We explore the role of non-ergodicity in the relationship between income inequality, the extent of concentration in the income distribution, and income mobility, the feasibility of an individual to ...change their position in the income rankings. For this purpose, we use the properties of an established model for income growth that includes 'resetting' as a stabilizing force to ensure stationary dynamics. We find that the dynamics of inequality is regime-dependent: it may range from a strictly non-ergodic state where this phenomenon has an increasing trend, up to a stable regime where inequality is steady and the system efficiently mimics ergodicity. Mobility measures, conversely, are always stable over time, but suggest that economies become less mobile in non-ergodic regimes. By fitting the model to empirical data for the income share of the top earners in the USA, we provide evidence that the income dynamics in this country is consistently in a regime in which non-ergodicity characterizes inequality and immobility. Our results can serve as a simple rationale for the observed real-world income dynamics and as such aid in addressing non-ergodicity in various empirical settings across the globe. This article is part of the theme issue 'Kinetic exchange models of societies and economies'.
A CUCKER-SMALE MODEL WITH NOISE AND DELAY ERBAN, RADEK; HAŠKOVEC, JAN; SUN, YONGZHENG
SIAM journal on applied mathematics,
01/2016, Volume:
76, Issue:
4
Journal Article
Peer reviewed
Open access
A generalization of the Cucker-Smale model for collective animal behavior is investigated. The model is formulated as a system of delayed stochastic differential equations. It incorporates two ...additional processes which are present in animal decision making, but are often neglected in modeling: (i) stochasticity (imperfections) of individual behavior and (ii) delayed responses of individuals to signals in their environment. Sufficient conditions for flocking for the generalized Cucker-Smale model are derived by using a suitable Lyapunov functional. As a by-product, a new result regarding the asymptotic behavior of delayed geometric Brownian motion is obtained. In the second part of the paper, results of systematic numerical simulations are presented. They not only illustrate the analytical results, but hint at a somehow surprising behavior of the system—namely, that the introduction of an intermediate time delay may facilitate flocking.
Gold prices have been of major interest for a lot of investors, analysts, and economists. Accordingly, a number of different modeling approaches have been used to forecast gold prices. In this ...manuscript, the geometric Brownian motion approach, used in the pricing of numerous types of assets, is used to forecast the prices of gold at yearly, monthly, and quarterly frequencies. This approach allows for simulating one-period-ahead prices and the associated probabilities. The expected prices obtained from the simulated prices and probabilities are found to provide reliable forecasts when compared with the observed yearly, monthly, and quarterly prices.
Guaranteed Lifetime Withdrawal Benefits (GLWBs) are an increasingly popular add-on to Variable Annuities, offering a guaranteed stream of payments for the remainder of the policyholder's life. GLWBs ...have typically been priced using numerical methods such as finite difference schemes or Monte Carlo simulations; obtaining accurate and precise solutions using these methods can be very computationally expensive. In this paper, we extend an existing method for analytic pricing of these policies to a more general fee structure. We introduce a novel variation on the commonly offered ratchet rider that more directly addresses policyholder motivation around lapse-and-reentry behaviour. We then modify our pricing method to accommodate this new rider and compare it to the existing annual ratchet with respect to a policyholder's incentive to lapse such a policy.
Understanding the statistical dynamics of growth and inequality is a fundamental challenge to ecology and society. Recent analyses of wealth and income in contemporary societies show that economic ...inequality is very dynamic and that individuals experience substantially different wealth growth rates over time. However, despite a fast-growing body of evidence for the importance of fluctuations, we still lack a general statistical theory for understanding the dynamical effects of heterogeneous growth across a population. Here we derive the statistical dynamics of correlated wealth growth rates in heterogeneous populations. We show that correlations between growth rate fluctuations at the individual level influence aggregate population growth, while only driving inequality on short time scales. We also find that growth rate fluctuations are a much stronger driver of long-term inequality than income volatility. Our findings show that the dynamical effects of statistical fluctuations in growth rates are critical for understanding the emergence of inequality over time and motivate a greater focus on the properties and endogenous origins of growth rates in stochastic environments.
•Agent growth rates and initial resources correlations affect aggregate growth rates.•Negative correlated growth rate assignments reduce aggregate growth.•Volatility and growth rate, resource covariance dominate intermediate time dynamics.•Variances in growth rates across the population dominate long time inequality.•The effects of heterogeneity on inequality outpace effects on growth.
The main question we would like to address in this paper is as follows: Given a geometric Brownian motion (GBM) as the underlying stock price model, what is the cumulative distribution function (CDF) ...for the trading profit or loss, call it g(t), when an affine feedback control strategy with stop-loss order is considered? Moreover, is it possible to obtain a closed-form characterization for the desired CDF for g(t) so that a theoretician or practical trader might be benefited from it? The answers to these questions are affirmative. In this paper, we provide a closed-form expression for the cumulative distribution function for the trading profit or loss. In addition, we show that the affine feedback controller with stop-loss order indeed generalizes the result without stop order in the sense of distribution function. Some simulations and illustrative examples are also provided as supporting evidence of the theory.
Classical option pricing schemes assume that the value of a financial asset follows a geometric Brownian motion (GBM). However, a growing body of studies suggest that a simple GBM trajectory is not ...an adequate representation for asset dynamics, due to irregularities found when comparing its properties with empirical distributions. As a solution, we investigate a generalisation of GBM where the introduction of a memory kernel critically determines the behaviour of the stochastic process. We find the general expressions for the moments, log-moments, and the expectation of the periodic log returns, and then obtain the corresponding probability density functions using the subordination approach. Particularly, we consider subdiffusive GBM (sGBM), tempered sGBM, a mix of GBM and sGBM, and a mix of sGBMs. We utilise the resulting generalised GBM (gGBM) in order to examine the empirical performance of a selected group of kernels in the pricing of European call options. Our results indicate that the performance of a kernel ultimately depends on the maturity of the option and its moneyness.
We present an asymptotic result for the Laplace transform of the time integral of the geometric Brownian motion F(θ,T)=Ee−θXT with XT=∫0TeσWs+(a−12σ2)sds, which is exact in the limit σ2T→0 at fixed ...σ2θT2 and aT. This asymptotic result is applied to pricing zero coupon bonds in the Dothan model of stochastic interest rates. The asymptotic result provides an approximation for bond prices which is in good agreement with numerical evaluations in a wide range of model parameters. As a side result we obtain the asymptotics for Asian option prices in the Black-Scholes model, taking into account interest rates and dividend yield contributions in the σ2T→0 limit.
Stop-loss contracts are the most commonly used reinsurance agreements in insurance whose important factors are the retention and the maximum (cap) values attained on the random loss, which may occur ...within the policy period. Therefore, determining and forecasting the loss amounts is an important issue for both the insurer and the reinsurer. Along with many approaches in actuarial literature, we propose a geometric Brownian motion (BM) with the time-varying parameters to capture the time-dependent loss amounts. We implement the time-influence on stop-loss contract in the frame of the stochastic model and find the analytical derivations of costs associated with reinsurance contract for reinsurer and insurer with constraints on time, loss amount, retention, and both retention and cap levels. Additionally, the analytical forms of exposure curves are depicted to determine the premium share between reinsurer and insurer under time, loss, retention, and both retention and cap constraints. An application of the proposed methodology on real-life data and the calibration of time-varying parameters using dynamic maximum likelihood estimator and simulations on the proposed model are performed. Finally, we forecast the claim amounts, expected costs, and exposure curves on time-varying parameters using the cubic spline extrapolation and the dynamic ARIMA with trend search. It is shown that the time-varying approach using the stochastic model copes with the behavior of the claims and assures fair share between insurer and reinsurer.
•A stochastic process with time varying parameters on stop-loss contract is implemented.•Analytical derivations of cost for reinsurer and insurer with related constraints are made.•Analytical forms of premium share using exposure curve are introduced.•Application on MTPL data and the calibration by dynamic MLE and simulations are made.•Forecasting by cubic spline extrapolation and dynamic ARIMA with trend search is done.
The so-called Benford’s laws are of frequent use to detect anomalies and regularities in data sets, particularly in election results and financial statements. However, primary financial market ...indices have not been much studied, if studied at all, within such a perspective.
This paper presents features in the distributions of S&P500 daily closing values and the corresponding daily log-returns over a long time interval, 03/01/1950 - 22/08/2014, amounting to 16265 data points. We address the frequencies of the first, second, and first two significant digits and explore the conformance to Benford’s laws of these distributions at five different (equal size) levels of disaggregation. The log-returns are studied for either positive or negative cases. The results for the S&P500 daily closing values are showing a remarkable lack of conformity, whatever the different levels of disaggregation. The causes of this non-conformity are discussed, pointing to the danger in taking Benford’s laws for granted in large databases, whence drawing “definite conclusions”. The agreements with Benford’s laws are much better for the log-returns. Such a disparity in agreements finds an explanation in the data set itself: the index’s inherent trends. To further validate this, daily returns have been simulated via the Geometric Brownian Motion and calibrating the simulations with the observed data averages and testing against Benford’s laws when the log-returns distribution’s standard deviation changes. One finds that the trend and the standard deviation of the distributions are relevant parameters in concluding about conformity with Benford’s laws.
•On 1950-2014 distributions of S&P500 daily closing values and daily log-returns.•Benford’s laws fail for S&P500 when five equally sized time intervals are considered.•Large financial time series are not more likely to be Benford’s laws compliant.•The Geometric Brownian Motion generates or not Benford’s laws compliant time series.•A heuristic procedure to study Geometric Brownian Motion and Benford’s laws relation.
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