Asymptotic optimal pricing with asymmetric risk and applications in finance

Abstract

In Part I, we study several problems of hedging and pricing of options related to a risk measure. Our main approach is the use of an asymmetric risk function and an asymptotic framework in which we obtain optimal solutions through nonlinear partial differential equations (PDE). In Chapter 1, we focus on pricing and hedging European options. We consider the optimization problem of the residual risk generated by a discrete-time hedging in the presence of an asymmetric risk criterion. Instead of analyzing the asymptotic behavior of the solution to the associated discrete problem, we study the integrated asymmetric measure of the residual risk in a Markovian framework. In this context, we show the existence of the asymptotic risk measure. Thus, we describe an asymptotically optimal hedging strategy via the solution to a fully nonlinear PDE. Chapter 2 is an application of the hedging method to the valuation problem of the power plant. Since the power plant generates maintenance costs whether it is on or off, we are interested in reducing the risk associated with its uncertain revenues by hedging with forwards contracts. We study the impact of a maintenance cost depending on the electricity price into the hedging strategy. In Part II, we consider several control problems associated with economy and finance. Chapter 3 is dedicated to the study of a McKean-Vlasov (MKV) problem class with common noise, called polynomial conditional MKV. We reduce this polynomial class by a Markov embedding to finite-dimensional control problems. We compare three different probabilistic techniques for numerical resolution of the reduced problem: quantization, control randomization and regress later. We provide numerous numerical examples, such as the selection of a portfolio under drift uncertainty. In Chapter 4, we solve dynamic programming equations associated with financial valuations in the energy market. We consider that a calibrated underlying model is not available and that a limited sample of historical data is accessible. In this context, we suppose that forward contracts are governed by hidden factors modeled by Markov processes. We propose a non-intrusive method to solve these equations through empirical regression techniques using only the log price history of observable futures contracts.

Isaque Pimentel

Quantitative Analyst, Consultant

Ph.D. in Applied Mathematics interested in Quantitative Finance and Data Science.