Positive Jumps Research Articles

Change Point Detection and Estimation of the Two-Sided Jumps of Asset Returns Using a Modified Kalman Filter

In the first part of the paper, the positive and negative jumps of NASDAQ daily (log-) returns and three of its stocks are estimated based on the methodology presented by Theodosiadou et al. 2016, where jumps are assumed to be hidden random variables. For that reason, the use of stochastic state space models in discrete time is adopted. The daily return is expressed as the difference between the two-sided jumps under noise inclusion, and the recursive Kalman filter algorithm is used in order to estimate them. Since the estimated jumps have to be non-negative, the associated pdf truncation method, according to the non-negativity constraints, is applied. In order to overcome the resulting underestimation of the empirical time series, a scaling procedure follows the stage of truncation. In the second part of the paper, a nonparametric change point analysis concerning the (variance–) covariance is applied to the NASDAQ return time series, as well as to the estimated bivariate jump time series derived after the scaling procedure and to each jump component separately. A similar change point analysis is applied to the three other stocks of the NASDAQ index.

On magnitude, asymptotics and duration of drawdowns for Lévy models

This paper considers magnitude, asymptotics and duration of drawdowns for some L\'{e}vy processes. First, we revisit some existing results on the magnitude of drawdowns for spectrally negative L\'{e}vy processes using an approximation approach. For any spectrally negative L\'{e}vy process whose scale functions are well-behaved at $0+$, we then study the asymptotics of drawdown quantities when the threshold of drawdown magnitude approaches zero. We also show that such asymptotics is robust to perturbations of additional positive compound Poisson jumps. Finally, thanks to the asymptotic results and some recent works on the running maximum of L\'{e}vy processes, we derive the law of duration of drawdowns for a large class of L\'{e}vy processes (with a general spectrally negative part plus a positive compound Poisson structure). The duration of drawdowns is also known as the "Time to Recover" (TTR) the historical maximum, which is a widely used performance measure in the fund management industry. We find that the law of duration of drawdowns qualitatively depends on the path type of the spectrally negative component of the underlying L\'{e}vy process.

On the free-precession candidate PSR B1828-11: Evidence for increasing deformation

We observe that the periodic variations in spin-down rate and beam-width of the radio pulsar PSR B1828-11 are getting faster. In the context of a free precession model, this corresponds to a decrease in the precession period $P_{\mathrm{fp}}$. We investigate how a precession model can account for such a decrease in $P_{\mathrm{fp}}$, in terms of an increase over time in the absolute biaxial deformation ($|\epsilon_{\mathrm{p}}|{\sim}10^{-8}$) of this pulsar. We perform a Bayesian model comparison against the 'base' precession model (with constant $\epsilon_{\mathrm{p}}$) developed in Ashton et al (2016), and we obtain decisive odds in favour of a time-varying deformation. We study two types of time-variation: (i) a linear drift with a posterior estimate of $\dot{\epsilon}_{\mathrm{p}}{\sim}10^{-18}\,\mathrm{s}^{-1}$ and odds of $10^{75}$ compared to the base-model, and (ii) $N$ discrete positive jumps in $\epsilon_{\mathrm{p}}$ with very similar odds to the linear $\epsilon_{\mathrm{p}}$-drift model. The physical mechanism explaining this behaviour is unclear, but the observation could provide a crucial probe of the interior physics of neutron stars. We also place an upper bound on the rate at which the precessional motion is damped, and translate this into a bound on a dissipative mutual friction-type coupling between the star's crust and core.

Growth-fragmentation processes and bifurcators

Markovian growth-fragmentation processes introduced by Bertoin model a system of growing and splitting cells in which the size of a typical cell evolves as a Markov process $X$ without positive jumps. We find that two growth-fragmentations associated respectively with two processes $X$ and $Y$ (with different laws) may have the same distribution, if $(X,Y)$ is a bifurcator, roughly speaking, which means that they coincide up to a bifurcation time and then evolve independently. Using this criterion, we deduce that the law of a self-similar growth-fragmentation is determined by its index of self-similarity and a cumulant function $\kappa $.

Splitting and time reversal for Markov additive processes

We consider a Markov additive process with a finite phase space and study its path decompositions at the times of extrema, first passage and last exit. For these three families of times we establish splitting conditional on the phase, and provide various relations between the laws of post- and pre-splitting processes using time reversal. These results offer valuable insight into the behaviour of the process, and while being structurally similar to the Lévy process case, they demonstrate various new features. As an application we formulate the Wiener–Hopf factorization, where time is counted in each phase separately and killing of the process is phase dependent. Restricting to the case of no positive jumps, we find concise formulas for these factors, and also characterize the time of last exit from the negative half-line. The latter result is obtained using three quite different approaches based on the established path decomposition theory, which further demonstrates its applicability.

Liquidity dynamics around intraday price jumps in Chinese stock market

Using 4128 single jumps detected from high frequency data of 220 individual stocks in SZ300P index, this paper investigates the liquidity dynamics around price jumps in Chinese market. Some interesting empirical results are obtained and the corresponding explanations are given. The frequency of positive jumps is quite higher than that of negative jumps. The trading volumes and average trade sizes are all in a high level around positive jumps. The relatively low liquidities around negative jumps show that negative jumps may be generated and enlarged by poor liquidity provision. The price reversal after price jumps is significant, and price reversal lasts longer after positive jumps. Moreover, the size and direction of jumps are significantly correlated with the returns and trades in the post-jump trading time. These findings are believed to be associated with the high proportion of retail investors and their herding behavior for price trend chasing.

The effect of investor sentiment on gold market return dynamics: Evidence from a nonparametric causality-in-quantiles approach

This paper explores the effect of investor sentiment on the intraday return dynamics in the gold market. We build on the recent evidence by Da et al. (2015) that the Financial and Economic Attitudes Revealed by Search (FEARS) index, as a proxy for investor sentiment, has predictive power over stock market returns and extend the analysis to gold intraday returns using a novel methodology developed by Balcilar et al. (2016) to examine nonlinear casual effects of sentiment on gold return and volatility. We find that the effect of investor sentiment is more prevalent on intraday volatility in the gold market, rather than daily returns. The sentiment effect, however, is channeled via the discontinuous (jump) component of intraday volatility, particularly at extreme quantiles, implying that extreme fear (confidence) contributes to positive (negative) volatility jumps in gold returns. The results suggest that measures of sentiment could be utilized to model volatility jumps in safe haven assets that are often hard to predict and have significant implications for risk management as well as the pricing of options.

Distributional representations and dominance of a Lévy process over its maximal jump processes

Distributional identities for a Lévy process $X_{t}$, its quadratic variation process $V_{t}$ and its maximal jump processes, are derived, and used to make “small time” (as $t\downarrow0$) asymptotic comparisons between them. The representations are constructed using properties of the underlying Poisson point process of the jumps of $X$. Apart from providing insight into the connections between $X$, $V$, and their maximal jump processes, they enable investigation of a great variety of limiting behaviours. As an application, we study “self-normalised” versions of $X_{t}$, that is, $X_{t}$ after division by $\sup_{0<s\le t}\Delta X_{s}$, or by $\sup_{0<s\le t}\vert \Delta X_{s}\vert $. Thus, we obtain necessary and sufficient conditions for $X_{t}/\sup_{0<s\le t}\Delta X_{s}$ and $X_{t}/\sup_{0<s\le t}\vert \Delta X_{s}\vert $ to converge in probability to 1, or to $\infty$, as $t\downarrow0$, so that $X$ is either comparable to, or dominates, its largest jump. The former situation tends to occur when the singularity at 0 of the Lévy measure of $X$ is fairly mild (its tail is slowly varying at 0), while the latter situation is related to the relative stability or attraction to normality of $X$ at 0 (a steeper singularity at 0). An important component in the analyses is the way the largest positive and negative jumps interact with each other. Analogous “large time” (as $t\to\infty$) versions of the results can also be obtained.

Sensitivity analysis of market and stock returns by considering positive and negative jumps

Purpose This paper aims to present the results of further investigating the Polimenis (2012) stochastic model, which aims to decompose the stock return evolution into positive and negative jumps, and a Brownian noise (white noise), by taking into account different noise levels. This paper provides a sensitivity analysis of the model (through the analysis of its parameters) and applies this analysis to Google and Yahoo returns during the periods 2006-2008 and 2008-2010, by means of the third central moment of Nasdaq index. Moreover, the paper studies the behavior of the calibrated jump sensitivities of a single stock as market skew changes. Finally, simulations are provided for the estimation of the jump betas coefficients, assuming that the jumps follow Gamma distributions. Design/methodology/approach In the present paper, the model proposed in Polimenis (2012) is considered and further investigated. The sensitivity of the parameters for the Google and Yahoo stock during 2006-2008 estimated by means of the third (central) moment of Nasdaq index is examined, and consequently, the calibration of the model to the returns is studied. The associated robustness is examined also for the period 2008-2010. A similar sensitivity analysis has been studied in Polimenis and Papantonis (2014), but unlike the latter reference, where the analysis is done while market skew is kept constant with an emphasis in jointly estimating jump sensitivities for many stocks, here, the authors study the behavior of the calibrated jump sensitivities of a single stock as market skew changes. Finally, simulations are taken place for the estimation of the jump betas coefficients, assuming that the jumps follow Gamma distributions. Findings A sensitivity analysis of the model proposed in Polimenis (2012) is illustrated above. In Section 2, the paper ascertains the sensitivity of the calibrated parameters related to Google and Yahoo returns, as it varies the third (central) market moment. The authors demonstrate the limits of the third moment of the stock and its mixed third moment with the market so as to get real solutions from (S1). In addition, the authors conclude that (S1) cannot have real solutions in the case where the stock return time series appears to have highly positive third moment, while the third moment of the market is significantly negative. Generally, the positive value of the third moment of the stock combined with the negative value of the third moment of the market can only be explained by assuming an adequate degree of asymmetry of the values of the beta coefficients. In such situations, the model may be expanded to include a correction for idiosyncratic third moment in the fourth equation of (S1). Finally, in Section 4, it is noticed that the distribution of the error estimation of the coefficients cannot be considered to be normal, and the variance of these errors increases as the variance of the noise increases. Originality/value As mentioned in the Findings, the paper demonstrates the limits of the third moment of the stock and its mixed third moment with the market so as to get real solutions from the main system of equations (S1). It is concluded that (S1) cannot have real solutions when the stock return time series appears to have highly positive third moment, while the third moment of the market is significantly negative. Generally, the positive value of the third moment of the stock combined with the negative value of the third moment of the market can only be explained by assuming an adequate degree of asymmetry of the values of the beta coefficients. In such situations, the model proposed should be expanded to include a correction for idiosyncratic third moment in the fourth equation of (S1). Finally, it is noticed that the distribution of the error estimation of the coefficients cannot be considered to be normal, and the variance of these errors increases as the variance of the noise increases.

Sparre Andersen identity and the last passage time

AbstractIt is shown that the celebrated result of Sparre Andersen for random walks and Lévy processes has intriguing consequences when the last time of the process in (-∞, 0], say σ, is added to the picture. In the case of no positive jumps this leads to six random times, all of which have the same distribution—the uniform distribution on [0, σ]. Surprisingly, this result does not appear in the literature, even though it is based on some classical observations concerning exchangeable increments.

Stabilization for Positive Markovian Jump Systems with Actuator Saturation

This paper considers stabilization for positive Markovian jump systems in the presence of actuator saturation. Firstly, we use a Lyapunov function approach and convex analysis to derive sufficient conditions for stochastic stability and positivity of the continuous- and discrete-time cases. Then, the state feedback controller design and the estimate of the domain of attraction are presented by solving a convex optimization problem with linear matrix inequalities. Finally, numerical examples are given to demonstrate the validity of the main results.

Explaining credit default swap spreads by means of realized jumps and volatilities in the energy market

This paper studies the relationship between credit default swap (CDS) spreads for the Energy sector and oil futures dynamics. Using data on light sweet crude oil futures from 2004 to 2013, which contains a crisis period, we examine the importance of volatility and jumps extracted from the futures in explaining CDS spread changes. The analysis is performed at an index level and by rating group; as well as for the pre-crisis, crisis and post-crisis periods. Our findings are consistent with Merton's theoretical framework. At an index level, futures jumps are important when explaining CDS spread changes, with negative jumps having higher impact during the crisis. The continuous volatility part is significant and positive, indicating that futures volatility conveys relevant information for the CDS market. As for the analysis per rating group, negative jumps have an increasing importance as the credit rating deteriorates and during the crisis period, while the results for positive jumps and futures volatility are mixed. Overall, the relation between the CDS market and the futures market is stronger during volatile periods and strengthened after the Global Financial Crisis.

Stochastic Schrödinger evolution over piecewise enlarged filtrations

This paper constructs a nonlinear filtering framework that admits appearances of new information processes at random times by introducing piecewise enlargements of filtrations and proposes a new energy-based Schrodinger evolution expressed as a stochastic differential equation on a complex Hilbert space. Each information process is modeled as the sum of a random variable taking the eigenvalues of a Hamiltonian and an independent Brownian bridge noise. It is shown that under a piecewise enlarged filtration, the wave function is a jump-diffusion process until it collapses at some terminal time. In between discontinuities, the dynamics of the state vector are governed by different Wiener processes and diffusion coefficients. This motivates the introduction of an inclusive chain of Kolmogorov probability spaces or a *-isomorphic chain of commutative von Neumann probability spaces, on which the quantum system evolves differently based on the number of active information processes. The expectation of the Hamiltonian at a given state is the solution of a second-order nonlinear differential equation determined by one of the possible regimes that the quantum system belongs to. It is shown that the collapse rate is a submartingale with positive jumps and the Shannon entropy process is a supermartingale with expected negative jumps when passing to higher-order probability spaces. The framework is extended to the case when the Hamiltonian is modeled as a function of a set of commutative operators, where each operator is associated with a different piecewise enlarged filtration.

Volatility Forecasting: Downside Risk, Jumps and Leverage Effect

We provide empirical evidence of volatility forecasting in relation to asymmetries present in the dynamics of both return and volatility processes. Using recently-developed methodologies to detect jumps from high frequency price data, we estimate the size of positive and negative jumps and propose a methodology to estimate the size of jumps in the quadratic variation. The leverage effect is separated into continuous and discontinuous effects, and past volatility is separated into “good” and “bad”, as well as into continuous and discontinuous risks. Using a long history of the S &amp; P500 price index, we find that the continuous leverage effect lasts about one week, while the discontinuous leverage effect disappears after one day. “Good” and “bad” continuous risks both characterize the volatility persistence, while “bad” jump risk is much more informative than “good” jump risk in forecasting future volatility. The volatility forecasting model proposed is able to capture many empirical stylized facts while still remaining parsimonious in terms of the number of parameters to be estimated.

Tightness and Convergence of Trimmed Lévy Processes to Normality at Small Times

Let $^{(r,s)}X_t$ be the L\'evy process $X_t$ with the $r$ largest positive jumps and $s$ smallest negative jumps up till time $t$ deleted and let $^{(r)}\widetilde X_t$ be $X_t$ with the $r$ largest jumps in modulus up till time $t$ deleted. Let $a_t \in \mathbb{R}$ and $b_t>0$ be non-stochastic functions in $t$. We show that the tightness of $({}^{(r,s)}X_t - a_t)/b_t$ or $({}^{(r)}\widetilde X_t - a_t)/b_t$ at $0$ implies the tightness of all normed ordered jumps, hence the tightness of the untrimmed process $(X_t -a_t)/b_t$ at $0$. We use this to deduce that the trimmed process $({}^{(r,s)}X_t - a_t)/b_t$ or $({}^{(r)}\widetilde X_t - a_t)/b_t$ converges to $N(0,1)$ or to a degenerate distribution if and only if $(X_t-a_t)/b_t $ converges to $N(0,1)$ or to the same degenerate distribution, as $t \downarrow 0$.

The determinants of CDS open interest dynamics

It has been argued that the CDS market may be a threat to financial stability. Such concern may stem from the counterparty risk assumed by market participants and the high sensitivity of these instruments to the business cycle. The open interest of the CDS market mirrors investors’ maximum exposure and captures aggregate inventory risk, liquidity risk, and trading activity. In this paper, we aim to identify the main determinants of the dynamics of two alternative measures of open interest, the gross and net notional amounts. Our results suggest that both asymmetry of information and divergence of opinions on firms’ future performance help explain the growth of the net notional amount of single-reference contracts, but systematic factors have a much greater influence. Net notional amount growth of different obligors co-varies in time and the dynamics of open interest is pro-cyclical. The CDS market expands following a positive stock market performance and contracts when large negative (positive) jumps in stock (CDS) prices are perceived by investors. In line with the market microstructure theory, funding costs and counterparty risk reduce CDS market players’ willingness to incur inventory risk, thus contracting gross notional amounts.

Finite-Time $$L_1$$ L 1 Control for Positive Markovian Jump Systems with Partly Known Transition Rates

The paper deals with the problem of finite-time $$L_1$$L1 control for positive Markovian jump systems with partly known transition rates. Firstly, by constructing a linear co-positive Lyapunov function, sufficient conditions for finite-time boundedness and $$L_1$$L1 finite-time boundedness of the open-loop system are developed. Then, an effective method is proposed for the construction of the state feedback controller. These sufficient criteria are derived in the form of linear programming. A key point of this paper is to extend the special requirement of completely known transition rates to more general form that covers completely known and completely unknown transition rates as two special cases. Finally, two examples are given, which include a mathematical model of virus mutation treatment to illustrate the validity of the obtained results.

The Role of News in Commodity Markets

In this paper, we give a broad overview of how commodity-related news affects several aspects of commodity markets. We examine the main commodity classes: energy, agriculturals and metals, as well as market responses to news: in terms of prices, returns, volatilities and fine features of prices, such as price jumps. Market responses are analysed for different latencies, ranging from minutes to days and longer horizons. We discuss how these insights can be used in trading strategies, investment decisions and risk management.In particular, we address the following questions: • What are the distinguishing features of commodity-related news?• How commodity prices react to positive and negative sentiment in news?• How we can combine news signals from several commodity markets into an overall commodity news index? What are the relationships of such a news index with well-known commodity price indices? • Can we improve volatility forecasts by including news variables in volatility models?• Which characteristics of commodity price movements – volatility, positive and negative jumps – cause and are caused by news?

Loss rates in the single-server queue with complete rejection

Consider the single-server queue in which customers are rejected if their total sojourn time would exceed a certain level $$K$$ . A basic performance measure of this system is the probability $$P_K$$ that a customer gets rejected in steady state. This paper presents asymptotic expansions for $$P_K$$ as $$K\rightarrow \infty $$ . If the service time $$B$$ is light-tailed and inter-arrival times are exponential, it is shown that the loss probability has an exponential tail. The proof of this result heavily relies on results on the two-sided exit problem for Lévy processes with no positive jumps. For heavy-tailed (subexponential) service times and generally distributed inter-arrival times, the loss probability is shown to be asymptotically equivalent to the trivial lower bound $$P(B>K)$$ .

Option pricing for an uncertain stock model with jumps

By means of uncertain differential equation, an uncertain stock model usually describes the evolution of the stock price which highly depends on human uncertainty. Considering the sudden drifts on the stock price which might be caused by war, policy or technology, this paper proposes an uncertain stock model with both positive jumps and negative jumps in form of uncertain differential equation with jumps. European option pricing formulas for the proposed stock model are derived, and its monotonicity with respect to the parameters such as initial price, expiration date, and strike price is also studied.