Evidence of Markov properties of high frequency exchange rate data

Renner, Christoph; Peinke, Joachim; Friedrich, R.

doi:10.1016/s0378-4371(01)00269-2

Cited by 82 publications

(53 citation statements)

References 26 publications

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“…Without loss of generality we take τ i < τ i+1 . It should be noted that the Markov property can be tested for a given data set [7,12,13]. In this case the joint probability density can be substantially simplified:…”

Section: Methodsmentioning

confidence: 99%

Multiscale reconstruction of time series

Nawroth

Peinke

2006

Physics Letters A

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A new method is proposed which allows a reconstruction of time series based on higher order multiscale statistics given by a hierarchical process. This method is able to model the time series not only on a specific scale but for a range of scales. It is possible to generate complete new time series, or to model the next steps for a given sequence of data. The method itself is based on the joint probability density which can be extracted directly from given data, thus no estimation of parameters is necessary. The results of this approach are shown for a real world dataset, namely for turbulence. The unconditional and conditional probability densities of the original and reconstructed time series are compared and the ability to reproduce both is demonstrated. Therefore in the case of Markov properties the method proposed here is able to generate artificial time series with correct n-point statistics.

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Section: Methodsmentioning

confidence: 99%

Multiscale reconstruction of time series

Nawroth

Peinke

2006

Physics Letters A

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“…Our analysis is motivated by part 3 above and by the possibility that drift and diffusion coefficients for the stochastic equation below might be extracted from empirical data, but here we infer the diffusion coefficient from the empirical distribution of part 3 combined with the standard requirement that average volatility should show Brownian-like behavior. So far, no one has exhibited drift and diffusion coefficients empirically for returns although interesting results have been obtained for small price differences [20,21]. For the delta hedge we do not need the drift coefficient, only the diffusion term.…”

Section: Dynamics Of Volatility Of Returns and Option Pricingmentioning

confidence: 99%

An empirical model of volatility of returns and option pricing

McCauley

Gunaratne

2003

Physica A: Statistical Mechanics and its Applications

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In a seminal paper in 1973, Black and Scholes argued how expected distributions of stock prices can be used to price options. Their model assumed a directed random motion for the returns and consequently a lognormal distribution of asset prices after a finite time. We point out two problems with their formulation. First, we show that the option valuation is not uniquely determined; in particular ,strategies based on the delta-hedge and CAPM (the Capital Asset Pricing Model) are shown to provide different valuations of an option. Second, asset returns are known not to be Gaussian distributed. Empirically, distributions of returns are seen to be much better approximated by an exponential distribution. This exponential distribution of asset prices can be used to develop a new pricing model for options that is shown to provide valuations that agree very well with those used by traders. We show how the Fokker-Planck formulation of fluctuations (i.e., the dynamics of the distribution) can be modified to provide an exponential distribution for returns. We also show how a singular volatility can be used to go smoothly from exponential to Gaussian returns and thereby illustrate why exponential returns cannot be reached perturbatively starting from Gaussian ones, and explain how the theory of 'stochastic volatility' can be obtained from our model by making a bad approximation. Finally, we show how to calculate put and call prices for a stretched exponential density.

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“…Another case of interest is the analysis of rough surfaces and interfaces (Jafari et al 2003;Waechter et al 2004), where the characterization of the roughness is of paramount importance for understanding the physical and chemical properties of these surfaces. Another example is the investigation of the complex time variation of the market (Renner et al 2001b).…”

Section: Introductionmentioning

confidence: 99%

Markov properties of solar granulation

Ramos¹

2008

A&A

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Aims. We estimate the minimum length on which solar granulation can be considered to be a Markovian process. Methods. We measure the variation in the bright difference between two pixels in images of the solar granulation for different distances between the pixels. This scale-dependent data is empirically analyzed to find the minimum scale on which the process can be considered Markovian.Results. The results suggest that the solar granulation can be considered to be a Markovian process on scales longer than r M = 300−500 km. On longer length scales, solar images can be considered to be a Markovian stochastic process that consists of structures of size r M . Smaller structures exhibit correlations on many scales simultaneously yet cannot be described by a hierarchical cascade in scales. An analysis of the longitudinal magnetic-flux density indicates that it cannot be a Markov process on any scale. Conclusions. The results presented in this paper constitute a stringent test for the realism of numerical magneto-hydrodynamical simulations of solar magneto-convection. In future exhaustive analyse, the non-Markovian properties of the magnetic flux density on all analyzed scales might help us to understand the physical mechanism generating the field that we detect in the solar surface.

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Evidence of Markov properties of high frequency exchange rate data

Cited by 82 publications

References 26 publications

Multiscale reconstruction of time series

Multiscale reconstruction of time series

An empirical model of volatility of returns and option pricing

Markov properties of solar granulation

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