Stochastic Volatility with Levy Processes: Calibration and Pricing

In this thesis, stochastic volatility models with Levy processes are treated in parameter calibration by the Carr-Madan fast Fourier transform (FFT) method and pricing through the partial integro-differential equation (PIDE) approach. First, different models where the underlying log stock price or volatility driven by either a Brownian motion or a Levy…


1 Introduction
1.1 Stochastic Volatility Models
1.2 Models of Interest
2 Standard Errors for Financial Market Data Analysis
2.1 Introduction
2.2 Relationship to Statistical Standard Error Calculations
2.2.1 The Case of Maximum Likelihood
2.2.2 Nonlinear Least Squares
2.3 Estimating Risk Neutral Models on Financial Market Data
2.3.1 Data Set and Calibration Methodology
2.3.2 Results of Parameter Estimations
2.3.3 Standard Errors for Parameter Estimations
2.3.4 Numerical Implementations
3 Stability on Calibration of Stochastic Volatility Models
3.1 Introduction
3.2 Methodology
3.3 S&P 500 Data and Results
3.4 Stability on Calibration of Stochastic Skew Model on FX Data
3.4.1 Stochastic Skew Model
3.4.2 Methodology
3.4.3 FX Data and Results
4 Operator Splitting Method for PIDEs
4.1 Introduction
4.2 Operator Splitting for 1-D PIDEs
4.2.1 Approximation of Small Jumps by Brownian Motion
4.2.2 Operator Splitting
4.2.3 Variational Formulation of Integral Operator
4.2.4 Space Discretization of Integral Operator
4.2.5 Results
4.3 Operator Splitting for 2-D PIDEs
4.3.1 Time Changed L´evy Process Model
4.3.2 Operator Splitting in Diffusion Step
4.3.3 Operator Splitting in Jump Step
4.3.4 Results
4.4 Operator Splitting for 3-D PIDEs
4.4.1 Stochastic Skew Model…


Author: Wu, Xianfang

Source: University of Maryland

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