Asymptotic analysis of new stochastic volatility models

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Title: Asymptotic analysis of new stochastic volatility models
Author(s): Shi, Fangwei
Item Type: Thesis or dissertation
Abstract: A good options pricing model should be able to fit the market volatility surface with high accuracy. While the standard continuous stochastic volatility models can generate volatility smiles consistent with market data for relatively larger maturities, these models cannot reproduce market smiles for small maturities, which have the well-observed `small-time explosion' feature. In this thesis we propose three new types of stochastic volatility models, and we focus on the small-time asymptotic behaviour of the implied volatility in these models. We show that these models can generate implied volatilities with explosion, hence they can theoretically provide a better fit to the market data. The thesis is organised as follows. Chapter 0 is the introduction. We briefly discuss the development and performance of standard continuous stochastic volatility models, and raise the small-time fitness issue of these traditional models. In Chapter 1 we propose the randomised Heston model and analyse its small and large time asymptotic behaviours. In particular, we show that any small-time explosion rate in between of [0, 1/2] for the implied variance can be captured by a suitable choice of the initial randomisation. In Chapter 2 we propose a fractional version of the Heston model and detail the small-time asymptotic behaviour of the implied volatility in this setting. We precise the link between the explosion rate and the Hurst parameter. Finally, in Chapter 3 we propose a new stochastic volatility model based on the recent work by Conus and Wildman in which the stock price can have past dependency. We show that in the case of a CIR variance process this model has similar behaviours to a fractional Heston environment.
Content Version: Open Access
Publication Date: Nov-2017
Date Awarded: Jun-2018
Advisor: Jacquier, Antoine
Sponsor/Funder: Imperial College London
Department: Mathematics
Publisher: Imperial College London
Qualification Level: Doctoral
Qualification Name: Doctor of Philosophy (PhD)
Appears in Collections:Mathematics PhD theses

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