Strict local martingales in continuous financial market models

University of Technology, Sydney. Faculty of Business.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)

Weak tail conditions for local martingales

© Institute of Mathematical Statistics, 2019. The following conditions are necessary and jointly sufficient for an arbitrary càdlàg local martingale to be a uniformly integrable martingale: (A) The weak tail of the supremum of its modulus is zero; (B) its jumps at the first-exit times from compact intervals converge to zero in L 1 on the events that those times are finite; and (C) its almost sure limit is an integrable random variable

Laplace transform identities for diffusions, with applications to rebates and barrier options

Using a simple integral identity, we derive general expressions for the Laplace transform of the transition density of the process, if killing or reflecting boundaries are specified. We also obtain a number of useful expressions for the Laplace transforms of some functions of first-passage times for the diffusion. These results are applied to the special case of squared Bessel processes with killing or reflecting boundaries. In particular, we demonstrate how the above-mentioned integral identity enables us to derive the transition density of a squared Bessel process killed at the origin, without the need to invert a Laplace transform. Finally, as an application, we consider the problem of pricing barrier options on an index described by the minimal market model

Quadratic hedging of basis risk

Research Paper Number: 225 Abstract: This paper examines a simple basis risk model based on correlated geometric Brownian motions. We apply quadratic criteria to minimize basis risk and hedge in an optimal manner. Initially, we derive the Follmer-Schweizer decomposition of a European claim. This allows pricing and hedging under the minimal martingale measure, corresponding to the local risk-minimizing strategy. Furthermore, since the mean-variance tradeoff process is deterministic in our setup, the minimal martingale- and variance-optimal martingale measures coincide. Consequently, the mean-variance optimal strategy is easily constructed. Simple closed-form pricing and hedging formulae for put and call options are derived. Due to market incompleteness, these formulae depend on the drift parameters of the processes. By making a further equilibrium assumption, we derive an approximate hedging formula, which does not require knowledge of these parameters. The hedging strategies are tested using Monte Carlo experiments, and are compared with recent results achieved using a utility maximization approach

Optimal prediction of the last-passage time of a transient diffusion

© 2014 Society for Industrial and Applied Mathematics We identify the integrable stopping time τ∗ with minimal L1-distance from the last-passage time γz associated with a given level z > 0, for an arbitrary nonnegative time-homogeneous transient diffusion X . We demonstrate that τ∗ is in fact the first time that X assumes a value outside a half-open interval [0, r∗). The upper boundary r∗ > z of this interval is characterized either as the solution for a one-dimensional optimization problem, or as part of the solution for a free-boundary problem. A number of concrete examples illustrate the result

A visual criterion for identifying Ito diffusions as martingales or strict local martingales

It is often important, in applications of stochastic calculus to financial modelling, to know whether a given local martingale is a martingale or a strict local martingale. We address this problem in the context of a time-homogenous diffusion process with a finite lower boundary, presented as the solution of a driftless stochastic differential equation. Our main theorem demonstrates that the question of whether or not this process is a martingale may be decided simply by examining the slope of a certain increasing function. Further results establish the connection between our theorem and other results in the literature, while a number of examples are provided to illustrate the use of our criterion

The effect of prolonged simulated non- gravitational environment on mineral balance in the adult male, volume 1 Final report

Effect of prolonged bed rest with simulated weightlessness on mineral balance in male adult - Vol.

Prevention of bone mineral changes induced by bed rest: Modification by static compression simulating weight bearing, combined supplementation of oral calcium and phosphate, calcitonin injections, oscillating compression, the oral diophosphonatedisodium etidronate, and lower body negative pressure

The phenomenon of calcium loss during bed rest was found to be analogous to the loss of bone material which occurs in the hypogravic environment of space flight. Ways of preventing this occurrence are investigated. A group of healthy adult males underwent 24-30 weeks of continuous bed rest. Some of them were given an exercise program designed to resemble normal ambulatory activity; another subgroup was fed supplemental potassium phosphate. The results from a 12-week period of treatment were compared with those untreated bed rest periods. The potassium phosphate supplements prevented the hypercalciuria of bed rest, but fecal calcium tended to increase. The exercise program did not diminish the negative calcium balance. Neither treatment affected the heavy loss of mineral from the calcaneus. Several additional studies are developed to examine the problem further

The effects of remodeling with heart failure on mode of initiation of ventricular fibrillation and its spatiotemporal organization

Purpose The effect of the heart failure substrate on the initiation of ventricular fibrillation (VF) and its resulting mechanism is not known. The objective of this study was to determine the effects of substrate on VF initiation and its spatiotemporal organization in the heart failure model. Methods Optical action potentials were recorded from LV wedge preparations either from structurally normal hearts (control, n = 11) or from congestive heart failure (CHF; n = 7), at the epicardial surface, endocardial surface which included a papillary muscle, and a transmural cross section. Action potential duration (APD80) was determined, and VF was initiated. A fast Fourier transform was calculated, and the dominant frequency (DF) was determined. Results The CHF group showed increased VF vulnerability (69 vs 26 %, p < 0.03), and every mapped surface showed an APD80 gradient which included islands of higher APDs on the transmural surface (M cells) which was not observed in controls. VF in the CHF group was characterized by stable, discrete, high-DF areas that correlated to either foci or spiral waves located on the transmural surface at the site of the papillary muscle. Overall, the top 10 % of DFs correlated to an APD of 101 ms while the bottom 10 % of DFs correlated to an APD of 126 ms (p < 0.01). Conclusions In the CHF model, APD gradients correlated with an increased vulnerability to VF, and the highest stable DFs were located on the transmural surface which was not seen in controls. This indicates that the CHF substrate creates unique APD and DF characteristics

On arbitrages arising from honest times

In the context of a general continuous financial market model, we study whether the additional information associated with an honest time gives rise to arbitrage profits. By relying on the theory of progressive enlargement of filtrations, we explicitly show that no kind of arbitrage profit can ever be realised strictly before an honest time, while classical arbitrage opportunities can be realised exactly at an honest time as well as after an honest time. Moreover, stronger arbitrages of the first kind can only be obtained by trading as soon as an honest time occurs. We carefully study the behavior of local martingale deflators and consider no-arbitrage-type conditions weaker than NFLVR.Comment: 25 pages, revised versio