Likelihood Estimation of Jump-Diffusions. Extensions from Diffusions to Jump-Diffusions, Implementation with Automatic Differentiation, and Applications

This thesis considers the problem of likelihood- based parameter estimation for time-homogeneous jump-diffusion processes. The problem is that there often is no analytic solution to the stochastic differential equations driving the process. Thus, the transition density of the process is unknown. In this thesis we build on the solution presented in Preston and Wood (2012), where the transition density of a time- homogeneous diffusion process is approximated by a saddlepoint approximation based on the approximated solution following from discretization schemes, which in turn stems from an Itô-Taylor expansion of the stochastic differential equation. The mathematical tools for understanding the method in Preston and Wood (2012) and the extended methods to jump- diffusions are developed. We reproduce the results found here, and extend the analysis with maximum likelihood estimation for benchmark processes such as the geometric Brownian motion, the Ornstein-Uhlenbeck process, the Cox- Ingersoll-Ross process, and the Merton model. We also investigate the use of the renormalized saddlepoint approximation in the context of maximum likelihood estimation. The implementation of the methods is carried out with the newly released parallel programming package, Template Model Builder, which uses automatic differentiation among other things. We therefore give an introduction to the basics of automatic differentiation in the context of our computational problems, and also extend the Template Model Builder package to e.g. allow for complex numbers. Thereafter we apply the methods developed in previous chapters to the analysis of stock prices modelled as nonlinear stochastic differential equations, with and without jumps. Finally we briefly analyse some models for stochastic volatility.Master i StatistikkMAMN-STATSTAT39

Maximum Likelihood and Gaussian Estimation of Continuous Time Models in Finance

Published in Handbook of financial time series, 2008, https://doi.org/10.1007/978-3-540-71297-8_22</p

Maximum likelihood and Gaussian estimation of continuous time models in finance

Ministry of Education, Singapore under its Academic Research Funding Tier

Adaptive Continuous time Markov Chain Approximation Model to General Jump-Diffusions

We propose a non-equidistant Q rate matrix formula and an adaptive numerical algorithm for a continuous time Markov chain to approximate jump-diffusions with affine or non-affine functional specifications. Our approach also accommodates state-dependent jump intensity and jump distribution, a flexibility that is very hard to achieve with other numerical methods. The Kologorov-Smirnov test shows that the proposed Markov chain transition density converges to the one given by the likelihood expansion formula as in Ait-Sahalia (2008). We provide numerical examples for European stock option pricing in Black and Scholes (1973), Merton (1976) and Kou (2002)

The History of the Quantitative Methods in Finance Conference Series. 1992-2007

This report charts the history of the Quantitative Methods in Finance (QMF) conference from its beginning in 1993 to the 15th conference in 2007. It lists alphabetically the 1037 speakers who presented at all 15 conferences and the titles of their papers.

Topics in volatility models

In this thesis I will present my PhD research work, focusing mainly on financial modelling of asset’s volatility and the pricing of contingent claims (financial derivatives), which consists of four topics: 1. Several changing volatility models are introduced and the pricing of European options is derived under these models; 2. A general local stochastic volatility model with stochastic interest rates (IR) is studied in the modelling of foreign exchange (FX) rates. The pricing of FX options under this model is examined through the use of an asymptotic expansion method, based on Watanabe-Yoshida theory. The perfect/partial hedging issues of FX options in the presence of local stochastic volatility and stochastic IRs are also considered. Finally, the impact of stochastic volatility on the pricing of FX-IR structured products (PRDCs) is examined; 3. A new method of non-biased Monte Carlo simulation for a stochastic volatility model (Heston Model) is proposed; 4. The LIBOR/swap market model with stochastic volatility and jump processes is studied, as well as the pricing of interest rate options under that model. In conclusion, some future research topics are suggested. Key words: Changing Volatility Models, Stochastic Volatility Models, Local Stochastic Volatility Models, Hedging Greeks, Jump Diffusion Models, Implied Volatility, Fourier Transform, Asymptotic Expansion, LIBOR Market Model, Monte Carlo Simulation, Saddle Point Approximation