Teaching an Old Dog New Tricks: Improved Estimation of the Parameters of Stochastic Differential Equations by Numerical Solution of the Fokker-Planck Equation

Many stochastic differential equations (SDEs) do not have readily available closed-form expressions for their transitional probability density functions (PDFs). As a result, a large number of competing estimation approaches have been proposed in order to obtain maximum-likelihood estimates of their parameters. Arguably the most straightforward of these is one in which the required estimates of the transitional PDF are obtained by numerical solution of the Fokker-Planck(or forward-Kolmogorov) partial differential equation. Despite the fact that this method produces accurate estimates and is completely generic, it has not proved popular in the applied literature. Perhaps this is attributable to the fact that this approach requires repeated solution of a parabolic partial differential equation to obtain the transitional PDF and is therefore computationally quite expensive. In this paper, three avenues for improving the reliability and speed of this estimation method are introduced and explored in the context of estimating the parameters of the popular Cox-Ingersoll-Ross and Ornstein-Uhlenbeck models. The recommended algorithm that emerges from this investigation is seen to offer substantial gains in reliability and computational time.stochastic di®erential equations, maximum likelihood, ¯nite di®erence, ¯nite element, cumulative

Developing analytical distributions for temperature indices for the purposes of pricing temperature-based weather derivatives

Temperature-based weather derivatives are written on an index which is normally defined to be a nonlinear function of average daily temperatures. Recent empirical work has demonstrated the usefulness of simple time-series models of temperature for estimating the payoffs to these instruments. This paper develops analytical distributions of temperature indices on which temperature derivatives are written. If deviations of daily temperature from its expected value is modelled as an Ornstein-Uhlenbeck process with time-varying variance, then the distributions of the temperature index on which the derivative is written is the sum of truncated, correlated Gaussian deviates. The key result of this paper is to provide an analytical approximation to the distribution of this sum, thus allowing the accurate computation of payoffs without the need for any simulation. A data set comprising average daily temperature spanning over a hundred years for four Australian cities is used to demonstrate the efficacy of this approach for estimating the payoffs to temperature derivatives. It is demonstrated that expected payoffs computed directly from historical records is a particulary poor approach to the problem when there are trends in underlying average daily temperature. It is shown that the proposed analytical approach is superior to historical pricing.Weather Derivatives, Temperature Models, Cooling Degree Days, Maximum Likelihood Estimation, Distribution for Correlated Variables

Estimating the Payoffs of Temperature-based Weather Derivatives

Temperature-based weather derivatives are written on an index which is normally defined to be a nonlinear function of average daily temperatures. Recent empirical work has demonstrated the usefulness of simple time-series models of temperature for estimating the payoffs to these instruments. This paper argues that a more direct and parsimonious approach is to model the time-series behaviour of the index itself, provided a sufficiently rich supply of historical data is available. A data set comprising average daily temperature spanning over a hundred years for four Australian cities is assembled. The data is then used to compare the actual payoffs of temperature-based European call options with the expected payoffs computed from historical temperature records and two time-series approaches. It is concluded that expected payoffs computed directly from historical records perform poorly by comparison with the expected payoffs generated by means of competing time-series models. It is also found that modeling the relevant temperature index directly is superior to modeling average daily temperatures.Temperature, Weather Derivatives, Cooling Degree Days, Time-series Models.

The Devil is in the Detail: Hints for Practical Optimisation

Finding the minimum of an objective function, such as a least squares or negative log-likelihood function, with respect to the unknown model parameters is a problem often encountered in econometrics. Consequently, students of econometrics and applied econometricians are usually well-grounded in the broad differences between the numerical procedures employed to solve these problems. Often, however, relatively little time is given to understanding the practical subtleties of implementing these schemes when faced with illbehaved problems. This paper addresses some of the details involved in practical optimisation, such as dealing with constraints on the parameters, specifying starting values, termination criteria and analytical gradients, and illustrates some of the general ideas with several instructive examples.gradient algorithms, unconstrained optimisation, generalised method of moments.

Testing for Time Dependence in Parameters

This paper proposes a new test based on a Fourier series expansion to approximate the unknown functional form of a nonlinear time-series model. The test specifically allows for structural breaks, seasonal parameters and time-varying parameters. The test is shown to have evry good size and power properties. However, it is not especially good in detecting nonlinearity in variables. As such, the test can help determine whether an observed rejection of the joint null hypothesis of linearity and time invariant parameters is due to time-varying coefficients of a nonliearity in variables.time varying parameters; fourier-series; nuisance parameters

Discrete time-series models when counts are unobservable

Count data in economics have traditionally been modeled by means of integer-valued autoregressive models. Consequently, the estimation of the parameters of these models and their asymptotic properties have been well documented in the literature. The models comprise a description of the survival of counts generally in terms of a binomial thinning process and an independent arrivals process usually specified in terms of a Poisson distribution. This paper extends the existing class of models to encompass situations in which counts are latent and all that is observed is the presence or absence of counts. This is a potentially important modification as many interesting economic phenomena may have a natural interpretation as a series of 'events' that are driven by an underlying count process which is unobserved. Arrivals of the latent counts are modeled either in terms of the Poisson distribution, where multiple counts may arrive in the sampling interval, or in terms of the Bernoulli distribution, where only one new arrival is allowed in the same sampling interval. The models with latent counts are then applied in two practical illustrations, namely, modeling volatility in financial markets as a function of unobservable 'news' and abnormal price spikes in electricity markets being driven by latent 'stress'.Integer-valued autoregression, Poisson distribution, Bernoulli distribution, latent factors, maximum likelihood estimation

Care as an alternative to killing? Reconceptualising veterinary end of life care for animals

This is the author accepted manuscript. The final version is available from Wiley via the DOI in this record.Palliative care is routinely offered to humans in the UK, while euthanasia remains illegal. The converse is true for nonhuman animals (henceforth animals). Indeed, euthanasia is widely accepted as the appropriate course of action for “suffering” animals, and for those whose behaviours or suspected ill health are thought to pose a threat to others. This article details examples of nonhuman death at a multi-faith ashram whose members vehemently oppose all forms of killing on religious grounds. Through exploring their efforts in palliative care for animals, and their emphasis on natural death as a means of respecting the sanctity of life, the practical, emotional and theoretical viability of caring for, instead of killing, other animals at the ends of their lives is considered. In the process, normative distinctions between different categories of animals, (including humans), and different approaches to end of life care (palliative care, euthanasia, natural death) are called into question. Indeed, paying mindful attention to the diverse ways in which individual animals are cared for as they die reveals the potential violence inherent in both palliative care leading to natural death, and euthanasia, blurring perceptions of good and bad death in both veterinary and human medicine

Momentum in Australian Stock Returns: An Update

It has been documented that a momentum investment strategy based on buying past well performing stocks while selling past losing stocks, is a profitable one in the Australian context particularly in the 1990s. The aim of this short paper is to investigate whether or not this feature of Australian stock returns is still evident. The paper confirms the presence of a medium-term momentum effect, but also provides some interesting new evidence on the importance of the size effect on momentum.Stock returns, Momentum portfolios, Size effect