Rough analysis with application in markets and related fields release_bjkijv2phnbptdwm2qh3kkfaxe

by Paul Peter Hager, Technische Universität Berlin, Peter K. Friz, Christian Bayer

Published by Technische Universität Berlin.

2021  

Abstract

We treat several topics related to stochastic processes and rough analysis. The problems we consider are motivated from the recent developments in rough volatility models and the recent proposition of rough price models in energy markets. Such models require the development of new methodology for pricing financial derivatives, as the Markov property in general does not hold true and the irregularity of the sample paths introduces numerical difficulties. An important concept from rough analysis that is central to the first and the second part of this thesis is the signature of a path. In the second and third part we consider stochastic control problems, as they appear for example in the pricing of American options. The fourth part focuses on limiting behavior of a rough model when the Hölder regularity vanishes. More specifically, in the first part we establish a universal functional relation for the signature cumulant in a general semimartingale context. Our work exhibits the importance of Magnus expansions in the algorithmic problem of computing expected signature cumulants, and further offers a far-reaching generalization of recent results on characteristic exponents dubbed diamond and cumulant expansions. The significance of the main result is illustrated in a variety of examples. In second part we propose a new method for solving optimal stopping problems that apply under minimal assumptions to the underlying process. We proof that the optimal stopping problem can be solved while restricting to stopping times that are parametrized by linear functionals of the signature associated to the underlying process. Further, using the log-signature as a feature set for a deep neural network we can efficiently solve the optimal stopping problem numerically. The methodology applies particularly to processes that fail to be either semimartingales or Markov processes, such as fractional Brownian motion, and can be used, in particular, for American-type option pricing. In the third part we consider a more general class of s [...]
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