Modeling stock prices by multifractional Brownian motion: an improved estimation of the pointwise regularity

Abstract

This paper deals with the problem of estimating the pointwise regularity of multifractional Brownian motion, assumed as a model of stock price dynamics. We (a) correct the shifting bias affecting a class of absolute moment-based estimators and (b) build a data-driven algorithm in order to dynamically check the local Gaussianity of the process. The estimation is therefore performed for three stock indices: the Dow Jones Industrial Average, the FTSE 100 and the Nikkei 225. Our findings show that, after the correction, the pointwise regularity fluctuates around 1/2 (the sole value consistent with the absence of arbitrage), but significant deviations are also observed.

Keywords:

• Multifractional Brownian motion
• Pointwise Hölder exponent
• Asset price dynamics
• time varying parameter
• Applied finance

JEL Classification :

• C1
• C2
• C5
• C13
• C22
• C51
• G1
• G14

Acknowledgement

The authors wish to thank the anonymous referees whose remarks significantly improved the quality of this paper.

Notes

†Remember that the Hölder exponent measures the degree of irregularity of the graph of a function. Given the function f(x), if there exists a constant C and a polynomial P _n of degree n < h such that |f(x) − P _n (x − x ₀)| ≤ C|x − x ₀| ^h , the Hölder exponent H(x ₀) is defined as the supremum of all hs such that the above relation holds. The polynomial P _n is often associated with the Taylor expansion of f around x ₀, but the relation is valid even if such an expansion does not exist.

†Basically, with regard to the process X sampled N times, the generalized quadratic variation is defined as

The variation serves to define the estimator , which, under certain assumptions on the function H _t , satisfies almost surely. The result stated for the infimum provides a way to estimate H itself at any point t ∈ (0, 1). In fact. choosing an appropriate (ε, N)-neighborhood of t, one can calculate the generalized quadratic variation V _ε,N and hence the estimator that satisfies (a.s.). Letting ε tend to zero, one obtains an estimate of H _t .

‡Here, H-sssi is the self-similar of parameter H with stationary increments (see Samorodnitsky and Taqqu 1994 for a detailed discussion).

§Indeed, Ayache and Taqqu (2005) provide sufficient conditions for a multifractional process with a random functional parameter to be self-similar (in the sense of its marginal distributions) or to have stationary increments.

†The detailed proofs recalled in this section can be found in Bianchi (2005).

†In order to simulate fBm we used the wavelet-based algorithm introduced by Sellan (1995), implemented by Abry and Sellan (1996) and revised by Bardet et al. (2003), in order to remove the many high-frequency components.

‡Remember that the function L is slowly varying at infinity if lim _t→∞(L(αt)/L(t)) = 1 for some α ∈ ℝ⁺.

†In the application, we use Scott's rule to determine the number of bins. Scott (1979) provides a formula for the optimal histogram bin width that asymptotically minimizes the integrated mean squared error: h _n  = 3.49 · σ · n ^−1/3, where σ is an estimate of the data standard deviation.

†Owing to the closeness of the values, taking one or three bins does not suffice to exclude, in principle, that the variability of the estimates could be ascribed solely to the estimator variance.