Establishes a microstructural foundation for a rough log-normal volatility model.
problem Developing a robust model for financial volatility under microstructural effects.
method Introduced a sequence of order-driven financial market models with Poisson process arrivals and analyzed their convergence to a log-normal rough volatility model.
result Weak convergence of price-volatility process to a log-normal rough volatility model with established weak error rates.
Critical volatility triggers log-normal to power-law transitions in interconnected systems.
problem Understanding the transition from log-normal to power-law distributions in interconnected systems.
method Analyzing an infinite option-on-option chain model, deriving a critical volatility threshold.
result A critical volatility threshold of approximately 250.66% for unconditional cases, dropping to 125.3% with selective survival.
We consider an interest rate model with log-normally distributed rates in the terminal measure in discrete time. Such models are used in financial practice as parametric versions of the Markov functional model, or as approximations to the log-normal Libor market model. We show that the model has two distinct regimes, a…
We propose a novel time discretization for the log-normal SABR model and derive its asymptotic properties.
problem Analyzing the log-normal SABR model's time-discretized behavior and implied volatility surface.
method We use the Euler-Maruyama scheme for time discretization and derive asymptotic properties in the limit of large number of time steps.
result We derive an exact representation of the implied volatility surface for arbitrary maturity and strike in the asymptotic regime.
In this paper we investigate general linear stochastic volatility models with correlated Brownian noises. In such models the asset price satisfies a linear SDE with coefficient of linearity being the volatility process. This class contains among others Black-Scholes model, a log-normal stochastic volatility model and H…
We derive the exact solution of a one-dimensional Markov functional model with log-normally distributed interest rates in discrete time. The model is shown to have two distinct limiting states, corresponding to small and asymptotically large volatilities, respectively. These volatility regimes are separated by a phase …
We study the dynamics of the normal implied volatility in a local volatility model, using a small-time expansion in powers of maturity T. At leading order in this expansion, the asymptotics of the normal implied volatility is similar, up to a different definition of the moneyness, to that of the log-normal volatility. …
We study the volatility of the S&P500 stock index from 1984 to 1996 and find that the volatility distribution can be very well described by a log-normal function. Further, using detrended fluctuation analysis we show that the volatility is power-law correlated with Hurst exponent α≅0.9.
Significantly reduces Monte Carlo runtime for rough Bergomi model pricing.
problem Calibrating rough Bergomi model with high runtime for Monte Carlo simulations.
method Novel composition of variance reduction methods for log-normal stochastic volatility models.
result Significant runtime reductions (20 times average) across different correlation regimes.
This work introduces a geometric approach to probability representation and option pricing.
problem Representing probability distributions geometrically for better understanding and approximation.
method Introducing a geometric representation of probability using implied volatility and geometric transformations.
result Any probability distribution on positive reals can be represented by a planar curve, facilitating approximation and analysis.
Bayesian inference and superstatistics model financial volatility dynamics across different timescales.
problem Modeling correlated volatility in financial time series with heavy tails and long memory.
method Superstatistical dynamics, Bayesian Inference, Metropolis-Hasting sampling.
result The log-Normal model is reliable for short timescales, while inverse-Gamma is preferred for long timescales.
Introduces new financial models using subordinated processes.
problem Modeling asset returns with behavioral finance considerations.
method Introduces multiple internally embedded financial time-clocks, subordinated to Brownian motion, with a behavioral subordinator.
result New log-price process with multiple embedded subordinations, requiring estimation of new parameters.
We propose a stochastic process for stock movements that, with just one source of Brownian noise, has an instantaneous volatility that rises from a type of statistical feedback across many time scales. This results in a stationary non-Gaussian process which captures many features observed in time series of real stock r…
A new tree model, GRST, improves option pricing without log-normality assumptions.
problem Limitations of CRR binomial trees in valuing securities with early exercise characteristics.
method Gaussian Recombining Split Tree (GRST) that generates a discrete probability mass function approximating a Gaussian distribution.
result Option prices from GRST align closely with market prices.
Proposes a new way to represent uncertainty using implied volatility.
problem Uncertainty in financial markets and biological systems.
method Mathematical analysis of various probability distributions.
result Representation of different probability distributions using BSM implied volatility.
Study on Kyle's model with stochastic liquidity impacts asset volatility.
problem Impact of stochastic volatility of noise trading on asset volatility.
method Construct equilibrium for continuous-time Kyle's model with stochastic liquidity.
result In equilibrium, Kyle's Lambda and its inverse are submartingales.
Study shows non-convergence of short-maturity expansion in SABR model.
problem Analyzing convergence of SABR model option prices.
method Analyzing analyticity properties of the payoff function and using the McKean kernel.
result Short-maturity expansion for SABR model is asymptotic and non-convergent.
Introduces log S-fBM model to unify rough and multifractal volatility.
problem Modeling volatility with varying roughness and multifractality.
method Develops log S-fBM family of random measures and proposes estimation methods.
result Demonstrates the estimation of Hurst exponent H and intermittency coefficient λ².
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.
The dynamics of prices in financial markets has been studied intensively both experimentally (data analysis) and theoretically (models). Nevertheless, a complete stochastic characterization of volatility is still lacking. What it is well known is that absolute returns have memory on a long time range, this phenomenon i…
Develops a GMM method to estimate roughness in stochastic volatility models.
problem Estimating roughness in stochastic volatility models with fractional Brownian motion.
method GMM approach for log-normal models with integrated variance and noisy realized variance.
result Consistent and asymptotically normal parameter estimator with bias correction.
Study on financial crises duration and volatility in US markets.
problem Duration of negative stock market returns and its impact on volatility.
method Survival models, log-normal distribution, continuous time analysis.
result Conditional probability of ending negative return spells increases up to 2-3 months after onset.
In this paper we study the possible microscopic origin of heavy-tailed probability density distributions for the price variation of financial instruments. We extend the standard log-normal process to include another random component in the so-called stochastic volatility models. We study these models under an assumptio…
Survey finds log-normal distribution fits better to market trends than stock prices.
problem Analyzing market-technical trend data for better trading systems.
method Statistical evaluation of market-technical trend variables.
result Log-normal assumption better fits market trend data than daily stock returns.
Develops formulas for pricing European quanto options in a local volatility FX-LIBOR model.
problem Pricing European quanto options in a local volatility FX-LIBOR model with skew/smile effects.
method Derives dynamics of foreign LIBOR rates, considers local volatility models, uses expansions around log-normal dynamics.
result Derives approximation formulas of Black-Scholes type with accurate error estimation.
Entropy corrections improve GBM's predictive accuracy for non-log-normal distributions.
problem Log-normal distribution limitations in GBM predictions.
method Entropy corrections to geometric Brownian motion (GBM).
result Improved predictive accuracy for non-log-normal distributions.
A new formula predicts stock prices using median instead of mean for skewed distributions.
problem Erroneous predictions from expected value in skewed stock price distributions.
method Uses geometric mean or median for log-normal distribution, especially for long-term outcomes.
result More realistic prediction for heavy-tailed distributions of stock price variations.
Alternative closed-form formula for spread call option prices under log-normal models.
problem Valuation of spread call options under log-normal models.
method Developed an alternative closed-form formula for spread call option prices.
result Our formula performs better for certain range of model parameters than existing closed-form formula.
Log-Normal Multiplicative Dynamics improves low-precision training of neural networks.
problem Training large neural networks with low precision is unstable.
method Derive a Bayesian learning rule with log-normal posterior distributions and multiplicative updates.
result LMD achieves stable and accurate training for Vision Transformer and GPT-2.
Paper derives option pricing formula for SVJ models with jumps.
problem Option pricing for models with finite activity jumps.
method Generic decomposition and explicit approximation formulas for SVJ models.
result Improved computation efficiency and insight into volatility smiles.
Study shows Merton model limits to Poisson process with log-normal intensity, improving default portfolio prediction.
problem Improving prediction of default portfolios using complex models.
method Applying Merton model with log-normal intensity function to Poisson process, discussing temporal correlation effects.
result Power decay model provides better generalization for long-term default portfolio data.
We study the effect of parameters uncertainties on a stochastic diffusion model, in particular the impact on the pricing of contingent claims, thanks to Dirichlet Forms methods. We apply recent techniques, developed by Bouleau, to hedging procedures in order to compute the sensitivities of SDE trajectories with respect…
Employing profits data of Japanese firms in 2003--2005, we kinematically exhibit the static log-normal distribution in the middle scale region. In the derivation, a Non-Gibrat's law under the detailed balance is adopted together with following two approximations. Firstly, the probability density function of profits gro…
Model non-stationary financial data using log-normal distributions and Langevin equations.
problem Modeling non-stationary volume-price distributions in finance.
method Model non-stationary volume-price distributions with a log-normal distribution. Derive Langevin equations from the series of log-normal parameters.
result Reconstructed statistics of volume-price distributions fit well empirical data.
We study the effect of parameter uncertainty on a stochastic diffusion model, in particular the impact on the pricing of contingent claims, using methods from the theory of Dirichlet forms. We apply these techniques to hedging procedures in order to compute the sensitivity of SDE trajectories with respect to parameter …
RL and DTSOC for final quadratic hedging performance studied.
problem Optimal hedging of European call options with and without transaction costs.
method Reinforcement Learning and Deep Trajectory-based Stochastic Optimal Control.
result RL and DTSOC perform similarly to variance-optimal hedging in various market models.
Gradually Truncated Log-normal distribution - Size distribution of firms Abstract Many natural and economical phenomena are described through power law or log- normal distributions. In these cases, probability decreases very slowly with step size compared to normal distribution. Thus it is essential to cut-off these di…
In the LIBOR market model, forward interest rates are log-normal under their respective forward measures. This note shows that their distributions under the other forward measures of the tenor structure have approximately log-normal tails.
New optimal portfolios derived for power and logarithmic utilities under log-normal returns.
problem Optimal portfolio weights for power and logarithmic utilities under log-normal returns.
method Closed-form expressions derived for optimal portfolio weights, proving mean-variance efficiency.
result Both optimal portfolios are mean-variance efficient and belong to the feasible set.
Two methods find typical sums of log-normal variates in GBM trajectories.
problem Finding typical sums of log-normal variates in GBM trajectories.
method Mapped to spin glasses and used Ito calculus.
result Qualitative and quantitative agreement between methods.
Study on Euler-discretized Hull-White model with volatility and asset price asymptotics.
problem Analyzing properties of the Hull-White model under time discretization.
method Discretization using Euler-Maruyama scheme, study of asymptotics in large time steps limit.
result Explicit expressions for growth rates of asset price moments, phase transition criteria.
Study on the geometric Dyson Brownian motion of non-square matrix products.
problem Understanding the spectrum of a product of non-square random matrices.
method Proportional depth-width limit followed by mean-field limit, solving Burgers equation.
result Free log-normal law is obtained in the identity-start case.
We introduce the formalism of generalized Fourier transforms in the context of risk management. We develop a general framework to efficiently compute the most popular risk measures, Value-at-Risk and Expected Shortfall (also known as Conditional Value-at-Risk). The only ingredient required by our approach is the knowle…
Paper improves basket option pricing for log-normal models.
problem Challenges in pricing basket options with negative weights.
method Moment matching and solving a unary cubic equation.
result Highly accurate closed form solution for basket options.
LNMC improves link prediction on social networks by considering log-normal degree distributions.
problem Link prediction in social networks with log-normal degree distributions.
method Log-Normal Matrix Completion (LNMC) using Alternating Direction Method of Multipliers.
result Up to 5% AUC increase over non-structured sparsity based methods.
Employing data on the assessed value of land in 1974--2007 Japan, we exhibit a quasistatically varying log-normal distribution in the middle scale region. In the derivation, a Non-Gibrat's law under the detailed quasi-balance is adopted together with two approximations. The resultant distribution is power-law with the …
The study examines the chaos of fractional Brownian fields as Hurst parameter approaches zero.
problem Understanding the chaos of fractional Brownian fields as their Hurst parameter tends to zero.
method Defining normalizing kernels and using Berestycki's ``good points'' approach to derive the limiting measure of multiplicative chaos.
result The limiting measure of multiplicative chaos converges to a log-correlated Gaussian field as the Hurst parameter approaches zero.
We study the statistical properties of volatility---a measure of how much the market is likely to fluctuate. We estimate the volatility by the local average of the absolute price changes. We analyze (a) the S&P 500 stock index for the 13-year period Jan 1984 to Dec 1996 and (b) the market capitalizations of the largest…