Static and dynamic modelling of distributions in finance and actuarial science

Abstract

In the last several decades, most of the research in financial and actuarial modelling concentrated on dynamically describing the time evolution of price levels of financial assets, based on which pricing, risk management, portfolio optimization, premium calculation and ruin analysis are conducted. In this thesis, distributions of financial assets and other random variables of interest are modelled to solve financial and actuarial problems. Modelling distributions gives us much more freedom for calculation and analysis, but correspondingly puts more constraints to the mathematical formulations of models.

The first part of the thesis shows a static method called SVM-Jacobi to approximate probability distributions by linear combinations of exponential distributions, associated with a comprehensive asymptotic analysis. In multivariate cases, the method also effectively works to provide approximations by linear combinations of products of independent exponential distributions. The proposed statistical method is particularly applicable and useful in quantitative finance and actuarial science, especially in modeling random time, like default time and remaining lifetime. Many pricing and hedging formulas have closed forms under exponential distributions. By approximating real-world distributions and implied distributions, it is feasible to use the closed-form formulas and fitted coefficients of SVM-Jacobi to approximate the prices and Greeks. In addition to the methodology and theoretical analysis, the first part of the thesis gives examples of pricing defaultable bonds, European options, credit default swaps and equity-linked death benefits, and calculating credit value adjustment of credit default swaps. Some numerical results also are presented for illustration.

In the second part, it shows a dynamic risk-neutral forward density model using Gaussian random fields to capture different aspects of market information from European options and volatility derivatives of a market index. The well-structured model is built in the framework of Heath–Jarrow–Morton philosophy and the Musiela parametrization with a user-friendly arbitrage-free condition. It reduces to the popular Geometric Brownian Model for the spot price of the market index and can be intuitively visualized to have a better view of the market trend. In addition, theorems are developed to show how our model drives local volatility and variance swap rates. Hence volatility futures and options can be priced taking the forward density implied by European options as the initialization input. And our model can be accordingly calibrated to the market prices of these volatility derivatives. An efficient algorithm is also developed for both simulating and pricing. And a simulation study is conducted with the market data.

The two techniques thoroughly cover two different aspects of modelling distributions and fully illustrate the effectiveness in finance and actuarial science.