  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.
  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 NIG Levy process, and reveals
a deeper relationship between integrated CIR processes and the IG process.
Within this framework, a `Riemann-Liouville-Heston' 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.
