"Fractional Brownian motion has been suggested as a model for the movement of log share prices which would allow long–range dependence between returns on different days. While this is true, it also allows arbitrage opportunities, which we demonstrate both indirectly and by constructing such an arbitrage. Nonetheless, it is possible by looking at a process similar to the fractional Brownian motion to model long–range dependence of returns while avoiding arbitrage."
"Problems in economy and finance have attracted the interest of statistical physicists all over the world. Fundamental problems pertain to the existence or not of long-, medium- or=and short-range power-law correlations in various economic systems, to the presence of financial cycles and on economic considerations, including economic policy. A method like the detrended fluctuation analysis is recalled emphasizing its value in sorting out correlation ranges, thereby leading to predictability at short horizon. The (m; k)-Zipf method is presented for sorting out short-range correlations in the sign and amplitude of the fluctuations. A well-known financial analysis technique, the so-called moving average, is shown to raise questions to physicists about fractional Brownian motion properties. Among spectacular results, the possibility of crash predictions has been demonstrated through the log-periodicity of financial index oscillations."
"A fractal approach is used to analyze financial time series, applying different degrees of time resolution, and the results are interrelated. Some fractal properties of foreign exchange (FX) data are found. In particular, the mean size of the absolute values of price changes follows a “fractal” scaling law (a power law) as a function of the analysis time interval ranging from a few minutes up to a year. In an autocorrelation study of intra-day data, the absolute values of price changes are seen to behave like the fractional noise ofMandelbrot and Van Ness rather than those of a GARCH process.
Intra-day FX data exhibit strong seasonal and autoregressive heteroskedasticity. This can be modeled with the help of new time scales, one of which is termed intrinsic time. These time scales are successfully applied to a forecasting model with a “fractal” structure for FX as well as interbank interest rates, the latter presenting market structures similar to the Foreign Exchange.
The goal of this paper is to demonstrate how the analysis of high-frequency data and the finding of fractal properties lead to the hypothesis of a heterogeneous market where different market participants analyze past events and news with different time horizons. This hypothesis is further supported by the success of trading models with different dealing frequencies and risk profiles. Intrinsic time is proposed for modeling the frame of reference of each component of a heterogeneousmarket."
"This paper discusses the analysis of the self-similarity of high-frequency DEM-USD exchange rate records and the 30 main German stock price records. Distributional selfsimilarity is found in both cases and some of its consequences are discussed."
"In this paper, we estimate the four parameters of the Pareto Stable probability distribution for six agricultural futures. The behavior of these estimates for different time-scaled distributions is consistent with the conjecture that the stochastic processes generating these agricultural futures returns are characterized by a fractal structure. In particular, we empirically verify that the six futures returns satisfy the property of statistical self-similarity. Moreover, we analyze the same time series by using the so called rescaled range analysis. This analysis is able to detect both the fractal structure and the presence of long-term dependence within the observations. We estimate and test the Hurst exponent using two methods: the classical and modified rescaled analysis. Finally, using Mandelbrot's result on the existence of a link between the characteristic exponent of a stable distribution and the Hurst exponent, we find further empirical confirmation that the processes generating agricultural futures returns are fractal."
"Stochastic long memory is established as a feature of the currency forward premia, Euroyen deposit rates, and Euroyen term premium series. The martingale model cannot be rejected for the spot, forward, and stock price series."
"This paper has two main objectives. First, it develops the multifractal formalism in a context suitable for both, measures and functions, deterministic as well as random, thereby emphasizing an intuitive approach. Second, it carefully discusses several examples, such as the binomial cascades and self-similar processes with a special eye on the use of wavelets. Particular attention is given to a novel class of multifractal processes which combine the attractive features of cascades and self-similar processes. Statistical properties of estimators as well as modelling issues are addressed."
"The purpose of this paper is to develop a fractional white noise calculus and to apply this to markets modeled by (Wick–) Itô type of stochastic differential equations driven by fractional Brownian motion BH(t); 1/2 < H < 1.
We show that if we use an Itô type of stochastic integration with respect to BH(t) (as developed in Ref. 8), then the corresponding Itô fractional Black–Scholes market has no arbitrage, contrary to the situation when the pathwise integration is used. Moreover, we prove that our Itô fractional Black–Scholes market is complete and we compute explicitly the price and replicating portfolio of a European option in this market. The results are compared to the classical results based on standard Brownian motion B(t)."
"This paper studies a classical extension of the Black and Scholes model for option pricing, often known as the Hull and White model. Our specification is that the volatility process is assumed not only to be stochastic, but also to have long-memory features and properties. We study here the implications of this continuous-time long-memory model, both for the volatility process itself as well as for the global asset price process. We also compare our model with some discrete time approximations. Then the issue of option pricing is addressed by looking at theoretical formulas and properties of the implicit volatilities as well as statistical inference tractability. Lastly, we provide a few simulation experiments to illustrate our results."
"We present a new framework for fractional Brownian motion in which processes with all indices can be considered under the same probability measure. Our results extend recent contributions by Hu, Øksendal, Duncan, Pasik–Duncan, and others. As an application we develop option pricing in a fractional Black–Scholes market with a noise process driven by a sum of fractional Brownian motions with various Hurst indices."
"This paper investigates the multifractal model of asset returns (MMAR), a class of continuous-time processes that incorporate the thick tails and volatility persistence exhibited by many financial time series. The simplest version of the MMAR compounds a Brownian motion with a multifractal time-deformation. Prices follow a semi-martingale, which precludes arbitrage in a standard two-asset economy. Volatility has long memory, and the highest finite moments of returns can take any value greater than 2. The local variability of a sample path is highly heterogeneous and is usefully characterized by the local Hölder exponent at every instant. In contrast with earlier processes, this exponent takes a continuum of values in any time interval. The MMAR predicts that the moments of returns vary as a power law of the time horizon. We confirm this property for Deutsche mark/U.S. dollar exchange rates and several equity series. We develop an estimation procedure and infer a parsimonious generating mechanism for the exchange rate. In Monte Carlo simulations, the estimated multifractal process replicates the scaling properties of the data and compares favorably with some alternative specifications."
"This paper provides a bibliographical guide to researchers and trac engineers who are interested in self-similar trac modeling and analysis. It lists some of the most recent network trac studies and includes surveys and research papers in the areas of data analysis, statistical inference, mathematical modeling, queueing and performance analysis. It also contains references to other areas of applications (e.g., hydrology, economics, geophysics, biology and biophysics) where similar developments have taken place and where numerous results have been obtained that can often be directly applied in the network trac context. Heavy tailed distributions, their relation to self-similar modeling, and corresponding estimation techniques are also covered in this guide."
"A Meyer-Tanaka formula involving weighted local time is derived for fractional Brownian motion and geometric fractional Brownian motion. The formula is applied to the study of the stop-loss-start-gain (SLSG) portfolio in a fractional Black-Scholes market. As a consequence, we obtain a fractional version of the Carr-Jarrow decomposition of the European call and put option prices into their intrinsic and time values."
"This paper studies properties of tests for long memory for general fourth order stationary sequences. We propose a rescaled variance test based on V/S statistic which is shown to have a simpler asymptotic distribution and to achieve a somewhat better balance of size and power than Lo’s (Econometrica 59 (1991) 1279) modi6ed R/S test and the KPSS test of Kwiatkowski et al. (J. Econometrics 54 (1992) 159). We investigate theoretical performance of R/S, KPSS and V/S tests under short memory hypotheses and long memory alternatives, providing a Monte Carlo study and a brief empirical example. Assumptions of the same type are used in both short and long memory cases, covering all persistent dependence scenarios. We show that the results naturally apply and the assumptions are well adjusted to linear sequences (levels) and to squares of linear ARCH sequences (volatility)."
"This paper presents the multifractal model of asset returns asset returns (“MMAR”), based upon the pioneering research into multifractal measures by Mandelbrot (1972, 1974). The multifractal model incorporates two elements of Mandelbrot's past research that are now well-known in nance. First, the MMAR contains long-tails, as in Mandelbrot (1963), which focused on Levy-stable distributions. In contrast to Mandelbrot (1963), this model does not necessarily imply infinite variance. Second, the model contains long-dependence, the characteristic feature of fractional Brownian Motion (FBM), introduced by Mandelbrot and van Ness (1968). In contrast to FBM, the multifractal model displays long dependence in the absolute value of price increments, while price increments themselves can be uncorrelated. As such, the MMAR is an alternative to ARCH-type representations that have been the focus of empirical research on the distribution of prices for the past fifteen years. The distinguishing feature of the multifractal model is multiscaling of the return distribution's moments under time-rescalings. We define multiscaling, show how to generate processes with this property, and discuss how these processes differ from the standard processes of continuous-time nance. The multifractal model implies certain empirical regularities, which are investigated in a companion paper."
"A simple construction that will be referred to as an error duration model is shown to generate fractional integration and long memory. An error-duration representation also exists for many familiar ARMA models, making error duration an alternative to autoregression for explaining dynamic persistence in economic variables. The results lead to a straightforward procedure for simulating fractional integration and establish a connection between fractional integration and common notions of structural change. Two examples show how the error-duration model could account for fractional integration in aggregate employment and in asset price volatility."
"In this paper we perform multifractal analyses of five daily Foreign Exchange (FX) rates. These techniques are currently used in turbulence to characterize scaling and intermittency. We show the multifractal nature of FX returns, and estimate the three parameters in the universal multifactal framework, which characterize all small and medium intensity #uctuations, at all scales. For large #uctuations, we address the question of hyperbolic (fat) tails of the distributions which are characterized by a fourth parameter, the tail index. We studied both the prices #uctuations and the returns, "nding no systematic di!erence in the scaling exponents in the two cases.
We discuss and compare our results with several recent studies, and show how the additive models are not compatible with data: Brownian, fractional Brownian, Le/vy, Truncated L/evy and fractional LeH vy models. We analyse in this framework the ARCH(1), GARCH(1,1) and HARCH (7) models, and show that their structure functions scaling exponents are undistinguishable from that of Brownian motion, which means that these models do not adequately describe the scaling properties of the statistics of the data.
Our results indicate that there might exist a multiplicative ‘flux of financial information’, which conditions small-scale statistics to large-scale values, as an analogy with the energy flux in turbulence."
"The correlation function of a financial index of the New York stock exchange, the S&P 500, is analyzed at 1min intervals over the 13-year period, Jan 84 – Dec 96. We quantify the correlations of the absolute values of the index increment. We find that these correlations can be described by two different power laws with a crossover time t× 600min. Detrended fluctuation analysis gives exponents 1 = 0.66 and 2 = 0.93 for t < t× and t > t× respectively. Power spectrum analysis gives corresponding exponents 1 = 0.31 and 2 = 0.90 for f > f× and f < f× respectively."
"We prove a Donsker type approximation theorem for the fractional Brownian motion in the case $H>1/2.$ Using this approximation we construct an elementary market model that converges weakly to the fractional analogue of the Black-Scholes model. We show that there exist arbitrage opportunities in this model. One such opportunity is constructed explicitly."
"The scaling properties of financial prices raise many questions. To provide background - appropriately so in the first issue of a new journal! - this paper, part I (sections 1 to 3), is largely a survey of the present form of some material that is well known yet repeatedly rediscovered. It originated in the author's work during the 1960s. Part II follows as sections 4 to 6, but can to a large extent be read separately. It is more technical and includes important material on multifractals and the `star equation'; part of it appeared in 1974 but is little known or appreciated - for reasons that will be mentioned. Part II ends by showing the direct relevance to finance of a very recent improvement on the author's original (1974) theory of multifractals."
SIMONSEN, I., M.H. JENSEN and A. JOHANSEN, 2002. Optimal investment horizons. The European Physical Journal B-Condensed Matter. [Cited by 10] (2.64/year)