This thesis develops a new framework for modelling price processes in finance, such as an
equity price or foreign exchange rate. This can be related to the conventional Ito calculus-based
framework through the time integral of a price’s squared volatility, or ‘cumulative
variance’. In the new framework, corresponding processes are strictly increasing, solve random
ordinary differential equations (ODEs), and are composed with geometric Brownian
motion. The new framework has no dependence on stochastic calculus, so processes can be
studied on a pathwise basis using probability-free ODE techniques and functional analysis.
The ODEs considered depend on continuous driving functions which are ‘spatially irregular’,
meaning they need not have any spatial regularity properties such as Holder continuity. They are however strictly increasing in time, thus temporally asymmetric. When sensible initial
values are chosen, initial value problem (IVP) solutions are also strictly increasing, and the
solution set of such IVPs is shown to contain all differentiable bijections on the non-negative
reals. This enables the modelling of any non-negative volatility path which is not zero over
intervals, via the time derivative of solutions. Despite this generality, new well-posedness
results establish the uniqueness of solutions going forwards in time. A condition is provided
which prohibits explosions, and then the IVPs’ solution map is shown to be continuous with
respect to uniform convergence over compacts.
Motivation to explore this framework comes from its connection with a time-changed Heston
volatility model. The framework shows how Heston price processes can converge to a generalisation of the normal-inverse Gaussian (NIG) Levy process, and reveals a deeper relationship between integrated Cox-Ingersoll-Ross (CIR) processes and the inverse Gaussian (IG) process. Within this framework, a ‘Riemann-Liouville-Heston’ (RLH) martingale model is
defined which generalises these relationships to fractional counterparts. This model’s implied
volatilities are simulated, and exhibit features characteristic of leading volatility models.