Strict local martingales in continuous financial market models
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It is becoming increasingly clear that strict local martingales play a distinctive and important role in stochastic finance. This thesis presents a detailed study of the effects of strict local martingales on financial modelling and contingent claim valuation, with the explicit aim of demonstrating that some of the apparently strange features associated with these processes are in fact quite intuitive, if they are given proper consideration. The original contributions of the thesis may be divided into two parts, the first of which is concerned with the classical probability-theoretic problem of deciding whether a given local martingale is a uniformly integrable martingale, a martingale, or a strict local martingale. With respect to this problem, we obtain interesting results for general local martingales and for local martingales that take the form of time-homogeneous diffusions in natural scale. The second area of contribution of the thesis is concerned with the impact of strict local martingales on stochastic finance. We identify two ways in which strict local martingales may appear in asset price models: Firstly, the density process for a putative equivalent risk-neutral probability measure may be a strict local martingale. Secondly, even if the density process is a martingale, the discounted price of some risky asset may be a strict local martingale under the resulting equivalent risk-neutral probability measure. The minimal market model is studied as an example of the first situation, while the constant elasticity of variance model gives rise to the second situation (for a particular choice of parameter values).
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