1. Multifractality. Theory and Evidence An Application to the Romanian Stock Market MSc Student: Cristina-Camelia Paduraru Supervisor: PhD Professor Moisa Altar

2. 2 Presentation contents Motivation Review of the Literature Basics of Multifractal Modeling Methodology to Detect Multifractality Data Main Results Conclusions Bibliography

3. 3 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.

4. 4 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.

5. 5 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.

6. 6 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.

7. 7 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. B H (t) and θ(t) are independent.

8. 8 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 τ X (q) - concave - has intercept τ X (0) = -1 Concavity of the scaling function => multifractality. Unifractal processes – linear scaling functions fully determined by H.

9. 9 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.

10. 10 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:

11. 11 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 q th moment exists.

12. 12 Taking logs: where. We plot logS q (Δt) vs log(Δt) for various values of q and Δt. Linearity of those functions => scaling. OLS estimations of the partition functions => τ X (q), the scaling function. Of particular interest: the value of q where This value of q identifies H:

13. 13 The Scaling Function Calvet, Fisher, Mandelbrot (1997) shows that the scaling function - is concave - has intercept τ X (0) = -1 and From the scaling function we estimate the multifractal spectrum through the Legendre transform:

14. 14 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.

15. 15 Plots of the Partition Functions SIF2

16. 16 Plots of the Partition Functions BRD

17. 17 Plots of the Partition Functions SNP

18. 18 Plots of the Partition Functions TEL

19. 19 Plots of the Scaling Functions

20. 20 Plots of the Scaling Functions

21. 21 Plots of the Scaling Functions

22. 22 Plots of the Scaling Functions

23. 23 Plots of the Multifractal Spectrum

24. 24 Plots of the Multifractal Spectrum

25. 25 Plots of the Multifractal Spectrum

26. 26 Plots of the Multifractal Spectrum

27. 27 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 α min. The minimum α corresponds to the most irregular instants on the price path, and thus the riskiest events for investors.

28. 28 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)

29. 29 Main Results We find: SIF2BRDSNPTEL 0.56 0.530.52 0.570.580.540.58 1.021.041.021.12 0.130.270.130.79

30. 30 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.

31. 31 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

32. 32 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