Motivated by successes of fast reverting volatility models, and the implicit dependence of `rough' processes on infinitesimal reversionary timescales, we establish a pathwise volatility framework which leads to a complete understanding of volatility trajectories' behaviour in the limit of infinitely-fast reversion. Towards this, we first establish processes that are weakly equivalent to Cox-Ingersoll-Ross (CIR) processes, but in contrast prove well-defined without reference to a probability measure. This provides an unusual example of Skorokhod's representation theorem. In particular, we become able to generalise Heston's model of volatility to an arbitrary degree, by sampling drivers \omega under any probability measure; a rough one if so desired. Our main analysisrelates to separable initial-value problems of type x'= \omega(x) + t - x + 1,%DIFDELCMD < \quad %%% x(0)=0, with \omega only assumed continuous (not Lipschitz, nor H\"older), solutions of which \varphi correspond to time-averages of volatility trajectories \varphi'. Such solutions are shown to exist, be unique and bijective for any \omega, essentially placing no constraints on corresponding volatility trajectories, except for their non-negativity. After bounding these solutions in time , we prove a rare type of convergence result, towards c\`adl\`ag exit-time limits, on Skorokhod' s M_1 topology. One immediate corollary of this limiting result is a weak connection between the time-averaged CIR process and  the inverse-Gaussian L\'evy subordinator .
%DIFDELCMD < \quad %%% 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. In the new framework, corresponding processes are strictly increasing, solve random ODEs, and are composed with geometric Brownian motion to obtain price processes. 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, IVP solutions are also strictly increasing, and the IVPs' solution set 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, and continuity of the IVPs' solution map. Motivation to explore this framework comes from its connection with the Heston volatility model. The framework explains how Heston price processes can converge to an interval-valued generalisation of the NIG Levy process, and reveals a deeper relationship between integrated CIR processes and the IG Levy process. Within this framework, a `Riemann-Liouville-Heston' martingale model is defined which generalises these relationships to fractional counterparts. Implied volatilities from this model are simulated, and exhibit features characteristic of leading `rough' volatility models .
D
Motivated by successes of fast reverting volatility models, and the implicit dependence of `rough' processes on infinitesimal reversionary timescales, we establish a pathwise volatility framework which leads to a complete understanding of volatility trajectories' behaviour in the limit of infinitely-fast reversion. Towards this, we first establish processes that are weakly equivalent to Cox-Ingersoll-Ross (CIR) processes, but in contrast prove well-defined without reference to a probability measure. This provides an unusual example of Skorokhod's representation theorem. In particular, we become able to generalise Heston's model of volatility to an arbitrary degree, by sampling drivers \omega under any probability measure; a rough one if so desired. Our main analysisrelates to separable initial-value problems of type x'= \omega(x) + t - x + 1,
R
x(0)=0, with \omega only assumed continuous (not Lipschitz, nor H\"older), solutions of which \varphi correspond to time-averages of volatility trajectories \varphi'. Such solutions are shown to exist, be unique and bijective for any \omega, essentially placing no constraints on corresponding volatility trajectories, except for their non-negativity. After bounding these solutions
This thesis develops a new framework for modelling price processes
R
time
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. In the new framework, corresponding processes are strictly increasing, solve random ODEs, and are composed with geometric Brownian motion to obtain price processes. 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'
R
we prove a rare type of convergence result, towards c\`adl\`ag exit-time limits, on Skorokhod' s M_1 topology. One immediate corollary of this limiting result is a weak connection between the time-averaged CIR process
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, IVP solutions are also strictly increasing,
A
the IVPs' solution set is shown to contain all differentiable bijections on
R
inverse-Gaussian L\'evy subordinator
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, and continuity of the IVPs' solution map. Motivation to explore this framework comes from its connection with the Heston volatility model. The framework explains how Heston price processes can converge to an interval-valued generalisation of the NIG Levy process, and reveals a deeper relationship between integrated CIR processes and the IG Levy process. Within this framework, a `Riemann-Liouville-Heston' martingale model is defined which generalises these relationships to fractional counterparts. Implied volatilities from this model are simulated, and exhibit features characteristic of leading `rough' volatility models