Modular average case analysis: Language implementation and extension

Motivated by accurate average-case analysis, MOdular Quantitative Analysis (MOQA) is developed at the Centre for Efficiency Oriented Languages (CEOL). In essence, MOQA allows the programmer to determine the average running time of a broad class of programmes directly from the code in a (semi-)automated way. The MOQA approach has the property of randomness preservation which means that applying any operation to a random structure, results in an output isomorphic to one or more random structures, which is key to systematic timing. Based on original MOQA research, we discuss the design and implementation of a new domain specific scripting language based on randomness preserving operations and random structures. It is designed to facilitate compositional timing by systematically tracking the distributions of inputs and outputs. The notion of a labelled partial order (LPO) is the basic data type in the language. The programmer uses built-in MOQA operations together with restricted control flow statements to design MOQA programs. This MOQA language is formally specified both syntactically and semantically in this thesis. A practical language interpreter implementation is provided and discussed. By analysing new algorithms and data restructuring operations, we demonstrate the wide applicability of the MOQA approach. Also we extend MOQA theory to a number of other domains besides average-case analysis. We show the strong connection between MOQA and parallel computing, reversible computing and data entropy analysis.

A study of the Hobson and Rogers volatility model

We firstly examine the model of Hobson and Rogers for the volatility of a financial asset such as a stock or share. The main feature of this model is the specification of volatility in terms of past price returns. The volatility process and the underlying price process share the same source of randomness and so the model is said to be complete. Complete models are advantageous as they allow a unique, preference independent price for options on the underlying price process. One of the main objectives of the model is to reproduce the `smiles' and `skews' seen in the market implied volatilities and this model produces the desired effect. In the first main piece of work we numerically calibrate the model of Hobson and Rogers for comparison with existing literature. We also develop parameter estimation methods based on the calibration of a GARCH model. We examine alternative specifications of the volatility and show an improvement of model fit to market data based on these specifications. We also show how to process market data in order to take account of inter-day movements in the volatility surface. In the second piece of work, we extend the Hobson and Rogers model in a way that better reflects market structure. We extend the model to take into account both first and second order effects. We derive and numerically solve the pde which describes the price of options under this extended model. We show that this extension allows for a better fit to the market data. Finally, we analyse the parameters of this extended model in order to understand intuitively the role of these parameters in the volatility surface.