arXiv subjects

Alexis Anagnostakis

Publications and source records attributed to Alexis Anagnostakis.

10 recordsLinked to original sources

arXiv2026

On the hedging problem in general 1D diffusion markets

We develop a PDE-based methodology for pricing and hedging European contingent claims in general one-dimensional diffusion markets characterized solely by their scale function and speed measure, possibly without a classical SDE representation, and with constant interest rate. We derive a hedging equation whose solution generates a self-financing hedging strategy and provide sufficient conditions on scale, speed, and interest rate, under which this strategy achieves the minimal hedging capital, expressed through the no free lunch with vanishing risk (NFLVR) condition. We further prove necessary and sufficient conditions for NFLVR and characterize the class of equivalent local martingale measures through an auxiliary diffusion whose scale and speed characteristics are determined by those of the real-world diffusion and by the interest rate. When the NFLVR condition fails, the framework may produce multiple hedging equations corresponding to non-minimal strategies, whose associated prices can exceed the minimal hedging capital. We illustrate both the effectiveness and limitations of the approach through numerical experiments involving diffusion models with irregular features.

q-fin.MF↗

arXiv2025

On the structure of increasing profits in a 1D general diffusion market with interest rates

In this paper, we investigate a financial market model consisting of a risky asset, modeled as a general diffusion parameterized by a scale function and a speed measure, and a bank account process with a constant interest rate. This flexible class of financial market models allows for features such as reflecting boundaries, skewness effects, sticky points, and slowdowns on fractal sets. For this market model, we study the structure of a strong form of arbitrage opportunity called increasing profits. Our main contributions are threefold. First, we characterize the existence of increasing profits in terms of an auxiliary deterministic signed measure $\nu$ and a canonical trading strategy $\theta$, both of which depend only on the deterministic parametric characteristics of our model, namely the scale function, the speed measure, and the interest rate. More precisely, we show that an increasing profit exists if and only if $\nu$ is nontrivial, and that this is equivalent to $\theta$ itself generating an increasing profit. Second, we provide a precise characterization of the entire set of increasing profits in terms of $\nu$ and $\theta$, and moreover characterize the value processes associated with increasing profits. Finally, we establish novel connections between no-arbitrage theory and the general theory of stochastic processes. Specifically, we relate the failure of the representation property for general diffusions to the existence of certain types of increasing profits whose value processes are dominated by the quadratic variation measure of a space-transformed version of the asset price process.

q-fin.MF↗

arXiv2025

General diffusions on metric graphs as limits of time-space Markov Chains

We introduce the Space-Time Markov Chain Approximation (STMCA) for a general diffusion process on a finite metric graph $\Gamma$. The STMCA is a doubly asymmetric (in both time and space) random walk defined on a subdivisions of $\Gamma$, with transition probabilities and conditional transition times that match, in expectation, those of the target diffusion. We derive bounds on the $p$-Wasserstein distances between the diffusion and its STMCA in terms of a thinness quantifier of the subdivision. This bound shows that convergence occurs at any rate inferior to $\frac{1}{4} \wedge \frac{1}{p} $ in terms of the the maximum cell size of the subdivision, for adapted subdivisions, at any rate inferior to $\frac{1}{2} \wedge \frac{2}{p} $. Additionally, we provide explicit analytical formulas for transition probabilities and times, enabling practical implementation of the STMCA. Numerical experiments illustrate our results.

math.PR↗

arXiv2025

On weak notions of no-arbitrage in a 1D general diffusion market with interest rates

We establish deterministic necessary and sufficient conditions for the no-arbitrage notions "no increasing profit" (NIP), "no strong arbitrage" (NSA) and "no unbounded profit with bounded risk" (NUPBR) in one-dimensional general diffusion markets. These are markets with one risky asset, which is modeled as a regular continuous strong Markov process that is also a semimartingale, and a riskless asset that grows exponentially at a constant rate $r\in \mathbb{R}$. All deterministic criteria are provided in terms of the scale function and the speed measure of the risky asset process. Our study reveals a variety of surprising effects. For instance, irrespective of the interest rate, NIP is not excluded by reflecting boundaries or an irregular scale function. In the case of non-zero interest rates, it is even possible that NUPBR holds in the presence of reflecting boundaries and/or skew thresholds. In the zero interest rate regime, we also identify NSA as the minimal no arbitrage notion that excludes reflecting boundaries and that forces the scale function to be continuously differentiable with strictly positive absolutely continuous derivative, meaning that it is of the same form as for a stochastic differential equation.

q-fin.MF↗

arXiv2025

General diffusions on the star graph as time-changed Walsh Brownian motion

We establish the representation of general regular diffusions on star-shaped graphs as time-changed Walsh Brownian motions. These are regular continuous Markov processes described locally by a family generalized second order differential operators defined on every edge and a gluing condition at the junction vertex. This allows us to prove two additional results: (i) A representation of diffusions with sticky gluing conditions as time-changes of diffusions governed by the same differential operators but with non-sticky gluing conditions. (ii) An occupation times formula for such diffusions, analogous to the classical It\^o--McKean formula for one-dimensional diffusions. Additionally, we prove two results of independent interest. First, conditions under which a diffusion on the star graph is Feller and Feller--Dynkin, extending classical results for one-dimensional diffusions. Second, the existence uniqueness of solutions to the Dirichlet problem on the unit disk of the star graph for a general diffusion operator and explicit expressions for its solution.

math.PR↗

arXiv2024

On the number of crossings and bouncings of a diffusion at a sticky threshold

In this paper, we study the asymptotic behavior of the number of crossings by a one-dimensional diffusion of a threshold where the process exhibits stickiness. We distinguish three types of crossings and show that to each type corresponds a distinct asymptotic regime for the respective number of crossings statistic. We introduce notions of bouncing as the symmetric counterparts to crossings and show that the corresponding number of bouncings statistics share the same asymptotic properties as their crossings counterparts. We first prove the results for sticky Brownian motion, then extend them to sticky-reflected Brownian motion (where only bouncing is possible) and to sticky diffusions. As an application, we propose consistent estimators for the stickiness parameter of sticky diffusions and sticky-reflected Brownian motion.

math.PR↗

arXiv2024

Sticky-threshold diffusions, local time approximation and parameter estimation

We study a class of high-frequency path functionals for one-dimensional diffusions with singular thresholds or boundaries, allowing for skewness, discontinuities in the diffusion coefficient, and stickiness or sticky reflection. These functionals, originally developed for local-time approximation in non-singular diffusions, are constructed from a test function and a diverging normalizing sequence. We establish convergence to local time, regardless of the nature of the singular threshold features, thereby extending several recent results on specific cases. Notably, our framework allows for any normalizing diverging sequence that is $o(n)$, where $n$ is the observation frequency, and thresholds at which several singular behaviors occur simultaneously. Combining the local-time approximation with occupation-time approximations, we construct consistent estimators of the stickiness and skewness parameters that remain valid in the presence of jump-discontinuities in the diffusion coefficient; this solves the estimation problem under the simultaneous presence of all these singular features.

math.PR↗

arXiv2023

Pricing and hedging for a sticky diffusion

We introduce a financial market model featuring a risky asset whose price follows a sticky geometric Brownian motion and a riskless asset that grows with a constant interest rate $r\in \mathbb R $. We prove that this model satisfies No Arbitrage (NA) and No Free Lunch with Vanishing Risk (NFLVR) only when $r=0 $. Under this condition, we derive the corresponding arbitrage-free pricing equation, assess replicability and representation of the replication strategy. We then show that all locally bounded replicable payoffs for the standard Black--Scholes model are also replicable for the sticky model. Last, we evaluate via numerical experiments the impact of hedging in discrete time and of misrepresenting price stickiness.

q-fin.MF↗

arXiv2022

General diffusion processes as the limit of time-space Markov chains

We prove the convergence of the law of grid-valued random walks, which can be seen as time-space Markov chains, to the law of a general diffusion process. This includes processes with sticky features, reflecting or absorbing boundaries and skew behavior. We prove that the convergence occurs at any rate strictly inferior to $(1/4) \wedge (1/p)$ in terms of the maximum cell size of the grid, for any $p$-Wasserstein distance. We also show that it is possible to achieve any rate strictly inferior to $(1/2) \wedge (2/p)$ if the grid is adapted to the speed measure of the diffusion, which is optimal for $p\le 4 $. This result allows us to set up asymptotically optimal approximation schemes for general diffusion processes. Last, we experiment numerically on diffusions that exhibit various features.

math.PR↗

arXiv2022

Functional convergence to the local time of a sticky diffusion

We establish the consistency of a local time approximation of a diffusion at a sticky threshold based on high-frequency observations. First, we prove the result for sticky Brownian motion, and then extend it to It\^o diffusions with a sticky point (SID). For this, we derive the pathwise formulation of an SID along with respective versions of key stochastic calculus results (It\^o formula, Girsanov theorem). Based on the local time approximation, we develop a consistent estimator for the stickiness parameter. We conclude with numerical experiments and assess statistical properties of the stickiness estimator and the local time approximation.

math.PR↗