based on a two-step debiasing procedure for the truncated realized quadratic variation of the process. Our Monte Carlo experiments indicate that the method outperforms other efficient alternatives in the literature in the setting covered by our theoretical framework.

econ.EM↗

arXiv2021

Market Making with Stochastic Liquidity Demand: Simultaneous Order Arrival and Price Change Forecasts

We provide an explicit characterization of the optimal market making strategy in a discrete-time Limit Order Book (LOB). In our model, the number of filled orders during each period depends linearly on the distance between the fundamental price and the market maker's limit order quotes, with random slope and intercept coefficients. The high-frequency market maker (HFM) incurs an end-of-the-day liquidation cost resulting from linear price impact. The optimal placement strategy incorporates in a novel and parsimonious way forecasts about future changes in the asset's fundamental price. We show that the randomness in the demand slope reduces the inventory management motive, and that a positive correlation between demand slope and investors' reservation prices leads to wider spreads. Our analysis reveals that the simultaneous arrival of buy and sell market orders (i) reduces the shadow cost of inventory, (ii) leads the HFM to reduce price pressures to execute larger flows, and (iii) introduces patterns of nonlinearity in the intraday dynamics of bid and ask spreads. Our empirical study shows that the market making strategy outperforms those which ignores randomness in demand, simultaneous arrival of buy and sell market orders, and local drift in the fundamental price.

q-fin.TR↗

arXiv2020

Optimal Iterative Threshold-Kernel Estimation of Jump Diffusion Processes

In this paper, we propose a new threshold-kernel jump-detection method for jump-diffusion processes, which iteratively applies thresholding and kernel methods in an approximately optimal way to achieve improved finite-sample performance. We use the expected number of jump misclassifications as the objective function to optimally select the threshold parameter of the jump detection scheme. We prove that the objective function is quasi-convex and obtain a new second-order infill approximation of the optimal threshold in closed form. The approximate optimal threshold depends not only on the spot volatility, but also the jump intensity and the value of the jump density at the origin. Estimation methods for these quantities are then developed, where the spot volatility is estimated by a kernel estimator with thresholding and the value of the jump density at the origin is estimated by a density kernel estimator applied to those increments deemed to contain jumps by the chosen thresholding criterion. Due to the interdependency between the model parameters and the approximate optimal estimators built to estimate them, a type of iterative fixed-point algorithm is developed to implement them. Simulation studies for a prototypical stochastic volatility model show that it is not only feasible to implement the higher-order local optimal threshold scheme but also that this is superior to those based only on the first order approximation and/or on average values of the parameters over the estimation time period.

math.ST↗

arXiv2019

Bayesian Inference on Volatility in the Presence of Infinite Jump Activity and Microstructure Noise

Volatility estimation based on high-frequency data is key to accurately measure and control the risk of financial assets. A Lévy process with infinite jump activity and microstructure noise is considered one of the simplest, yet accurate enough, models for financial data at high-frequency. Utilizing this model, we propose a "purposely misspecified" posterior of the volatility obtained by ignoring the jump-component of the process. The misspecified posterior is further corrected by a simple estimate of the location shift and re-scaling of the log likelihood. Our main result establishes a Bernstein-von Mises (BvM) theorem, which states that the proposed adjusted posterior is asymptotically Gaussian, centered at a consistent estimator, and with variance equal to the inverse of the Fisher information. In the absence of microstructure noise, our approach can be extended to inferences of the integrated variance of a general Itô semimartingale. Simulations are provided to demonstrate the accuracy of the resulting credible intervals, and the frequentist properties of the approximate Bayesian inference based on the adjusted posterior.

math.ST↗

arXiv2017

Third-Order Short-Time Expansions for Close-to-the-Money Option Prices under the CGMY Model

A third-order approximation for close-to-the-money European option prices under an infinite-variation CGMY Lévy model is derived, and is then extended to a model with an additional independent Brownian component. The asymptotic regime considered, in which the strike is made to converge to the spot stock price as the maturity approaches zero, is relevant in applications since the most liquid options have strikes that are close to the spot price. Our results shed new light on the connection between both the volatility of the continuous component and the jump parameters and the behavior of option prices near expiration when the strike is close to the spot price. In particular, a new type of transition phenomenon is uncovered in which the third order term exhibits two distinct asymptotic regimes depending on whether $Y\in(1,3/2)$ or $Y\in(3/2,2)$. Unlike second order approximations, the expansions herein are shown to be remarkably accurate so that they can actually be used for calibrating some model parameters. For illustration, we calibrate the volatility $σ$ of the Brownian component and the jump intensity $C$ of the CGMY model to actual option prices.

q-fin.PR↗

arXiv2017

Optimal placement of a small order in a diffusive limit order book

We study the optimal placement problem of a stock trader who wishes to clear his/her inventory by a predetermined time horizon t, by using a limit order or a market order. For a diffusive market, we characterize the optimal limit order placement policy and analyze its behavior under different market conditions. In particular, we show that, in the presence of a negative drift, there exists a critical time t0>0 such that, for any time horizon t>t0, there exists an optimal placement, which, contrary to earlier work, is different from one that is placed "infinitesimally" close to the best ask, such as the best bid and second best bid. We also propose a simple method to approximate the critical time t0 and the optimal order placement.

q-fin.TR↗

arXiv2017

Optimum thresholding using mean and conditional mean square error

We consider a univariate semimartingale model for (the logarithm of) an asset price, containing jumps having possibly infinite activity (IA). The nonparametric threshold estimator of the integrated variance IV proposed in Mancini 2009 is constructed using observations on a discrete time grid, and precisely it sums up the squared increments of the process when they are below a threshold, a deterministic function of the observation step and possibly of the coefficients of X. All the threshold functions satisfying given conditions allow asymptotically consistent estimates of IV, however the finite sample properties of the truncated realized variation can depend on the specific choice of the threshold. We aim here at optimally selecting the threshold by minimizing either the estimation mean square error (MSE) or the conditional mean square error (cMSE). The last criterion allows to reach a threshold which is optimal not in mean but for the specific volatility (and jumps paths) at hand. A parsimonious characterization of the optimum is established, which turns out to be asymptotically proportional to the Lévy's modulus of continuity of the underlying Brownian motion. Moreover, minimizing the cMSE enables us to propose a novel implementation scheme for approximating the optimal threshold. Monte Carlo simulations illustrate the superior performance of the proposed method.

q-fin.ST↗

arXiv2017

Change-point detection for Lévy processes

Since the work of Page in the 1950s, the problem of detecting an abrupt change in the distribution of stochastic processes has received a great deal of attention. In particular, a deep connection has been established between Lorden's minimax approach to change-point detection and the widely used CUSUM procedure, first for discrete-time processes, and subsequently for some of their continuous-time counterparts. However, results for processes with jumps are still scarce, while the practical importance of such processes has escalated since the turn of the century. In this work we consider the problem of detecting a change in the distribution of continuous-time processes with independent and stationary increments, i.e. Lévy processes, and our main result shows that CUSUM is indeed optimal in Lorden's sense. This is the most natural continuous-time analogue of the seminal work of Moustakides for sequentially observed random variables that are assumed to be i.i.d. before and after the change-point. From a practical perspective, the approach we adopt is appealing as it consists in approximating the continuous-time problem by a suitable sequence of change-point problems with equispaced sampling points, and for which a CUSUM procedure is shown to be optimal.

math.PR↗

arXiv2017

Short-Time Expansions for Call Options on Leveraged ETFs Under Exponential Lévy models With Local Volatility

In this article, we consider the small-time asymptotics of options on a \emph{Leveraged Exchange-Traded Fund} (LETF) when the underlying Exchange Traded Fund (ETF) exhibits both local volatility and jumps of either finite or infinite activity. Our main results are closed-form expressions for the leading order terms of off-the-money European call and put LETF option prices, near expiration, with explicit error bounds. We show that the price of an out-of-the-money European call on a LETF with positive (negative) leverage is asymptotically equivalent, in short-time, to the price of an out-of-the-money European call (put) on the underlying ETF, but with modified spot and strike prices. Similar relationships hold for other off-the-money European options. In particular, our results suggest a method to hedge off-the-money LETF options near expiration using options on the underlying ETF. Finally, a second order expansion for the corresponding implied volatility is also derived and illustrated numerically.

q-fin.MF↗