Fractional Brownian Motion

The Omniscient, yet Lazy, Investor ArXiv ID: 2510.24467 “View on arXiv” Authors: Stanisław M. S. Halkiewicz Abstract We formalize the paradox of an omniscient yet lazy investor - a perfectly informed agent who trades infrequently due to execution or computational frictions. Starting from a deterministic geometric construction, we derive a closed-form expected profit function linking trading frequency, execution cost, and path roughness. We prove existence and uniqueness of the optimal trading frequency and show that this optimum can be interpreted through the fractal dimension of the price path. A stochastic extension under fractional Brownian motion provides analytical expressions for the optimal interval and comparative statics with respect to the Hurst exponent. Empirical illustrations on equity data confirm the theoretical scaling behavior. ...

On the rate of convergence of estimating the Hurst parameter of rough stochastic volatility models ArXiv ID: 2504.09276 “View on arXiv” Authors: Unknown Abstract In [“Han & Schied, 2023, \textit{“arXiv 2307.02582”}”], an easily computable scale-invariant estimator $\widehat{"\mathscr{R"}}^s_n$ was constructed to estimate the Hurst parameter of the drifted fractional Brownian motion $X$ from its antiderivative. This paper extends this convergence result by proving that $\widehat{"\mathscr{R"}}^s_n$ also consistently estimates the Hurst parameter when applied to the antiderivative of $g \circ X$ for a general nonlinear function $g$. We also establish an almost sure rate of convergence in this general setting. Our result applies, in particular, to the estimation of the Hurst parameter of a wide class of rough stochastic volatility models from discrete observations of the integrated variance, including the rough fractional stochastic volatility model. ...

Estimation of bid-ask spreads in the presence of serial dependence ArXiv ID: 2407.17401 “View on arXiv” Authors: Unknown Abstract Starting from a basic model in which the dynamic of the transaction prices is a geometric Brownian motion disrupted by a microstructure white noise, corresponding to the random alternation of bids and asks, we propose moment-based estimators along with their statistical properties. We then make the model more realistic by considering serial dependence: we assume a geometric fractional Brownian motion for the price, then an Ornstein-Uhlenbeck process for the microstructure noise. In these two cases of serial dependence, we propose again consistent and asymptotically normal estimators. All our estimators are compared on simulated data with existing approaches, such as Roll, Corwin-Schultz, Abdi-Ranaldo, or Ardia-Guidotti-Kroencke estimators. ...

Modeling a Financial System with Memory via Fractional Calculus and Fractional Brownian Motion ArXiv ID: 2406.19408 “View on arXiv” Authors: Unknown Abstract Financial markets have long since been modeled using stochastic methods such as Brownian motion, and more recently, rough volatility models have been built using fractional Brownian motion. This fractional aspect brings memory into the system. In this project, we describe and analyze a financial model based on the fractional Langevin equation with colored noise generated by fractional Brownian motion. Physics-based methods of analysis are used to examine the phase behavior and dispersion relations of the system upon varying input parameters. A type of anomalous marginal glass phase is potentially seen in some regions, which motivates further exploration of this model and expanded use of phase behavior and dispersion relation methods to analyze financial models. ...

A Note on Optimal Liquidation with Linear Price Impact ArXiv ID: 2402.14100 “View on arXiv” Authors: Unknown Abstract In this note we consider the maximization of the expected terminal wealth for the setup of quadratic transaction costs. First, we provide a very simple probabilistic solution to the problem. Although the problem was largely studied, as far as we know up to date this simple and probabilistic form of the solution has not appeared in the literature. Next, we apply the general result for the numerical study of the case where the risky asset is given by a fractional Brownian Motion and the information flow of the investor can be diversified. ...

The Euler Scheme for Fractional Stochastic Delay Differential Equations with Additive Noise ArXiv ID: 2402.08513 “View on arXiv” Authors: Unknown Abstract In this paper we consider the Euler-Maruyama scheme for a class ofstochastic delay differential equations driven by a fractional Brownian motion with index $H\in(0,1)$. We establish the consistency of the scheme and study the rate of convergence of the normalized error process. This is done by checking that the generic rate of convergence of the error process with stepsize $Δ_{“n”}$ is $Δ_{“n”}^{"\min{H+\frac{1"}{“2”},3H,1}}$. It turned out that such a rate is suboptimal when the delay is smooth and $H>1/2$. In this context, and in contrast to the non-delayed framework, we show that a convergence of order $H+1/2$ is achievable. ...

Rough volatility: evidence from range volatility estimators ArXiv ID: 2312.01426 “View on arXiv” Authors: Unknown Abstract In Gatheral et al. 2018, first posted in 2014, volatility is characterized by fractional behavior with a Hurst exponent $H < 0.5$, challenging traditional views of volatility dynamics. Gatheral et al. demonstrated this using realized volatility measurements. Our study extends this analysis by employing range-based proxies to confirm their findings across a broader dataset and non-standard assets. Notably, we address the concern that rough volatility might be an artifact of microstructure noise in high-frequency return data. Our results reveal that log-volatility, estimated via range-based methods, behaves akin to fractional Brownian motion with an even lower $H$, below $0.1$. We also affirm the efficacy of the rough fractional stochastic volatility model (RFSV), finding that its predictive capability surpasses that of AR, HAR, and GARCH models in most scenarios. This work substantiates the intrinsic nature of rough volatility, independent of the microstructure noise often present in high-frequency financial data. ...

Estimating the roughness exponent of stochastic volatility from discrete observations of the integrated variance ArXiv ID: 2307.02582 “View on arXiv” Authors: Unknown Abstract We consider the problem of estimating the roughness of the volatility process in a stochastic volatility model that arises as a nonlinear function of fractional Brownian motion with drift. To this end, we introduce a new estimator that measures the so-called roughness exponent of a continuous trajectory, based on discrete observations of its antiderivative. The estimator has a very simple form and can be computed with great efficiency on large data sets. It is not derived from distributional assumptions but from strictly pathwise considerations. We provide conditions on the underlying trajectory under which our estimator converges in a strictly pathwise sense. Then we verify that these conditions are satisfied by almost every sample path of fractional Brownian motion (with drift). As a consequence, we obtain strong consistency theorems in the context of a large class of rough volatility models, such as the rough fractional volatility model and the rough Bergomi model. We also demonstrate that our estimator is robust with respect to proxy errors between the integrated and realized variance, and that it can be applied to estimate the roughness exponent directly from the price trajectory. Numerical simulations show that our estimation procedure performs well after passing to a scale-invariant modification of our estimator. ...

Fractal properties, information theory, and market efficiency ArXiv ID: 2306.13371 “View on arXiv” Authors: Unknown Abstract Considering that both the entropy-based market information and the Hurst exponent are useful tools for determining whether the efficient market hypothesis holds for a given asset, we study the link between the two approaches. We thus provide a theoretical expression for the market information when log-prices follow either a fractional Brownian motion or its stationary extension using the Lamperti transform. In the latter model, we show that a Hurst exponent close to 1/2 can lead to a very high informativeness of the time series, because of the stationarity mechanism. In addition, we introduce a multiscale method to get a deeper interpretation of the entropy and of the market information, depending on the size of the information set. Applications to Bitcoin, CAC 40 index, Nikkei 225 index, and EUR/USD FX rate, using daily or intraday data, illustrate the methodological content. ...