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

Logical presentations of domains

Bibliography: pages 168-174.This thesis combines a fairly general overview of domain theory with a detailed examination of recent work which establishes a connection between domain theory and logic. To start with, the theory of domains is developed with such issues as the semantics of recursion and iteration; the solution of recursive domain equations; and non-determinism in mind. In this way, a reasonably comprehensive account of domains, as ordered sets, is given. The topological dimension of domain theory is then revealed, and the logical insights gained by regarding domains as topological spaces are emphasised. These logical insights are further reinforced by an examination of pointless topology and Stone duality. A few of the more prominent categories of domains are surveyed, and Stone-type dualities for the objects of some of these categories are presented. The above dualities are then applied to the task of presenting domains as logical theories. Two types of logical theory are considered, namely axiomatic systems, and Gentzen-style deductive systems. The way in which these theories describe domains is by capturing the relationships between the open subsets of domains

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.

Means-Tested Income Support, Portfolio Choice and Decumulation in Retirement

We investigate the impact of means tested public income transfers on post-retirement decumulation and portfolio choice using theoretical simulations and panel data on Australian Age Pensioners. Means tested public pension payments in Australia have broad coverage and give insight into the incentive responsiveness of well-off, as well as poorer households. Via numerical solutions to a discrete time, finite horizon dynamic programming problem, we simulate the optimal consumption and portfolio allocation strategies for a retired household subject to assets and income tests. Relative to benchmark, means tested households should optimally decumulate faster early in retirement, and choose more risky portfolios. Panel data tests on inferred wealth for pensioner households show evidence of more rapid spending early in retirement. However they also show that better-off households continue to accumulate, even when facing a steeper implicit tax rate on wealth than applies to poorer households. Wealthier households also hold riskier portfolios. Results from tests for Lorenz dominance of the panel wealth distribution show no decrease in wealth inequality over the five years of the study.retirement wealth; life-cycle saving; public pension; portfolio choice

How suitable are equity release mortgages as investments for pension funds?

This article examines the claim that equity release mortgages, the U.K. equivalent of reverse mortgages in the U.S., are suitable investments for pension funds. We present valuation, stress test and scenario analysis results that suggest that equity release mortgages are unsuitable for pension funds because: (i) they bear returns that are typically below the risk-free rate; (ii) they are not hedges for annuity books, let alone good hedges; and (iii) they are heavily exposed to house price risk, which annuity books are not. Our results suggest that equity release mortgages meet none of these criteria to be suitable for pension funds and are almost entirely dominated by risk-free government bonds. We offer an explanation for why investors appear to be unaware of the low returns on equity release mortgages