Jump Size Distribution Research Articles

Exact calculation of the mean first-passage time of continuous-time random walks by nonhomogeneous Wiener–Hopf integral equations

We study the mean first-passage time (MFPT) for asymmetric continuous-time random walks in continuous-space characterised by waiting-times with finite mean and by jump-sizes with both finite mean and finite variance. In the asymptotic limit, this well-controlled process is governed by an advection-diffusion equation and the MFPT results to be finite when the advecting velocity is in the direction of the boundary. We derive a nonhomogeneous Wiener–Hopf integral equation that allows for the exact calculation of the MFPT by avoiding asymptotic limits and it emerges to depend on the whole distribution of the jump-sizes and on the mean-value only of the waiting-times, thus it holds for general non-Markovian random walks. Through the case study of a quite general family of asymmetric distributions of the jump-sizes that is exponential towards the boundary and arbitrary in the opposite direction, we show that the MFPT is indeed independent of the jump-sizes distribution in the opposite direction to the boundary. Moreover, we show also that there exists a length-scale, which depends only on the features of the distribution of jumps in the direction of the boundary, such that for starting points near the boundary the MFPT depends on the specific whole distribution of jump-sizes, in opposition to the universality emerging for starting points far-away from the boundary.

LAMN property for jump diffusion processes with discrete observations on a fixed time interval

We consider a one-dimensional stochastic differential equation with jumps driven by a Brownian motion and an independent Lévy process with finite Lévy measure, whose drift and diffusion coefficients depend on an unknown parameter. Under smoothness and non-degeneracy assumptions on the drift and diffusion coefficients and integrability assumption of jump size distribution, we prove the Local Asymptotic Mixed Normality property when the solution process is observed discretely at high frequency on a fixed time interval. The proof is essentially based on Malliavin calculus techniques and an analysis of the jump structure of the Lévy process.

Jump Dynamics and Leverage Effect: Evidences from Energy Exchange Traded Fund (ETFs)

Abstract This paper is concerned with the behavior of energy ETF prices. It applies three models: autoregressive moving average (ARMA) and generalized autoregressive conditional heteroskedasticity (GARCH), along with their revised forms, ARMA–Exponential-GARCH, Glosten-Jagannathan-Runkle (GJR), and GARCH diffusion process with jump models. This study looks at the volatility behavior and jumps dynamics of Energy and Master Limited Partnership's (MLP) ETFs. The results show that ARMA-GARCH is appropriate for modeling energy and MLP ETFs. Both ETFs offer positive leverage and asymmetric volatility. The results show that the jump model with a GARCH volatility specification has an actual amount of jump presence and time variation in the jump size distribution. The conclusion of the ARMA - EGARCH model gives evidence of the reverse leverage effect. The leverage term positively influences the conditional variance, while the asymmetry coefficient for the GJR model is positive and significant. These results reveal that both Energy and MLPs ETF have high volatility. JEL classification numbers: F3. Keywords: Energy ETFs, MLPs, ARMA-GARCH model, Volatility Asymmetry, Leverage and Jump Effect.

Large-scale behavior of a particle system with mean-field interaction: Traveling wave solutions

AbstractWe use probabilistic methods to study properties of mean-field models, which arise as large-scale limits of certain particle systems with mean-field interaction. The underlying particle system is such that n particles move forward on the real line. Specifically, each particle ‘jumps forward’ at some time points, with the instantaneous rate of jumps given by a decreasing function of the particle’s location quantile within the overall distribution of particle locations. A mean-field model describes the evolution of the particles’ distribution when n is large. It is essentially a solution to an integro-differential equation within a certain class. Our main results concern the existence and uniqueness of—and attraction to—mean-field models which are traveling waves, under general conditions on the jump-rate function and the jump-size distribution.

Shot-noise cojumps: Exact simulation and option pricing

We consider a parsimonious framework of jump-diffusion models for price dynamics with stochastic price volatilities and stochastic jump intensities in continuous time. They account for conditional heteroscedasticity and also incorporate key features appearing in financial time series of price volatilities and jump intensities, such as persistence of contemporaneous jumps (cojumps), mean reversion and feedback effects. More precisely, the stochastic variance and stochastic intensity are jointly modelled by a generalised bivariate shot-noise process sharing common jump arrivals with any non-negative jump-size distributions. This framework covers many classical and important models in the literature. The main contribution of this paper is that, we develop a very efficient scheme for its exact simulation based on perfect decomposition where neither numerical inversion nor acceptance/rejection scheme is required, which means that it is not only accurate but also the efficiency would not be sensitive to the parameter choice. Extensive numerical implementations and tests are reported to demonstrate the accuracy and effectiveness of this scheme. Our algorithm substantially outperforms the classical discretisation scheme. Moreover, we unbiasedly estimate the prices of discrete-barrier European options to show the applicability and flexibility of our algorithms.

Rotational dynamics and transition mechanisms of surface-adsorbed proteins

Assembly of biomolecules at solid–water interfaces requires molecules to traverse complex orientation-dependent energy landscapes through processes that are poorly understood, largely due to the dearth of in situ single-molecule measurements and statistical analyses of the rotational dynamics that define directional selection. Emerging capabilities in high-speed atomic force microscopy and machine learning have allowed us to directly determine the orientational energy landscape and observe and quantify the rotational dynamics for protein nanorods on the surface of muscovite mica under a variety of conditions. Comparisons with kinetic Monte Carlo simulations show that the transition rates between adjacent orientation-specific energetic minima can largely be understood through traditional models of in-plane Brownian rotation across a biased energy landscape, with resulting transition rates that are exponential in the energy barriers between states. However, transitions between more distant angular states are decoupled from barrier height, with jump-size distributions showing a power law decay that is characteristic of a nonclassical Levy-flight random walk, indicating that large jumps are enabled by alternative modes of motion via activated states. The findings provide insights into the dynamics of biomolecules at solid–liquid interfaces that lead to self-assembly, epitaxial matching, and other orientationally anisotropic outcomes and define a general procedure for exploring such dynamics with implications for hybrid biomolecular–inorganic materials design.

Self-exciting jump processes and their asymptotic behaviour

ABSTRACT The purpose of this paper is to investigate properties of self-exciting jump processes where the intensity is given by an SDE, which is driven by a finite variation stochastic jump process. The value of the intensity process immediately before a jump may influence the jump size distribution. We focus on properties of this intensity function, and show that for each fixed point in time, , a scaling limit of the intensity process converges in distribution, and the limit equals the strong solution of the square-root diffusion process (Cox–Ingersoll–Ross process) at t. As a particular example, we study the case of a linear intensity process and derive explicit expressions for the expectation and variance in this case.

Stochastic volatility: A tale of co-jumps, non-normality, GMM and high frequency data

In this article we introduce a linear–quadratic volatility model with co-jumps and show how to calibrate this model to a rich dataset. We apply GMM and more specifically match the moments of realized power and multi-power variations, which are obtained from high-frequency stock market data. Our model incorporates two salient features: the setting of simultaneous jumps in both return process and volatility process and the superposition structure of a continuous linear–quadratic volatility process and a Lévy-driven Ornstein–Uhlenbeck process. We compare the quality of fit for several models, and show that our model outperforms the conventional jump diffusion or Bates model. Besides that, we find evidence that the jump sizes are not normally distributed and that our model performs best when the distribution of jump-sizes is only specified through certain (co-) moment conditions. Monte Carlo experiments are employed to confirm this.

Adaptation to climate change: Extreme events versus gradual changes

Global climate change will lead to increased frequency and severity of extreme weather events, on top of the gradual changes in temperature and precipitation. We develop a real options model of adaptation to climate change, capturing the different effects of gradual changes, represented by a Brownian motion process, and extreme events, represented by Poisson jumps with a hyper-exponential jump size distribution. We compare adaptation decisions under the increased frequency, severity, and tail thickness of extreme events as well as gradual changes. We find that while the adaptation incentives are higher in response to gradual changes, the probability of carrying out adaptation activities is higher in response to extreme events. The catalyst effects of extreme events become more significant when the tail distribution of the extreme events becomes heavier.

News as sources of jumps in stock returns: Evidence from 21 million news articles for 9000 companies

Material news events can be potentially important sources of jumps in stock returns. We collect 21 million news articles associated with more than 9000 publicly-traded companies and use textual analyses to derive measures to summarize the news. We find that stock return jumps (including time-variation in jump-size distributions and jump intensity) are significantly related to news flow frequency and content and those effects increase substantially over the last few decades. The sensitivity of jump probability to news is stronger for firms with higher media visibility, analyst coverage, and institutional ownership. This sensitivity also varies across different news categories.

Effect of Grain Shape on Quasi‐Static Fluid‐Fluid Displacement in Porous Media

AbstractWe study how grain shapes impact multiphase flow in porous media in the quasi‐static regime using an extended pore‐network model. The algorithm allows the explicit determination of different types of pore‐scale instabilities and tracks the interface motion during the fluid‐fluid displacement process. It also includes the volume capacitance model, such that both the evolution of capillary pressure signal and sizes of Haines jumps can be captured. Further, it considers the pinning of menisci at sharp edges of grains, through which the distribution of effective contact angles can be obtained. Simulations are carried out across a wide range of wetting conditions for different particle shapes. Our results show that the effective contact angle distribution during displacement widens as the grain becomes more angular, which consequently modifies the macroscopic fluid invasion morphology. By analyzing various characteristic metrics during displacement, including capillary pressure signal, Haines jump size distribution, and fractal dimension, our results highlight the profound influence of particle shape on the multiphase flow.

Statistical arbitrage in jump-diffusion models with compound Poisson processes

We prove the existence of statistical arbitrage opportunities for jump-diffusion models of stock prices when the jump-size distribution is assumed to have finite moments. We show that to obtain statistical arbitrage, the risky asset holding must go to zero in time. Existence of statistical arbitrage is demonstrated via ‘buy-and-hold until barrier’ and ‘short until barrier’ strategies with both single and double barrier. In order to exploit statistical arbitrage opportunities, the investor needs to have a good approximation of the physical probability measure and the drift of the stochastic process for a given asset.

Hermite expansion of transition densities and European option prices for multivariate diffusions with jumps

This paper shows that a small-time Hermite expansion is feasible for multivariate diffusions. By introducing an innovative quasi-Lamperti transform, which unitizes the diffusion matrix at the initial time, we derive explicit recursive formulas for the expansion coefficients of transition densities and European option prices for multivariate diffusions with jumps in return. These immediately available explicit formulas, particularly for the irreducbile, nonaffine, time-inhomogeneous model with different types of jump-size distribution, is new to the literature. The explicit formulas can lead to real-time derivatives pricing and hedging as well as model calibration. Extensive numerical experiments illustrate the accuracy and effectiveness of our approach.

Risk-Free Investments and Their Comparison with Simple Risky Strategies in Pension Insurance Model: Solving Singular Problems for Integro-Differential Equations

A collective pension insurance (life annuity) model is investigated in the case of risk-free investments, i.e., when the whole surplus of an insurance company at each time is invested in risk-free asset (bank account). This strategy is compared with previously studied simple risky investment strategies, according to which, irrespective of the surplus of an insurance company, a constant positive fraction of this surplus at each time consists of risky assets (stocks), while the remaining fraction is invested in a bank account. The strategies are compared in terms of a traditional solvency criterion, namely, the survival probability. The original insurance model is dual to the classical Cramer–Lundberg model: the variation in the surplus over the portfolio of same-type contracts is described by the sum of a decreasing deterministic linear function corresponding to total pension payments and a compound Poisson process with positive jumps corresponding to the income gained by the company at the moments of transferring policyholders' property. In the case of an exponential jump size distribution and risk-free investments, it is shown that the survival probability regarded as a function of the initial surplus defined on the nonnegative real half-line is a solution of a singular problem for an integro-differential equation with a non-Volterra integral operator. The solution of the stated problem is obtained, its properties are analytically examined, and numerical examples are given. Examples are used to compare the influence exerted by risky and risk-free investments on the survival probability in the given model.

A splitting strategy for the calibration of jump-diffusion models

We present a detailed analysis and implementation of a splitting strategy to identify simultaneously the local volatility surface and the jump-size distribution from quoted European prices. The underlying model consists of a jump-diffusion driven asset with time- and price-dependent volatility. Our approach uses a forward Dupire-type partial integro-differential equation for the option prices to produce a parameter-to-solution map. The ill-posed inverse problem for this map is then solved by means of a Tikhonov-type convex regularisation. The proofs of convergence and stability of the algorithm are provided together with numerical examples that illustrate the robustness of the method both for synthetic and real data.

Approximate Hedging with Constant Proportional Transaction Costs in Financial Markets with Jumps

We study a problem of option replication under constant proportional transaction costs in models where stochastic volatility and jumps are combined to capture the market's important features. Assuming some mild condition on the jump size distribution, we show that transaction costs can be approximately compensated by applying the Leland adjusting volatility principle and the asymptotic property of the hedging error due to discrete readjustments. In particular, the jump risk can be approximately eliminated, and the results established in continuous diffusion models are recovered. The study also confirms that, for the case of constant trading cost rate, the approximate results established by Kabanov and Safarian [Finance Stoch., 1 (1997), pp. 239--250] and by Pergamenschikov [Ann. Appl. Probab., 13 (2003), pp. 1099--1118] are still valid in jump-diffusion models with deterministic volatility.

Evaluation of energy forward curves with jumps under the general Levy process

In this study, we evaluate the relationship between the forward rates and the future delivery period with the consideration of the Levy process for a time-inhomogeneous exponential jump-diffusion process and model the forward curve. This is a large variety of stylized features observed in the Samuelson effect of increasing volatilities close to maturity. However, a new method based on characteristic functions is used to estimate the jump component in a finite-activity Levy process, which includes the jump frequency and the jump size distribution which enables ´ the further investigation of the properties of estimators without the presence of high frequency data ∆. Numerical implementation of the approach was applied on sample electricity data of about 10,000 observations between the period of 2 years and then a seasonalized forecast for an extra year was implemented to normalize the volatility in forwards contracts.

Hermite Expansion for Transition Densities of Irreducible Diffusions with An Application to Option Pricing

This paper shows that a small-time Hermite expansion is feasible for multivariate diffusions. By introducing an innovative quasi-Lamperti transform, which unitizes the diffusion matrix at the initial time, we derive explicit recursive formulas for the expansion coefficients of transition densities and European option prices for multivariate diffusions with jumps in return. These immediately available explicit formulas, particularly for the irreducbile, nonaffine, time-inhomogeneous model with different types of jump-size distribution, is new to the literature. The explicit formulas can lead to real-time derivatives pricing and hedging as well as model calibration. Extensive numerical experiments illustrate the accuracy and effectiveness of our approach.

Efficient and flexible model-based clustering of jumps in diffusion processes

Jump–diffusion processes involving diffusion processes with discontinuous movements, called jumps, are widely used to model time-series data that commonly exhibit discontinuity in their sample paths. The existing jump–diffusion models have been recently extended to multivariate time-series data. The models are, however, still limited by a single parametric jump-size distribution that is common across different subjects. Such strong parametric assumptions for the shape and structure of a jump-size distribution may be too restrictive and unrealistic for multiple subjects with different characteristics. This paper thus proposes an efficient Bayesian nonparametric method to flexibly model a jump-size distribution while borrowing information across subjects in a clustering procedure using a nested Dirichlet process. For efficient posterior computation, a partially collapsed Gibbs sampler is devised to fit the proposed model. The proposed methodology is illustrated through a simulation study and an application to daily stock price data for companies in the S&P 100 index from June 2007 to June 2017.

News as Sources of Jumps in Stock Returns: Evidence From 21 Million News Articles for 9000 Companies

Material news events can be potentially important sources of jumps in stock returns. We collect 21 million news articles associated with more than 9,000 publicly-traded companies and use textual analyses to derive measures to summarize the news. We find that stock return jumps (including time-variation in jump-size distributions and jump intensity) are significantly related to news flow frequency and content and those effects increase substantially over the last few decades. The sensitivity of jump probability to news is stronger for firms with higher media visibility, analyst coverage, and institutional ownership. This sensitivity also varies across different news categories. The primary working dataset based on Factiva news can be downloaded using this link: https://www.dropbox.com/s/62lt6uq1t4n6gcr/MainData_Factiva_Public.zip?dl=0.