Multiscale volatility analysis via malliavin calculus

İnkaya, B. Alper
In this thesis, we study multifractal stochastic processes and stability properties of stochastic processes with the aim of analyzing the multiscale characteristics of dynamic risk premiums present in financial asset prices. Multifractal processes are first defined to model the statistical properties of turbulent flows and characterized by the scale-invariance property, which implies volatility clustering, long-range dependency and multiplicative instead of additive behavior. The multifractal characterization of a dataset can be obtained, also, via the multifractal spectrum, the singularity spectrum and the generalized dimensions. The complex dynamics of financial markets resembling chaos recently gave rise to the development of multifractal models in finance. In the present study we aim to relate the multifractal behaviour of markets to the existence of multiscale risk premiums. We employ Malliavin calculus techniques to analyze the dynamics of the instantaneous risk premiums by estimating the pricevolatility feedback effect rate, which is defined as the expansion rate of the rescaled variation resulting from the perturbation of the stochastic process. Throughout our study, we discover that the price-volatility feedback effect rate is the local Lyapunov exponent of the perturbation resulting in the change of measure. The fundamental indicator of chaotic dynamics is generally accepted to be the sensitive dependency to initial conditions, which can be measured via the Lyapunov exponents. The local Lyapunov exponents (LLE) characterize the finite-time behaviour of the expansion rates. We analyze the dimensional properties of the price-volatility feedback effect rate to show the existence of multiscale risk premiums in financial return series. The generalized dimensions constitutes the basis of our study as they allow for the analysis of perturbations of multifractal processes and LLEs. To bring the multifractal framework and Malliavin calculus techniques together, we first perform multifractal analysis of the empirical datasets. Then, we estimate the instantaneous volatilities and the price-volatility feedback effect rate series of the datasets using the recently defined Fourier series method. Additionally, analyze the multifractal characteristics of the instantaneous volatilities, while the usual multifractal analysis assumes multifractality of absolute returns. To demonstrate the existence of multiscale risk premiums, we perform dimensional analysis of both the return and the estimated instantaneous price-volatility feedback effect rate series. We conclude with the observation that the generalized dimensions spectrums of both series coincide, which suggests that the existence of scale-dependent non-linear type of behavior of the risk premiums in financial asset prices.