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Analytic solutions are found for prices of variance and volatility swaps under a new time-dependent stochastic model for the dynamics of variance. The main features of the new stochastic differential equation are (1) an empirically validated cν3/2 diffusion term and (2) a free function of time as a moving target in a reversion term, allowing additional flexibility for model calibration against market data.

[extract] In recent years the Australian National Electricity Market (the NEM) has been undergoing the transition to a fully deregulated marketplace. In December 1998, for the first time, the wholesale price of electricity was subject to market forces. The NEM includes the Queensland, New South Wales, Victorian, South australian, Tasmanian and Australian Capital Territory electricity markets. Almost 8 million end users are supplied by the world's longest interconnected power system, and the NEM trades up to $7 billion of electricity annually.

In this thesis, we establish the range of applicability of Lie's classical symmetry method, and various of its generalisations, in constucting new exact solutions to topical nonlinear partial differential equations, including reaction-diffusion equations, boundary layer equations and the poorly understood degenerate nonlinear diffusion equations. In addition, some established ad-hoc methods of nonlinear superposition and equation splitting are related to properties of Lie symmetry algebras. These equation solving methods are thereby incorporated in systematic symmetry-finding algorithms. Lie symmetry analysis of degenerate diffusion equations, in which the diffusivity depends on both concentration and concentration gradient, uncovers an interesting class of integrable equations. Using both linear transform methods and separation of variables, these nonlinear equations are solved subject to initial and boundary conditions on both infinite and finite domains. Solutions on the finite domain evolve towards a discontinuous jump. Also, relevant exact solutions are constructed for nonintegrable models. These solutions include the possibility of strong degeneracy with a step initial condition remaining discontinuous for a finite time. Every solution of a linear equation with constant coefficients is invariant under some classical symmetry. Strictly nonclassical symmetries are rare for nonlinear diffusion equations, a fact which is made evident by proving the equivalence of the nonclassical symmetry determining equations to their classical counterparts. However, new nonclassical solutions are constructed for 2+1-dimensional reaction-diffusion equations, including a case with an Arrhenius reaction term. The higher order symmetry method of generalised conditional symmetries is carried out on a class of degenerate diffusion equations with and without reaction terms. Recently discovered new solutions to the boundary layer equations are shown to follow from classical symmetry red