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Keywords = correlated non-stationary Gaussian increments

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57 pages, 4880 KB  
Article
Analytical Pricing of Volatility-Linked Financial Derivatives Under the Sub-Mixed Fractional Brownian Motion Framework in a No-Arbitrage Complete Market
by Sanae Rujivan, Touch Toem and Angelo E. Marasigan
Fractal Fract. 2026, 10(2), 125; https://doi.org/10.3390/fractalfract10020125 - 14 Feb 2026
Viewed by 1246
Abstract
This paper develops a unified analytical approach for pricing a broad class of volatility-linked financial derivatives under the sub-mixed fractional geometric Brownian motion model. The proposed framework captures key empirical features of financial markets, including correlated non-stationary Gaussian increments and long-memory dependence, while [...] Read more.
This paper develops a unified analytical approach for pricing a broad class of volatility-linked financial derivatives under the sub-mixed fractional geometric Brownian motion model. The proposed framework captures key empirical features of financial markets, including correlated non-stationary Gaussian increments and long-memory dependence, while preserving the semimartingale property required for arbitrage-free pricing. We present the exact distribution of the realized variance as a quadratic form of correlated non-stationary Gaussian increments, which leads to a closed-form expression for the cumulative distribution function via a Laguerre-series expansion. These distributional results enable analytical pricing formulas for an extensive family of volatility-linked derivatives. Monte Carlo simulations confirm the accuracy and computational efficiency of the proposed formulas, while numerical investigations illustrate the significant impact of non-stationarity, long-memory effects, and the Hurst parameter on derivative values. These results contribute to a deeper theoretical understanding and more effective computational methods for pricing nonlinear volatility derivatives in markets characterized by persistent temporal dependence and non-stationary stochastic dynamics. Full article
(This article belongs to the Special Issue Application of Fractal Processes and Fractional Derivatives in Finance, 3rd Edition)
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33 pages, 752 KB  
Article
Flux and First-Passage Time Distributions in One-Dimensional Integrated Stochastic Processes with Arbitrary Temporal Correlation and Drift
by Holger Nobach and Stephan Eule
Mathematics 2025, 13(19), 3163; https://doi.org/10.3390/math13193163 - 2 Oct 2025
Viewed by 1082
Abstract
The arrival of tracers at boundaries with defined distances from the origin of their motion in stochastically fluctuating advection processes is investigated. The advection model is a stationary one-dimensional integrated stochastic process with an arbitrary a priori known correlation and with possible mean [...] Read more.
The arrival of tracers at boundaries with defined distances from the origin of their motion in stochastically fluctuating advection processes is investigated. The advection model is a stationary one-dimensional integrated stochastic process with an arbitrary a priori known correlation and with possible mean drift. The current (direction-sensitive), the total flux (direction-insensitive) of tracers through a non-absorbing boundary, and the first-passage times of the tracers at an absorbing boundary are derived depending on the correlation function of the carrying flow velocity. While the general derivations are universal with respect to the distribution function of the advection’s increments, the current and the total flux are explicitly derived for a Gaussian distribution. The first-passage time is derived implicitly through an integral that is solved numerically in the present study. No approximations or restrictions to special cases of the advection process are used. One application is one-dimensional Gaussian turbulence, where the one-dimensional random velocity carries tracer particles through space. Finally, subdiffusive or superdiffusive behavior can temporarily be reached by such a stochastic process with an adequately designed correlation function. Full article
(This article belongs to the Special Issue Advances in Statistical Approaches with Applications for Multivariate Data Analysis)
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