New model uses generalized fractional Brownian motion for stock price prediction.
problem Traditional models fail to accurately predict stock price fluctuations.
method Introduces generalized fractional Brownian motion as a new stochastic process for price modeling.
result Validates the new model for option pricing and risk assessment.
Paper defines multi-dimensional fractional Brownian motion under volatility uncertainty.
problem Volatility uncertainty in fractional Brownian motion.
method Definition and study of multi-dimensional fractional Brownian motion (G-fBm) with Hurst index.
result First results on stochastic calculus for G-fBm with Hurst index > 0.5.
Modeling financial markets with memory using fractional calculus and Brownian motion.
problem Capturing memory effects in financial markets using stochastic models.
method Fractional Langevin equation with colored noise generated by fractional Brownian motion.
result Anomalous marginal glass phase observed in some regions of the system.
A new model captures option price dynamics using sub-fractional Brownian motion.
problem Capturing the complex price dynamics of financial options.
method Developed a CEV model driven by a mixed sub-fractional Brownian motion.
result Empirical tests show the model effectively captures option price dynamics.
This paper derives the non-analytic solution to the Fokker-Planck equation of fractional Brownian motion using the method of Laplace transform. Sequentially, by considering the fundamental solution of the non-analytic solution, this paper obtains the transition probability density function of the random variable that i…
The paper extends Merton model to price equity warrants under subdiffusive fractional Brownian motion of the short rate.
problem Equity warrant pricing under subdiffusive fractional Brownian motion of the short rate.
method The paper applies subdiffusive mechanism to analyze equity warrant in a fractional Brownian motion environment, deriving a pricing formula for equity warrant.
result The paper provides a pricing formula for equity warrants under subdiffusive fractional Brownian motion model of the short rate.
Rough volatility models are becoming increasingly popular in quantitative finance. In this framework, one considers that the behavior of the log-volatility process of a financial asset is close to that of a fractional Brownian motion with Hurst parameter around 0.1. Motivated by this, we wish to define a natural and re…
Replacing Black-Scholes' driving process, Brownian motion, with fractional Brownian motion allows for incorporation of a past dependency of stock prices but faces a few major downfalls, including the occurrence of arbitrage when implemented in the financial market. We present the development, testing, and implementatio…
A new option pricing model uses a time-varying Hurst exponent for more accurate financial predictions.
problem Inaccurate modeling of financial time series due to constant memory parameter limitations.
method Modeling price fluctuations with multifractional Brownian motion and deriving option pricing formula.
result Empirical performance shows the multifractional model fits market quotes better than standard models.
We consider so-called regular invertible Gaussian Volterra processes and derive a formula for their prediction laws. Examples of such processes include the fractional Brownian motions and the mixed fractional Brownian motions. As an application, we consider conditional-mean hedging under transaction costs in Black-Scho…
The paper provides approximations for pricing Asian options using a mixed fractional Brownian motion with jumps.
problem Pricing Asian options under a mixed fractional Brownian motion with jumps.
method Approximate closed-form solutions for arithmetic Asian options and power options.
result Analytical formulas for pricing arithmetic Asian options and power options are derived.
New rough stochastic volatility models using log-modulated fractional Brownian motion.
problem Analyzing rough stochastic volatility models over the range 0≤H<1/2. method Introducing log-modulated fractional Brownian motion (log-fBm) to handle H=0 and analyze over the full range. result Obtained skew asymptotics of log(1/T)−pTH−1/2 as To0 for H≥0, no flattening of skew as Ho0. New formulas forecast fractional Brownian motion for financial trading.
problem Forecasting financial log-prices following fractional Brownian motion.
method Theoretical formulas for accuracy metrics in fBm forecasting.
result Optimal trading strategies in fBm framework identified.
We survey some new progress on the pricing models driven by fractional Brownian motion \cb{or} mixed fractional Brownian motion. In particular, we give results on arbitrage opportunities, hedging, and option pricing in these models. We summarize some recent results on fractional Black & Scholes pricing model with trans…
G-framework is presented by Peng [41] for measure risk under uncertainty. In this paper, we define fractional G-Brownian motion (fGBm). Fractional G-Brownian motion is a centered G-Gaussian process with zero mean and stationary increments in the sense of sub-linearity with Hurst index H∈(0,1). This process has sta…
This paper extends Heston model to fractional Brownian motion for option pricing.
problem Developing a new financial model for option pricing with fractional Brownian motion.
method Extending Malliavin differentiability to fractional Heston-type model.
result Proves fractional Heston-type model is Malliavin differentiable and derives option pricing expressions.
The mixed-fractional CEV model improves CDS pricing by accounting for default risk.
problem Improving the pricing of Credit Default Swaps (CDS) by accounting for default risk.
method Using a mixed-fractional Brownian motion to model the Constant Elasticity of Variance (CEV) model.
result The mixed-fractional CEV model yields more realistic CDS spreads and default probabilities.
Study rough volatility models using path-dependent PDEs and fractional Brownian motions.
problem Modeling and analyzing rough volatility in financial markets.
method Showed conditional expectations are unique classical solutions to path-dependent PDEs derived from functional Itô formula. Leverage these to study weak rates of convergence for discretized stochastic integrals.
result Obtained optimal weak error rates for approximating log-stock prices in rough volatility models.
Model rough volatility using RDEs with correlated Brownian motion and fractional Brownian motion.
problem Modeling rough volatility with correlated stochastic processes.
method Developed a method to lift Brownian motion and rough paths, applying it to fractional Brownian motion to model rough volatility.
result Calibrated a new rough volatility model to market data.
Analyzed a generalized voter model with power-law herding intensity, revealing anomalous diffusion and long-range memory.
problem Anomalous diffusion and long-range memory in a generalized voter model.
method Derived analytical expressions for moments and first passage time distribution, confirmed numerically.
result The model exhibits long-range memory indicators despite being a Markov model.
FDBM models use fractional Brownian motion to model complex stochastic processes.
problem Capturing memory effects and long-range dependencies in stochastic processes.
method Developed a generative diffusion bridge framework using a Markovian approximation of fractional Brownian motion.
result FDBM outperforms standard models in predicting future states and unpaired data translation.
CFTM uses fractional Brownian motion for dynamic topic modeling.
problem Identifying long-term dependency or roughness in topic and word distributions over time.
method Continuous Time Fractional Topic Model (cFTM) incorporating fractional Brownian motion.
result cFTM captures long-term dependency or roughness in topic and word distributions.
This study deals with the problem of pricing compound options when the underlying asset follows a mixed fractional Brownian motion with jumps. An analytic formula for compound options is derived under the risk neutral measure. Then, these results are applied to value extendible options. Moreover, some special cases of …
Improved volatility models for option pricing with weak error rates.
problem Improving volatility models to fit market data better.
method Developed a weak convergence analysis for the Euler method applied to linear rough volatility models.
result Proved weak convergence rates of 1/2 + H for linear models and 1 for quadratic payoffs.
The aim of this paper is to evaluate geometric Asian option by a mixed fractional subdiffusive Black-Scholes model. We derive a pricing formula for geometric Asian option when the underlying stock follows a time changed mixed fractional Brownian motion. We then apply the results to price Asian power options on the stoc…
The paper proposes estimators for bid-ask spreads with and without serial dependence.
problem Estimating bid-ask spreads in financial markets with and without serial dependence.
method The authors propose moment-based estimators for bid-ask spreads, considering both geometric Brownian motion and geometric fractional Brownian motion for price dynamics, and Ornstein-Uhlenbeck process for microstructure noise.
result The estimators are consistent and asymptotically normal, and perform well compared to existing approaches on simulated data.
Instantaneous volatility of logarithmic return in the lognormal fractional SABR model is driven by the exponentiation of a correlated fractional Brownian motion. Due to the mixed nature of driving Brownian and fractional Brownian motions, probability density for such a model is less studied in the literature. We show i…
The long-term dependence of Bitcoin (BTC), manifesting itself through a Hurst exponent H>0.5, is exploited in order to predict future BTC/USD price. A Monte Carlo simulation with 104 geometric fractional Brownian motion realisations is performed as extensions of historical data. The accuracy of statistical inferen…
Study pricing derivatives in markets with long-range dependence and jumps.
problem Deriving pricing formulas for derivatives in markets with long-range dependence and jumps.
method Developed a fractional integro-partial differential equation (PIDE) and used semigroup theory and finite-difference schemes for numerical solutions.
result Closed-form pricing formula for European options and numerical solution for general options.
Quantum probability theory constructs Martingales for non-Brownian financial models.
problem Constructing Martingales for financial models using fractional Brownian motion.
method Quantum probability theory and Wick product.
result Quantum probability framework allows for Martingale construction without Brownian integrals.
We develop a variational framework for SDEs driven by fractional noise.
problem Capturing long-term dependencies in SDEs driven by fractional noise.
method Markov approximation of fractional Brownian motion, variational inference, neural networks.
result Efficient variational inference of posterior path measures for neural-SDEs.
The study tackles rough noise in high-frequency financial data using fractional Brownian motion.
problem Impediments to analyzing high-frequency financial data due to noise.
method Assuming an efficient price process as a continuous Itô semimartingale, the study derives consistent estimators and confidence intervals for roughness parameters and volatilities.
result The rough noise model explains divergence rates in volatility signature plots over time and between assets.
We introduce Hermite fractional financial markets, where market uncertainties are described by multidimensional Hermite motions. Hermite markets include as particular cases financial markets driven by multivariate fractional Brownian motion and multivariate Rosenblatt motion. Conditions for no-arbitrage and market comp…
In this work we introduce Heath-Jarrow-Morton (HJM) interest rate models driven by fractional Brownian motions. By using support arguments we prove that the resulting model is arbitrage free under proportional transaction costs in the same spirit of Guasoni [Math. Finance 16 (2006) 569-582]. In particular, we obtain a …
The Black-Scholes implied volatility skew at the money of SPX options is known to obey a power law with respect to the time-to-maturity. We construct a model of the underlying asset price process which is dynamically consistent to the power law. The volatility process of the model is driven by a fractional Brownian mot…
The paper evaluates integrals for fBm with various Hurst indices.
problem Evaluating integrals for stochastic processes with fractional Brownian motion for different Hurst indices.
method Analytic continuation from complex analysis to extend integral domain.
result Integral formulas for fBm with Hurst indices H∈(0,1) are derived. We continue the analysis of our previous paper (Czichowsky/Schachermayer/Yang 2014) pertaining to the existence of a shadow price process for portfolio optimisation under proportional transaction costs. There, we established a positive answer for a continuous price process S=(St)0≤t≤T satisfying the condi…
Study evaluates discretized arbitrage strategies in fractional financial markets.
problem Serial correlation in financial markets with fractional Brownian motion.
method Revisit and transfer Shiryaev and Salopek's strategies to a real-world setting, distretizing dynamics and introducing transaction costs.
result Both strategies are promising with respect to terminal portfolio values and loss probabilities.
It has been recently shown that rough volatility models, where the volatility is driven by a fractional Brownian motion with small Hurst parameter, provide very relevant dynamics in order to reproduce the behavior of both historical and implied volatilities. However, due to the non-Markovian nature of the fractional Br…
mfBm models and forecasts volatility with different Hurst exponents and correlations.
problem Modeling and forecasting volatility with varying Hurst exponents and correlations.
method Multivariate fractional Brownian motion (mfBm) with component-wise Hurst exponents, novel estimation method, time-reversibility test.
result mfBm reduces forecasting errors compared to a one-dimensional model and outperforms HAR model.
Non-Markovian point process shows power-law scaling, similar to nonlinear Markovian process.
problem Understanding the scaling behavior of non-Markovian point processes.
method Analyzed a confined fractional Brownian motion-driven point process and compared it to a nonlinear Markovian process.
result A nonlinear Markovian process can reproduce the power-law scaling behavior of a non-Markovian point process.
We consider conditional-mean hedging in a fractional Black-Scholes pricing model in the presence of proportional transaction costs. We develop an explicit formula for the conditional-mean hedging portfolio in terms of the recently discovered explicit conditional law of the fractional Brownian motion.
The paper introduces a new stochastic volatility model with long-term memory and jumps.
problem Developing a model for variance and volatility swaps with long-term memory and jumps.
method Fractional Barndorff-Nielsen and Shephard model incorporating long-term memory and jumps.
result Arbitrage-free prices for variance and volatility swaps derived for the new model.
Deep learning predicts path-dependent processes from historical data.
problem Predicting path-dependent processes using historical data.
method Nonparametric regression with deep neural networks.
result Deep learning method converges to theoretical predictions as observation frequency increases.
While absence of arbitrage in frictionless financial markets requires price processes to be semimartingales, non-semimartingales can be used to model prices in an arbitrage-free way, if proportional transaction costs are taken into account. In this paper, we show, for a class of price processes which are not necessaril…
Paper develops Euler scheme for fractional delay diff. eqs with additive noise.
problem Developing a consistent Euler-Maruyama scheme for fractional stochastic delay diff. eqs.
method Euler-Maruyama scheme for fractional Brownian motion with additive noise.
result Achieved convergence rate of H+1/2 for smooth delays when H>1/2.
Proposes a new model for equity options calibration.
problem Calibration of joint SPX/VIX options.
method Replaces fractional Brownian motion with grey Brownian motion.
result Shows potential advantages and calibration results for new model.
Develops a bi-variate stochastic framework to model mortality and interest rates with long-range dependence.
problem Captures long-range dependence and instantaneous correlation in mortality and interest rates.
method Mixed fractional Brownian motions, analytical solutions, risk-neutral measure, sequential parameter estimation.
result Explicit pricing of zero-coupon bonds and extreme mortality bonds, practical implications for pricing and risk management.