Volterra Volatility Models : Option Pricing and Hedging

Financial markets have extremely complex behavior that cannot be fully modeled using classical approaches. In particular, numerous empirical studies show that market volatility exhibits some form of long-range dependence and has time-varying Hölder regularity with prominent periods of “roughness” (i.e., of Hölder order ≈ 0.1). These two properties are far beyond the capabilities of classical Brownian diffusions and it is challenging to reproduce them simultaneously in one model. In the existing literature, the phenomenons of long-range dependence and roughness mentioned above are often addressed by using fractional Brownian motion. However, in this case, these two features turn out to be mutually exclusive and cannot be grasped simultaneously. Furthermore, existing stochastic models based on fractional Brownian motion pose additional challenges of the technical kind: they tend to produce prices with moment explosions (and hence are not applicable to pricing some widespread derivatives); they may have volatilities that hit zero (or even become negative) which results in problems with transitioning between physical and pricing measures; they often lack efficient numerical algorithms for derivative pricing, hedging, etc. In this book, we introduce a novel class of stochastic processes driven by general Hölder noises that allows for a very broad flexibility in the noises (to account for both roughness and long-range dependence simultaneously) and grasps the unconventional behavior of market volatility. We also present a variety of associated numerical methods and propose practically feasible algorithms for various applications, such as pricing of derivatives (including options with discontinuous payoffs) and quadratic hedging.

Giulia Di Nunno is a professor at the University of Oslo and an adjunct professor at Norwegian School of Economics (NHH). She received her PhD in mathematical statistics in 2003 from the University of Pavia. Her main research fields are stochastic analysis and control with financial modelling in view. She has worked in portfolio optimisation, risk evaluation, pricing and hedging, stochastic volatility modelling, and more recently, numerical methods and machine learning in finance. Di Nunno is the author and co-author of about 80 scientific works, including a monograph on Malliavin calculus for Levy Processes with Applications to Finance and she is an editor of various volumes, all published by Springer. She is active internationally within knowledge societies, research projects, and works as an editor for several leading journals. She has made significant contributions in promoting mathematics in education and research globally as well as promoting women in mathematics. She is also the recipient of the 2019 ICIAM Su Buchin Prize.

Yuliya Mishura received her PhD in probability and statistics from Kyiv University in 1978 and completed her postdoctoral degree in probability and statistics (Habilitation) in 1990. She is currently a professor of the Department of Probability, Statistics and Actuarial Mathematics at Taras Shevchenko National University of Kyiv. Having broad and varied scientific interests, she is the author/coauthor of more than 320 research papers and more than 20 books. Her research interests include theory and statistics of stochastic processes, stochastic differential equations, stochastic finance, stochastic analysis, functional limit theorems, entropies of probability distributions and stochastic systems and other applications of stochastics. She was the invited speaker of many international congresses and conferences and the organizer of a series of conferences. She is the Editor-in-Chief of the journal Theory of Probability and Mathematical Statistics, Coeditor-in-Chief of the journal Modern Stochastics: Theory and Applications and the associated editor of several journals. She was a team leader and participant in many international research projects.

Anton Yurchenko-Tytarenko completed his PhD at the Faculty of Mathematics and Natural Sciences, University of Oslo in 2022. He specializes in probability theory, statistics, and stochastic analysis, with research interests in financial modeling, particularly volatility, stochastic systems with memory, machine learning and numerical methods for random processes. He is currently a Senior Power Market Analyst at Statkraft Energi AS, where he focuses on modeling and forecasting prices of power market assets.