Abstract
This contribution surveys the main characteristics of two stochastic processes that generalize the fractional Brownian motion: the multifractional Brownian motion and the multifractional processes with random exponent. A special emphasis will be devoted to the meaning and to the applications that they can have in finance. If fractional Brownian motion is by now very well-known and studied as a model of the price dynamics, multifractional processes are yet widely unknown in the field of quantitative finance, mainly because of their nonstationarity. Nonetheless, in spite of their complex structure, such processes deserve consideration for their capability to seize the stylized facts that most of the current models cannot account for. In addition, their functional parameter provides an insightful and parsimonious interpretation of the market mechanism, and is able to unify in a single model two opposite approaches such as the theory of efficient markets and the behavioral finance.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
Remind that the random process \(Y_{t,\tau}\), \(\tau>t\), is a stochastic discount factor (or pricing kernel) if (a) \(\mathbb{P}(Y_{t,t}=1,Y_{t,\tau}>0)=1\), and (b) \(S_t=\mathbb{E}^\mathbb{P}_t[Y_{t,\tau}S_\tau]\).
- 2.
The definition of Hölder exponent will be given in paragraph 2; here, we just recall that it measures the degree of irregularity of the graph of a function.
- 3.
The distinction between these two ways of measuring the local regularity is not just a mathematical detail, since-as proved in (Ayache 2013)-while the trajectories of an arbitrary centered Gaussian process share the same global (uniform) as well as local Hölder regularity, the same property does not hold with respect to the pointwise Hölder regularity.
- 4.
We remind that the stochastic process \(\left\{X(t,\omega)\right\}_{t\in T}\) is said self-similar of parameter H if for any \(a>0\) and \(t\in T\), it is \(\{X(at,\,\omega )\}\overset{d}{\mathop{=}}\,\,\{{{a}^{H}}X(t,\,\omega )\}\), where the equality holds for the finite-dimensional distributions of \(\left\{X(t,\omega)\right\}\).
- 5.
- 6.
The fBm can be also defined in terms of random wavelet series as
$$\bar{B}(x,H) = \sum_{j=-\infty}^{+\infty} \sum_{k \in \mathbb{Z}} 2^{-jH} \epsilon_{j,k} \left(\Psi(2^jx-k,H) -\Psi(-k,H) \right)$$(10)where \(\left\{\epsilon_{j,k}\right\}_{(j,k)\in \mathbb{Z}^2}\) is a sequence of independent \(N(0,1)\) random variables; \(\Psi \in C^\infty(\mathbb{R}\times(0,1))\) is well-localized in the first variable and uniformly localized in H, which means that, for all \((n,p)\in \mathbb{N}^2\), \(\sup \left\{(1+|x|)^p|(\partial_x^{(n)}\Psi)(x,H)|:(x,H)\in \mathbb{R}\times(0,1)\right\}<+\infty.\) This representation-which allows to define the MPRE (see below) also when its random exponent depends on the Brownian measure-is almost surely uniformly convergent in \((x,H)\) on each compact subset of \(\mathbb{R}\times(0,1)\).
- 7.
The corrections mainly concerned the revision of classical definitions of arbitrage and self-financing condition (Hu and Oksendal 2003; Elliott and Van Der Hoek 2003); the regularization of the weighting kernel of the fBm (Rogers 1997; Cheridito 2001), the introduction of transaction costs (Guasoni 2006 or of delays in transaction times Cheridito 2003).
- 8.
Remind that the function h(t) is Hölderian of order β on each compact interval \(J \subset \mathbb{R}\) if, for each \(t,s \in J\) and for \(c>0\), it holds \(|h(t)-h(s)| \leq c|t-s|^\beta\), where \(\beta> \max_{t \in J}h(t)\).
- 9.
Indeed, (Ayache and Taqqu 2005) provide sufficient conditions for a multifractional process with random functional parameter to be self-similar (in the sense of its marginal distributions) or to have stationary increments.
- 10.
To avoid ambiguity, when necessary, we write explicitly ω for the stochastic process.
- 11.
Basically, with regard to the process X sampled at N times, the generalized quadratic variation is defined as \(V_N=\sum_{p=0}^{N-2}\left(X\left(\frac{p+2}{N}\right)-2X\left(\frac{p+1}{N}\right)+X\left(\frac{p}{N}\right)\right)^2\); the variation serves to define the estimator \(\hat{h}_N=\frac{1}{2}\left(1-\frac{lnV_N}{lnN}\right)\) which, under some assumptions on the function h(t), satisfies \(lim_{N\rightarrow+\infty}\hat{h}_N=inf_{t\in(0,1)}h(t)\) almost surely. The result stated for the infimum provides the way to estimate h itself at any point \(t\in(0,1)\); in fact choosing a proper \((\epsilon,N)\)-neighborhood of t one can calculate the generalized quadratic variation \(V_{\epsilon,N}\) and hence the estimator \(\hat{h}_{\epsilon,N}\), that satisfies \(lim_{N\rightarrow+\infty}\hat{h}_{\epsilon,N}=inf_{|s-t|<\epsilon}h(s)\) (a.s). Letting ϵ tend to zero one gets an estimate of h(t).
- 12.
The variance in (23) follows from assuming a smooth h(t). In fact, from (16) it follows:
$$\begin{aligned} Var\left(X_{t}-X_{s}\right)&=\mathbb{E}\left(X_{t}-X_{s}\right)^{2} -\left(\mathbb{E}(X_{t}-X_{s})\right) ^{2}= \mathbb{E}\left(X_{t}^{2}+X_{s}^{2}-2X_{t}X_{s}\right) = \\ &= t^{2h(t)}+s^{2h(s)}-D(h(t),h(s))\left(t^{h(t)+h(s)}+s^{h(t)+h(s)}-|t-s|^{h(t)+h(s)}\right)\\ &= t^{h(t)}(t^{h(t)}-D(h(t),h(s))t^{h(s)})+s^{h(s)}(s^{h(s)}-D(h(t),h(s))s^{h(t)}) + \\ & \hspace{6cm} + D(H_{t},H_{s})|t-s|^{h(t)+h(s)}.\end{aligned}$$Since \(\lim_{|h(t)-h(s)| \rightarrow 0}D(h(t),h(s))=1\), whenever \(h(t)\approx h(s)\), one has \(Var\left(X_{t}-X_{s}\right) \approx |t-s|^{2h(t)}\). Assuming that the mBm is sampled in discrete time over n points with \(Var(X_{n}-X_{0})=K^{2}\) entails therefore
$$\begin{aligned}Var\left(X_{\frac{t+q}{n-1}}-X_{\frac{t}{n-1}}\right) \cong K^{2}\left | \frac{t+q}{n-1}-\frac{t}{n-1} \right | ^{2h\left(\frac{t}{n-1}\right)} = K^{2}\left(\frac{q}{n-1}\right) ^{2h\left(\frac{t}{n-1}\right)}\end{aligned}$$that the variance in (23)
- 13.
For \(h \not= \frac{1}{2}\), the variance of \(h_{\delta,q,n,1}^k(t)\) is hard to deduce because of the term σ2 that appears in the variance of (30).
- 14.
Function L is slowly varying at infinity if \(\lim_{t \to \infty} \frac{L(\alpha t)}{L(t)}=1 \text{ for some} \alpha \in \mathbb{R^+}\)
- 15.
To avoid confusion with the notation, here we indicate the order of the absolute moment by s (instead of k used for \(h_{\delta,q,n,K}^{k}(t)\)).
- 16.
Using the two-sided Lilliefors test at different significance levels, Bianchi and Pianese 2008 show that, consistently with the mBm/MPRE model, the empirical significance values of three main stock indexes converge to the nominal ones as δ decreases. Although this suggests to maintain δ as short as possible in order to ensure the normality of data, a trade-off problem arises because the estimator’s variance increases as the length of the window decreases. The discussion of this effect leads the authors in (Bianchi and Pianese 2008) to set \(\delta=30\).
- 17.
This assumption is not restrictive, since it has just been shown that k = 2 minimizes the estimator’s variance (see Remark 14).
- 18.
Likewise the notation already introduced, from now on the subscript will refer to the discrete time sampling.
- 19.
Using a time changing δ (see Remark 14) produces very similar results, hence to save the homogeneity of the results we have chosen a fixed window of proper length.
- 20.
The Jarque-Bera test quantifies the distance from the Gaussian distribution in terms of skewness and kurtosis and then computes a single p value using the sum of these discrepancies.
References
Arneodo, A., E. Bacry, and J. F. Muzy. 1998. Random cascades on wavelet dyadic trees. Journal of Mathematical Physics 39:4142–4164.
Ayache, A.. 2000. The generalized multifractional Brownian motion can be multifractal. Publications du Laboratoire de statistique et probabilités 22.
Ayache, A. 2013. Continuous gaussian multifractional processes with random pointwise Hölder regularity. Journal of Theoretical Probability 26:72–93.
Ayache, A., and J. Lévy Véhel. 2000. The generalized multifractional brownian motion. Statistical Inference for Stochastic Processes 3:7–18.
Ayache, A., and J. Lévy Véhel. 2004. On the identification of the pointwise Hölder exponent of the generalized multifractional Brownian motion. Stochastic Processes and their Applications 111:119–156.
Ayache, A., and M.S. Taqqu. 2005. Multifractional processes with random exponent. Publicacions Matemà tiques 49:459–486.
Ayache, A., S. Cohen, and J. Lévy Véhel. 2000. The covariance structure of multifractional Brownian motion, with application to long range dependence. In ICASSP '00: Proceedings of the Acoustics, Speech, and Signal Processing, 2000. on IEEE International Conference, IEEE Computer Society, Washington, DC, USA, ISBN 0-7803-6293-4, pp. 3810–3813, doi:http://dx.doi.org/10.1109/ICASSP.2000.860233.
Ayache, A., S. Benassi, S. Cohen, and J. Lévy Véhel. 2005. Regularity and identification of generalized multifractional Gaussian processes. Séminaire de Probabilités XXXVIII, Lecture Notes in Math 1857:290–312.
Ayache, A., P. R. Bertrand, and J. Lévy Véhel. 2007. A central limit theorem for the quadratic variations of the step fractional Brownian motion. Statistical Inference for Stochastic Processes 10:1–27.
Baillie, R. T.. 1996. Long memory processes and fractional integration in econometrics. Journal of Econometrics 73:5–59.
Bansal, R., and A. Yaron. 2004. Risks for the long-run: A potential resolution of asset pricing puzzles. Journal of Finance 59:1481–1509.
Bayraktar, E., U. Horst, and R. Sircar. 2006. A limit theorem for financial markets with inert investors. Mathematics of Operations Research 31:789–710.
Benassi, A., S. Jaffard, and D. Roux. 1997. Gaussian processes and pseudodifferential elliptic operators. Revista Mathematica Iberoamericana 13:19–89.
Benassi, A., S. Cohen, and J. Istas. 1998. Identifying the multifractional function of a Gaussian process. Statistics and Probability Letters 39:337–345.
Benassi, A., S. Cohen, J. Istas, and S. Jaffard. 1998. Identification of filtered white noises. Stochastic Processes and their Applications 75:31–49.
Benassi, A., P. Bertrand, S. Cohen, and J. Istas. 1999. Identification d’un processus gaussien multifractionnaire avec des ruptures sur la fonction d'échelle. Comptes rendus de l’Académie des sciences. Série 1, Mathématique 329:435–440.
Benassi, Albert, Pierre Bertrand, Serge Cohen, and Jacques Istas. 2000. Identification of the hurst index of a step fractional brownian motion. Statistical Inference for Stochastic Processes 3:101–111.
Bender, C., T. Sottinen, and E. Valkeila. 2007. Arbitrage with fractional Brownian motion? Theory of Stochastic Processes 13:23–34.
Bertrand, Pierre. 2005. Financial modelling by multiscale fractional Brownian motion. In Fractals in Engineering, eds. Jacques Lévy Véhel and Evelyne Lutton, 181–196. London: Springer.
Bertrand, Pierre, Abdelkader Hamdouni, and Samia Khadhraoui. 2012. Modelling NASDAQ series by sparse multifractional Brownian motion. Methodology and Computing in Applied Probability 14:107–124.
Bianchi, S.. 2005. A cautionary note on the detection of multifractal scaling in finance and economics. Applied Economics Letters 12:775–780.
Bianchi, S. 2005. Pathwise identification of the memory function of the multifractional Brownian motion with application to finance. International Journal of Theoretical and Applied Finance 8:255–281.
Bianchi, S., and A. Pianese. 2007. Modelling stock price movements: Multifractality or multifractionality? Quantitative Finance 19:301–319.
Bianchi, S., and A. Pianese. 2008. Multifractional properties of stock indices decomposed by filtering their pointwise Hölder regularity. International Journal of Theoretical and Applied Finance 11:567–595.
Bianchi, S., and A. Pantanella. 2010. Pointwise regularity exponents and market cross-correlations. International Review of Business Research Papers 6:39–51.
Bianchi, S., and A. Pantanella. 2011. Pointwise regularity exponents and well-behaved residuals in stock markets. International Journal of Trade, Economics and Finance 2:52–60.
Bianchi, S., A. Pantanella, and A. Pianese. 2013. Modeling stock prices by multifractional Brownian motion: An improved estimation of the pointwise regularity. Quantitative Finance 13:1317–1330.
Calvet, L., and A. Fischer. 2002. Multifractality in asset returns: Theory and evidence. The Review of Economics and Statistics 84:381–406.
Calvet, L., A. Fischer, and B. B. Mandelbrot. 1997. A multifractal model of asset returns. Cowles Foundation Discussion Paper, Yale University, 1164.
Calvet, Laurent E., and Adlai J. Fisher. 2008. Multifractal volatility: Theory, forecasting, and pricing. Elsevier: Academic Press.
Campbell, J.Y., and J. H. Cochrane. 1999. By force of habit: A consumption-based explanation of aggregate stock market behavior. Journal of Political Economy 107:205–251.
Chan, G., and A. Wood. 1998. Simulation of multifractional Brownian motion. Proceedings in Computational Statistics, pp. 233–238.
Cheridito, P. 2001. Regularizing fractional Brownian motion with a view towards stock price modelling. Ph.D. thesis, Eidgenöss Technische Hochschule Zürich, Swiss Federal Institute of Technology.
Cheridito, P. 2003. Arbitrage in fractional brownian motion models. Finance and Stochastics 7:533–553.
Coeurjolly, J. F. 2000. Inférence statistique pour les mouvements browniens fractionnaires et multifractionnaires, Ph. D. Thesis.
Coeurjolly, J. F. 2005. Identification of the multifractional Brownian motion. Bernoulli 11:987–1008.
Coeurjolly, J. F.. 2008. Hurst exponent estimation of locally self-similar Gaussian processes using sample quantiles. Annals of Statistics 36:1404–1434.
Cohen, S. 1999. Chapter I. In From self-similarity to local self-similarity: The estimation problem, pages 3–16. ed. M. Dekking, J. Lévy-Véhel, E. Lutton, and C. Tricot, 3–16. Springer Berlin/Heidelberg
Cont, R. 2001. Empirical properties of asset returns: Stylized facts and statistical issues. Quantitative Finance 1:223–236.
Costa, Rogério L., and G. L. Vasconcelos. 2003. Long-range correlations and nonstationarity in the Brazilian stock market. Physica A: Statistical Mechanics and its Applications 329:231–248.
Dasgupta, A., and G. Kallianpur. 2000. Arbitrage opportunities for a class of Gladyshev processes. Applied Mathematics and Optimization 41:377–385.
Decreusefond, L., and S. A. Üstünel. 1999. Stochastic analysis of the fractional brownian Motion. Potential Analysis 10:177–214.
Delbaen, F., and W. Schachermayer. 1994. A general version of the fundamental theorem of asset pricing. Mathematische Annalen 300:463–520.
Elliott, R. J., and J. Van Der Hoek. 2003. A general fractional white noise theory and applications to finance. Mathematical Finance 13:301–330.
Fama, E. F. 1970. Efficient capital markets: A review of theory and empirical work. The Journal of Finance 25:383–417.
Fama, E.F., and K.R. French. 1988. Permanent and temporary components of stock prices. Journal of Political Economy 22:246–273.
Fama, E. F.. 1998. Market efficiency, long-term returns, and behavioral finance. Journal of Financial Economics 49 (3): 283–306.
Guasoni, P.. 2006. No arbitrage under transaction cost, with fractional Brownian motion and beyond. Mathematical Finance 16:569–582.
Henry, M., and P. Zaffaroni. 2003. Chapter III In The long range dependence paradigm for macroeconomics and finance, ed. Birkhäuser P. Doukhan, G. Oppenheim, and M.S. Taqqu, 417–438.
Hu, Y., and B. Øksendal. 2003. Fractional white noise calculus and applications to finance. Infinite Dimensional Analysis, Quantum Probability and Related Topics 6:1–32.
Hunt, G. A.. 1951. Random fourier transforms. Transactions of the American Mathematica 71:38–69.
Istas, J., and G. Lang. 1997. Quadratic variations and estimation of the local Hölder index of a Gaussian process. Annales de l’Institut Henri Poincare (B) Probability and Statistics 33:407–436.
Jarque, C. M., and A. K. Bera. 1987. A test for normality of observations and regression residuals. International Statistical Review 55:163–172.
Protter, P., R. A. Jarrow, and H. Sayit. 2009. No arbitrage without semimartingales. The Annals of Applied Probability 19:596–616.
Kahneman, D., and A. Tversky 1979. Prospect theory: An analysis of decision under risk. Econometrica 47 (2): 263–292.
Kolmogorov, A. N. 1940. Wienershe spiralen und einige andere interessante Kurven im Hilbertishen Raum. DAS of the URSS (Nat. Sciences) 26:115–118.
Kolwankar, K. M. and J. Lévy Véhel. 2002. A time domain characterization of the fine local regularity of functions. Journal of Fourier Analysis and Applications 8:319–334.
Lamperti, J. 1962. Semi-stable stochastic process. Transactions of The American Mathematical Society 104:62–78.
Lévy Véhel, J., and O. Barriére. 2008. Local Hölder regularity-based modeling of RR intervals. Proceedings of the 21th IEEE international symposium on computer-based medical systems, pp. 75–80.
Lo, A. W. 2004. The adaptive markets hypothesis: Market efficiency from an evolutionary perspective. Journal of Portfolio Management 30:15–29.
Lo, A. W. 2005. Reconciling efficient markets with behavioral finance: The adaptive market hypothesis. Journal of Investment Consulting 7 (2): 21–44.
Lo, A. W.. 2012. Adaptive markets and the new world order. Financial Analysts Journal 68:18–29.
Loutridis, S. J.. 2007. An algorithm for the characterization of time-series based on local regularity. Physica A Statistical Mechanics and its Applications 381:383–398.
Lustig, H. and S. Van Nieuwerburgh. 2005. Consumption Insurance and risk premia: An empirical perspective. Journal of Finance 60:1167–1219.
Mandelbrot, B. B., and J. W. Van Ness. 1968. Fractional brownian motions, Fractional noises and applications. SIAM Review 10:422–437.
Papanicolaou, G. C., and K. Sbibølna. 2002. Wavelet based estimation of local Kolmogorov turbulence. In Long-range dependence: Theory and applications. Birkhäuser, pp. 473–506.
Péltier, R.S., and J. Lévy Véhel. 1994. A new method for estimating the parameter of fractional Brownian motion. Technical report, CCSd/HAL: E-articles server (based on gBUS) [http://hal.ccsd.cnrs.fr/oai/oai.php] (France), http://hal.inria.fr/inria-00074279/en/.
Péltier, R. S., and J. Lévy Véhel. 1995. Multifractional Brownian Motion: Definition and preliminary results, Technical report, INRIA, citeseer. ist.psu.edu/peltier95multifractional.html.
Riedi, R. H. 2002. In Multifractal processes, ed. Birkhäuser P. Doukhan. G. Oppenheim, and M. Taqqu, pp. 625–715.
Rogers, L. C. G. 1997. Arbitrage with fractional Brownian motion. Mathematical Finance 7:95–105.
Sewell, M. 2007. Behavioural finance. http://www.behaviouralfinance.net/behavioural-finance.pdf.
Hersh, Shefrin, and Meir Statman. 1994. Behavioral capital asset pricing theory. The Journal of Financial and Quantitative Analysis 29:323–349.
Hersh Shefrin and Meir Statman. 2000. Behavioral portfolio theory. The Journal of Financial and Quantitative Analysis 35:127–151.
Shefrin, H. M. and R. H. Thaler. 1988. Hyothesis The Behavioral life-cycle. Economic Inquiry 26:609–643.
Shiller, R. J. 1984. Stock prices and social dynamics. Technical report, Brookings Papers on Economic Activity.
Shiryayev, A. N. 1998. On arbitrage and replication for fractal models. Technical report, Department of Mathematical Sciences, University of Aarhus, Denmark.
Stoev, A. S., and M. S. Taqqu. 2006. How rich is the class of multifractional Brownian motions? Stochastic Processes and their Applications 116:200–221.
Summers, L. H.. 1986. Does the stock market rationally reflect fundamental values? Journal of Finance 41:591–601.
Tapiero, Charles. 2010. Risk finance and asset pricing. New York: Wiley.
Tapiero, C. S. 2013. Re-Engineering risks and the future of finance, 233–250. London: EURO-MONEY Books.
Tapiero, C.S., and P. Vallois. 2000. The inter-event range process and testing for chaos in time series. Neural Network World 10 (1–2) 89–99.
Tapiero, C. S., and P. Vallois. 2007. Memory-based persistence in a counting random walk process. Physica A: Statistical Mechanics and its Applications 386 (1): 303–317.
Vallois, P., and C. S. Tapiero. 1996. Run length statistics and the Hurst exponent in random and birth-death random walks. Chaos, Solitons & Fractals 7 (9): 1333–1341.
Vallois, P., and C. S. Tapiero. 1997. Range reliability in random walks, Mathematical methods of operations research. Physica A: Statistical Mechanics and its Applications 45 (3): 325–345.
Vallois, P., and C. S. Tapiero. 2001. The range inter-event process in a symmetric birth death random walk. Applied Stochastic Models in Business and Industry 17 (3): 293–306.
Vallois, P., and C. S. Tapiero. 2008. Volatility estimators and the inverse range process in a random volatility random walk and Wiener processes. Physica A: Statistical Mechanics and its Applications 387 (11): 2565–2574.
Yaglom, A. M. 1958. Correlation theory of processes with random stationary n thincrements. American Mathmathical Society Translations. 2:87–141.
Yaglom, A. M. 1965. Stationary gaussian processes satisfying the strong mixing condition and best predictable functionals. In Proceedings international research seminar. p. 241–252.
Yen, G. 2008. Efficient market hypothesis (EMH): Past, present and future. Review of Pacific Basin Financial Markets and Policies 11:305–329.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Bianchi, S., Pianese, A. (2015). Asset Price Modeling: From Fractional to Multifractional Processes. In: Bensoussan, A., Guegan, D., Tapiero, C. (eds) Future Perspectives in Risk Models and Finance. International Series in Operations Research & Management Science, vol 211. Springer, Cham. https://doi.org/10.1007/978-3-319-07524-2_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-07524-2_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-07523-5
Online ISBN: 978-3-319-07524-2
eBook Packages: Business and EconomicsBusiness and Management (R0)