The roughness exponent and its application in finance

Abstract

Rough phenomena and trajectories are prevalent across mathematics, engineering, physics, and the natural sciences. In quantitative finance, sparked by the observation that the historical realized volatility is rougher than a Brownian martingale, rough stochastic volatility models were recently proposed and studied extensively. Unlike classical diffusion volatility models, the volatility process in a rough stochastic volatility model is driven by an irregular process. The roughness of the volatility process plays a pivotal role in governing the behavior of such models.

This thesis aims to explore the concept of roughness and estimate the roughness of financial time series in a strictly pathwise manner. To this end, we introduce the notion of the roughness exponent, which is a pathwise measure of the degree of roughness. A substantial portion of this thesis focuses on the model-free estimation of this roughness exponent for both price and volatility processes. Towards the end of this thesis, we study the Wiener–Young Φ-variation of classical fractal functions, which can be regarded as a finer characterization than the roughness exponent.

Chapter 2 introduces the roughness exponent and establishes a model-free estimator for the roughness exponent based on direct observations. We say that a continuous real-valued function x admits the roughness exponent R if the pth variation of x converges to zero for p>1/R and to infinity for p<1/R . The main result of this chapter provides a mild condition on the Faber–Schauder coefficients of x under which the roughness exponent exists and is given as the limit of the classical Gladyshev estimates. This result can be viewed as a strong consistency result for the Gladyshev estimator in an entirely model-free setting because no assumption whatsoever is made on the possible dynamics of the function x. Nonetheless, we show that the condition of our main result is satisfied for the typical sample paths of fractional Brownian motion with drift, and we provide almost-sure convergence rates for the corresponding Gladyshev estimates. In this chapter, we also discuss the connections between our roughness exponent and Besov regularity and weighted quadratic variation. Since the Gladyshev estimators are not scale-invariant, we construct several scale-invariant modifications of our estimator. Finally, we extend our results to the case where the p^th variation of x is defined over a sequence of unequally spaced partitions.

Chapter 3 considers the problem of estimating the roughness exponent of the volatility process in a stochastic volatility model that arises as a nonlinear function of a fractional Brownian motion with drift. To this end, we establish a new estimator based on the Gladyshev estimator that estimates the roughness exponent of a continuous function x, but based on the observations of its antiderivative y. We identify conditions on the underlying trajectory x under which our estimates converge in a strictly pathwise sense. Then, we verify that these conditions are satisfied by almost every sample path of fractional Brownian motion with drift. As a consequence, we obtain strong consistency of our estimator in the context of a large class of rough volatility models. Numerical simulations are implemented to show that our estimation procedure performs well after passing to a scale-invariant modification of our estimator.

Chapter 4 highlights the rationale of constructing the estimator from the Gladyshev estimator. In this chapter, we study the problem of reconstructing the Faber–Schauder coefficients of a continuous function from discrete observations of its antiderivative. This problem arises in the task of estimating the roughness exponent of the volatility process of financial assets but is also of independent interest. Our approach starts with formulating this problem through piecewise quadratic spline interpolation. We then provide a closed-form solution and an in-depth error analysis between the actual and approximated Faber–Schauder coefficients. These results lead to some surprising observations, which also throw new light on the classical topic of quadratic spline interpolation itself: They show that the well-known instabilities of this method can be located exclusively within the final generation of estimated Faber–Schauder coefficients, which suffer from non-locality and strong dependence on the initial value and the given data. By contrast, all other Faber–Schauder coefficients depend only locally on the data, are independent of the initial value, and admit uniform error bounds. We thus conclude that a robust and well-behaved estimator for our problem can be obtained by simply dropping the final-generation coefficients from the estimated Faber–Schauder coefficients.

Chapter 5 studies the Wiener–Young Φ-variation of classical fractal functions with a critical degree of roughness. In this case, the functions have vanishing p^th variation for all p>q but are also of infinite p^th variation for p< q for some q≥1 . We partially resolve this apparent puzzle by showing that these functions have finite, nonzero, and linear Wiener–Young Φ-variation along the sequence of certain partitions. For instance, functions of bounded variation admit vanishing p^th variation for any p>1. On the other hand, Weierstrass and Takagi–van der Waerden functions have vanishing p^th variation for p>1 but are also nowhere differentiable and hence not of bounded variation. As a result, power variation and the roughness exponent fail to distinguish the difference of degree of roughness for these functions. However, we can individuate these functions by showing that the Weierstrass and Takagi–van der Waerden functions admit a nontrivial and linear Φ-variation along the sequence of b-adic partitions, where Φ_q (x)=x/(-log ⁡x)^1/2. Moreover, for q>1 , we further develop a probabilistic approach so as to identify functions in the Takagi class that have linear and nontrivial Φ_q-variation for a prescribed Φ_q . Furthermore, for each fixed q>1 , the collection of functions Φ_q forms a wide class of increasing functions that are regularly varying at zero with an index of regular variation q.