A Visual Classification of Local Martingales

This paper considers the problem of when a local martingale is a martingale or a universally integrable martingale, for the case of time-homogeneous scalar diffusions. Necessary and suffcient conditions of a geometric nature are obtained for answering this question. These results are widely applicable to problems in stochastic finance. For example, in order to apply risk-neutral pricing, one must first check that the chosen density process for an equivalent change of probability measure is in fact a martingale. If not, risk-neutral pricing is infeasible. Furthermore, even if the density process is a martingale, the possibility remains that the discounted price of some security could be a strict local martingale under the equivalent risk-neutral probability measure. In this case, well-known identities for option prices, such as put-call parity, may fail. Using our results, we examine a number of basic asset price models, and identify those that suffer from the above-mentioned difficulties.diffusions; first-passage times; Laplace transforms; local martingales; ordinary differential equations

Quadratic Hedging of Basis Risk

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.Option hedging; incomplete markets; basis risk; local risk minimization; mean-variance hedging

Benchmarking and Fair Pricing Applied to Two Market Models

This paper considers a market containing both continuous and discrete noise. Modest assumptions ensure the existence of a growth optimal portfolio. Non-negative self-financing trading strategies, when benchmarked by this portfolio, are local martingales under the real-world measure. This justifies the fair pricing approach, which expresses derivative prices in terms of real-world conditional expectations of benchmarked payoffs. Two models for benchmarked primary security accounts are presented, and fair pricing formulas for some common contingent claims are derived.growth optimal portfolio; benchmark approach; fair pricing; Merton model; minimal market model

Three-Dimensional Brownian Motion and the Golden Ratio Rule

Let X =(Xt)t=0 be a transient diffusion processin (0,8) with the diffusion coeffcient s> 0 and the scale function L such that Xt ?8 as t ?8 ,let It denote its running minimum for t = 0, and let ? denote the time of its ultimate minimum I8 .Setting c(i,x)=1-2L(x)/L(i) we show that the stopping time minimises E(|? - t|- ?) over all stopping times t of X (with finite mean) where the optimal boundary f* can be characterised as the minimal solution to staying strictly above the curve h(i)= L-1(L(i)/2) for i > 0. In particular, when X is the radial part of three-dimensional Brownian motion, we find that where ? =(1+v5)/2=1.61 ... is the golden ratio. The derived results are applied to problems of optimal trading in the presence of bubbles where we show that the golden ratio rule offers a rigourous optimality argument for the choice of the well known golden retracement in technical analysis of asset prices.

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)