Multivariate Stochastic Volatility Models and Large Deviation Principles

Archil Gulisashvili

Chapter 1 in Reviews in Modern Quantitative Finance, 2024, pp 1-96 from World Scientific Publishing Co. Pte. Ltd.

Abstract: We establish a comprehensive sample path large deviation principle (LDP) for log-price processes associated with multivariate time-inhomogeneous stochastic volatility models. Examples of models for which the new LDP holds include Gaussian models, non-Gaussian fractional models, mixed models, models with reflection, and models in which the volatility process is a solution to a Volterra-type stochastic integral equation. The sample path and small-noise LDPs for log-price processes are used to obtain large deviation-style asymptotic formulas for the distribution function of the first exit time of a log-price process from an open set, multidimensional binary barrier options, call options, Asian options, and the implied volatility. Such formulas capture leading order asymptotics of the above-mentioned important quantities arising in the theory of stochastic volatility models. We also prove a sample path LDP for solutions to Volterra-type stochastic integral equations with predictable coefficients depending on auxiliary stochastic processes.

Keywords: Quantitative Finance; Financial Engineering; Mathematical Finance; Computational Finance; Computational Methods; Computational Problems; Pricing of Securities; Trading; Market Microstructures; Risk Theory; Queuing Theory; Asset Management Technique; Liability Management Technique; Risk Measures; Solvency; Financial Instability; Fintech; Cryptocurrencies; Financial Machine Learning; Artificial Intelligence; Fintech; Quantum Computing; Distributed Ledgers; Econophysics (search for similar items in EconPapers)
JEL-codes: C C02 C6 C61 (search for similar items in EconPapers)
Date: 2024
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