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Multifractality. Theory and Evidence An Application to the Romanian Stock Market. MSc Student: Cristina-Camelia Paduraru Supervisor: PhD Professor Moisa Altar. Presentation contents. Motivation Review of the Literature Basics of Multifractal Modeling Methodology to Detect Multifractality
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Multifractality. Theory and EvidenceAn Application to the Romanian Stock Market MSc Student: Cristina-Camelia Paduraru Supervisor: PhD Professor Moisa Altar
Presentation contents • Motivation • Review of the Literature • Basics of Multifractal Modeling • Methodology to Detect Multifractality • Data • Main Results • Conclusions • Bibliography
Motivation • The major discrepancies between the Bachelier model and actual financial data: - long memory in the absolute values of returns - long tails relative to the Gaussian. • The Multifractal Model of Asset Returns (MMAR) – Mandelbrot, Calvet, Fisher (1997) – accounts for these empirical regularities of financial time series and adds scale consistency.
Literature Review • Mandelbrot, Calvet, Fisher (1997) – the MMAR is developed – the focus is on the scaling property: the moments of the returns scale as a power law of the time horizon. • Calvet, Fisher, Mandelbrot (1997) – the focus is on the local properties of the multifractal processes. • Fisher, Calvet, Mandelbrot (1997) – an empirical investigation of the MMAR – evidence of multifractality in Deutschemark/US Dollar currency exchange rates • Calvet and Fisher (2002, 2008) – simplified version of the MMAR.
MMAR incorporates: • fat (long) tails - Mandelbrot (1963), but the MMAR does not necessarily imply infinite variance. • long dependence - fractional Brownian motion (FBM), Mandelbrot and Van Ness (1968). MMAR displays long dependence in the absolute value of price increments, while price increments themselves can be uncorrelated. • the concept of trading time - Mandelbrot and Taylor (1967): explicit modeling of the relationship between unobserved natural time-scale of the returns process, and clock time.
Multifractal processes bridge the gap between Itô and Jump diffusions • Itô diffusions - increments that grow locally at the rate (dt)1/2 throughout their sample paths. • FBM - local growth rates of order (dt)H , where H invariant over time (the Hurst exponent). • Multifractals - a multiplicity of local growth rates for increments: (dt)α(t), where α(t) represents the Hölder exponent. • Jump diffusions have α(t) = 0.
Construction of the MMAR Consider the price of a financial asset P(t) on [0, T] and the log-price process: • 1. • 2. θ(t) - the cumulative distribution function of a multifractal measure μ defined on [0, T]. • 3. BH(t) and θ(t) are independent.
Under Conditions 1 – 3: • X(t) - multifractal process with stationary increments; the moments of returns scale as a power law of the frequency of observation: as t→0. • The scaling function τX(q) - concave - has intercept τX(0) = -1 Concavity of the scaling function => multifractality. Unifractal processes – linear scaling functions fully determined by H.
Hölder Exponent • Let g be a function defined on the neighborhood of a given date t. The number , is called the Hölder exponent of g at t. • Describes the local variability of the function at a point in time.
Multifractal Spectrum • Describes the distribution of local Hölder exponents in a multifractal process. • The multifractal spectrum f(α) is the Legendre transform of the scaling function τ(q). • Between the spectrum of the log-price process and the spectrum of the trading time we have:
Testing for Multifractality • log-price series • Partitioning [0, T] into integer N intervals of length Δt, we define the partition function: • X(t) – multifractal => the addends are identically distributed; the scaling law yields: , when the qth moment exists.
Taking logs: where . • We plot logSq(Δt) vs log(Δt) for various values of q and Δt. • Linearity of those functions => scaling. • OLS estimations of the partition functions => τX(q), the scaling function. • Of particular interest: the value of q where • This value of q identifies H:
The Scaling Function • Calvet, Fisher, Mandelbrot (1997) shows that the scaling function - is concave - has intercept τX(0) = -1 and • From the scaling function we estimate the multifractal spectrum through the Legendre transform:
The Data • High frequency data sets – all transaction prices with transaction time during the period Jan, 2007 – May, 2009 for four Romanian securities listed at the Bucharest Stock Exchange: SIF2, BRD, SNP and TEL. • We have 328,555 transactions for SIF2 179,617 transactions for SNP 178,562 transactions for BRD 68,289 transactions for TEL.
Main Results • The partition functions – approximately linear => scaling in the moments of returns • The scaling functions – concave => evidence for multifractality. • Of particular interest: the value of q where • This value of q identifies H: • All scaling functions have intercept -1. • Each of the scaling functions is asymptotically linear, with a slope approximately equal to αmin.The minimum α corresponds to the most irregular instants on the price path, and thus the riskiest events for investors.
Main Results • The multifractal spectrum is also concave and its maximum is approximately 1 in all of the four cases. • The estimated multifractal spectrum: approximately quadratic => the limit lognormal multifractal measure for modeling the trading time. • Calvet, Fisher, and Mandelbrot (1997)
Main Results • We find:
Conclusions • We found evidence of multifractal scaling in 4 Romanian securities prices. • Using a methodology based on scaling function and multifractal spectrum => we recovered the MMAR components. • The estimated multifractal spectrum: approximately quadratic. • We found slight persistence in the analyzed data. • The scaling property holds from 4 days to one year. No intraday scaling! => We can model our series with multifractal processes at large time scales.
Bibliography • Calvet, L.E., A.J. Fisher, and B.B. Mandelbrot (1997) “Large Deviations and the Distribution of Price Changes”, Cowles Foundation Discussion Paper No. 1165; Sauder School of Business Working Paper • Calvet, L.E. and A.J. Fisher (1999), “A Multifractal Model of Assets Returns”, New York University Working Paper No. FIN-99-072 • Calvet, L.E. and A.J. Fisher (2002), “Multifractality in Asset Returns: Theory and Evidence”, Review of Economics and Statistics 84, 381-406 • Calvet, L.E. and A.J. Fisher (2008), “Multifractal Volatility: Theory, Forecasting, and Pricing”, Elsevier • Fama, E.F. (1963), “Mandelbrot and the Stable Paretian Hypothesis”, Journal of Business 36, 420-429 • Fillol, J. (2003) "Multifractality: Theory and Evidence an Application to the French Stock Market", Economics Bulletin 3, 1−12 • Fisher, A.J., L.E. Calvet, and B.B. Mandelbrot (1997), “Multifractality of Deutschemark / US Dollar Exchange Rates”, Cowles Foundation Discussion Paper No. 1166; Sauder School of Business Working Paper • Mandelbrot, B.B. (1963), “The Variation of Certain Speculative Prices”, Journal of Business 36, 394-419 • Mandelbrot, B.B. (1967), “The Variation of the Prices of Cotton, Wheat, and Railroad Stocks, and of some Financial Rates”, Journal of Business 40, 393-413
Bibliography 2 • Mandelbrot, B.B., and H.M. Taylor (1967), “On the Distribution of Stock Price Differences”, Operations Research 15, 1057-1062 • Mandelbrot, B.B., and J.W. Van Ness (1968), “Fractional Brownian Motions, Fractional Noises and Applications”, SIAM (Society for Industrial and Applied Mathematics) Review 10, 422-437 • Mandelbrot , B.B. (1972), “Possible refinement of the lognormal hypothesis concerning the distribution of energy dissipation in intermittent turbulence”, Statistical Models and Turbulence 12, 333-351 • Mandelbrot, B.B., A.J. Fisher, and L.E. Calvet (1997), “A Multifractal Model of Asset Returns”, Cowles Foundation Discussion Paper No. 1164; Sauder School of Business Working Paper • Mandelbrot, B.B. (2001), “Scaling in Financial Prices: Tails and Dependence”, Quantitative Finance 1, 113-123 • Mandelbrot, B.B. (2001), “Scaling in Financial Prices: Multifractals and the Star Equation”, Quantitative Finance 1, 124-130 • Mandelbrot, B.B. (2001), “Scaling in Financial Prices: Cartoon Brownian Motions in Multifractal Time”, Quantitative Finance 1, 427-440 • Mandelbrot, B.B. (2001), “Scaling in Financial Prices: Multifractal Concentration”, Quantitative Finance 1, 641-649 • Mandelbrot, B.B., and Richard L. Hudson (2004), “The (mis) Behavior of Markets”, Basic Books