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In the last decade the subordinated processes have become popular and have found many practical applications. Therefore in this paper we examine two processes related to time-changed (subordinated) classical Brownian motion with drift (called arithmetic Brownian motion). The first one, so called normal tempered stable, is related to the tempered stable subordinator, while the second one–to the inverse tempered stable process. We compare the main properties (such as probability density functions, Laplace transforms, ensemble averaged mean squared displacements) of such two subordinated processes and propose the parameters’ estimation procedures. Moreover we calibrate the analyzed systems to real data related to indoor air quality.  相似文献   

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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.  相似文献   

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The classical financial models are based on the standard Brownian diffusion-type processes. However, in the exhibition of some real market data (like interest or exchange rates) we observe characteristic periods of constant values. Moreover, in the case of financial data, the assumption of normality is often unsatisfied. In such cases the popular Vasi?ek model, that is a mathematical system describing the evolution of interest rates based on the Ornstein–Uhlenbeck process, seems not to be applicable. Therefore, we propose an alternative approach based on a combination of the popular Ornstein–Uhlenbeck process with a stable distribution and subdiffusion systems that demonstrate such characteristic behavior. The probability density function of the proposed process can be described by a Fokker–Planck type equation and therefore it can be examined as an extension of the basic Ornstein–Uhlenbeck model. In this paper, we propose the parameters’ estimation method and calibrate the subordinated Vasi?ek model to the interest rate data.  相似文献   

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The earliest model of stock prices based on Brownian diffusion is the Bachelier model. In this paper we propose an extension of the Bachelier model, which reflects the subdiffusive nature of the underlying asset dynamics. The subdiffusive property is manifested by the random (infinitely divisible) periods of time, during which the asset price does not change. We introduce a subdiffusive arithmetic Brownian motion as a model of stock prices with such characteristics. The structure of this process agrees with two-stage scenario underlying the anomalous diffusion mechanism, in which trapping random events are superimposed on the Langevin dynamics. We find the corresponding fractional Fokker-Planck equation governing the probability density function of the introduced process. We construct the corresponding martingale measure and show that the model is incomplete. We derive the formulas for European put and call option prices. We describe explicit algorithms and present some Monte-Carlo simulations for the particular cases of α-stable and tempered α-stable distributions of waiting times.  相似文献   

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Statistical analysis of financial data mostly focused on testing the validity of Brownian motion (Bm). Analyses performed on several time series have shown deviation from the Bm hypothesis, that is at the base of the evaluation of many financial derivatives. We analyze the behavior of performance measures based on maximum drawdown movements (MDD(T)), testing their stability when the underlying process deviates from the Bm hypothesis. In particular we consider the fractional Brownian motion (fBm), and fluctuations estimated empirically on raw market data. The case study of the rising part of speculative bubbles is reported.  相似文献   

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The measure of long-term memory is important for the study of economic and financial time series. This paper estimates the Hurst exponent from a Scaled Variance Ratio model for 17 commodity price series under the efficient market null H0:H=0.5. The distribution about the estimates of H are obtained from 90%, 95% and 99% confidence intervals generated from 20,000 Monte Carlo replications of a geometric Brownian motion. The results show that the scaled variance ratio provides a very good and stable estimate of the Hurst exponent, but the estimates can be quite different from the measure obtained from rescaled range or RS analysis. In general commodity prices are consistent with the underlying assumption of a geometric Brownian motion.  相似文献   

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One century after Einstein's work, Brownian motion still remains both a fundamental open issue and a continuous source of inspiration for many areas of natural sciences. We first present a discussion about stochastic and deterministic approaches proposed in the literature to model the Brownian motion and more general diffusive behaviors. Then, we focus on the problems concerning the determination of the microscopic nature of diffusion by means of data analysis. Finally, we discuss the general conditions required for the onset of large scale diffusive motion.  相似文献   

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In this paper we introduce two models of stochastic processes driven by Brownian motion and fractional Brownian motion subordinated with tempered α-stable waiting times. By using a new integro-differential operator we obtain the generalized Fokker-Planck type equations associated with these subordinated stochastic processes.  相似文献   

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