Schofield, MJ; (2015) Price feedback and hybrid diffusions in finance. Doctoral thesis , UCL (University College London).

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Abstract

It is well-known that the probabilistic behaviour of financial asset returns is not captured well by the classical Black-Scholes model. The true behaviour will never be perfectly captured in any model, but insight is continually being obtained into our understanding of more sophisticated and realistic models. Much research has been published recently exploring the use of L\'{e}vy process models, which maintain the original \emph{independent stationary increments} assumption present in the Black-Scholes model, but incorporate jumps in the modelling. This investigation seeks to motivate a new class of models, throwing out the stationary increments hypothesis. We argue that certain techniques of trading decision-making are not independent of previous price movements, and the returns, being driven by the trade order flow, will reflect that. From here, we develop two particular such models, which are both diffusion models, and study them for their probabilistic behaviour. The first of these models is a hybrid of the arithmetic and geometric Brownian motions, which has transition probabilities expressible in terms of a spectral expansion involving Legendre functions. The second is a hybrid of the arithmetic Brownian motion and the Cox-Ingersoll-Ross process, and its spectral expansions involve the confluent hypergeometric functions. Having developed these expressions in sufficient detail to do so, we consider the calculation of value-at-risk and expected shortfall in these two models.

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