Paul V. Johnson

Investment Lags

In this paper we use a mixture of numerical methods including finite difference and body fitted co-ordinates to form a robust stable numerical scheme to solve the investment lag model presented in the paper by Bar-Ilan and Strange (1996). This allows us to apply our methodology to models with different stochastic processes that does not have analytic solutions.

The expected lifetime of an extraction project

When a mining company begins extraction from a finite resource, it does so in the presence of numerous uncertainties. One key uncertainty is the future price of the commodity being extracted, since a large enough drop in price can make a resource no longer cost-effective to extract, resulting in the mine being closed down. By specifying a stochastic price process, and implementing a financial-type model which leads to the use of partial differential equations, this paper creates the framework for efficiently capturing the probability of a mine remaining open throughout its planned extraction period, and derives the associated expected lifetime of extraction. An approximation to the abandonment price is described, which enables a closed-form solution to be derived for the probability of operational success and expected lifetime. This approximation compares well with the full solution obtained using a semi-Lagrangian numerical technique.

The Determination of a Dynamic Cut-Off Grade for the Mining Industry

Prior to extraction from a mine, a pit is usually divided up into 3-D ‘blocks’ which contain varying levels of estimated ore-grades. From these, the order (or ‘pathway’) of extraction is decided, and this order of extraction can remain unchanged for several years. However, because commodity prices are uncertain, once each block is extracted from the mine, the company must decide in real-time whether the ore grade is high enough to warrant processing the block further in readiness for sale, or simply to waste the block. This paper first shows how the optimal cut-off ore grade—the level below which a block should be wasted—is not simply a function of the current commodity price and the ore grade, but also a function of the ore-grades of subsequent blocks, the costs of processing, and the bounds on the rates of processing and extraction. Secondly, the paper applies a stochastic price uncertainty, and shows how to derive an efficient mathematical algorithm to calculate and operate a dynamic optimal cut-off grade criterion throughout the extraction process, allowing the mine operator to respond to future market movements. The model is applied to a real mine composed of some 60,000 blocks, and shows that an extra 10% of value can be created by implementing such an optimal regime.

Flexible decision making in the wake of large scale nuclear emergencies: Long-term response

We develop a decision-making model that describes optimal protection and recovery strategies for a single economic location affected by radioactive release from the nearby Nuclear Power Plant. The initial period of release and deposition is characterised by high degrees of uncertainty, which is likely to lead to precautionary emergency measures being carried out regardless of the actual dangers to the public, and therefore it is excluded from the optimisation problem. Instead, the analysis is performed on the timescale of weeks, months, years and decades after the accident, implying that the problem is largely deterministic if one disregards long-term economic uncertainties. It is on these longer timescales that economically-driven decisions could be made on whether or not to implement various protection and recovery measures, which include relocation, remediation, repopulation and food banning. Our model allows one to find the joint cost-minimal strategy across the set of measures, providing certain spatial and temporal flexibilities are permitted. Several qualitatively different strategies are identified, including those with no relocation and delayed remediation. Which strategy is optimal depends on the initial radiation levels, the rates and costs of the individual actions, and the preferred economic valuation of the relevant health effects associated with radiation. Our main message is that in many possible settings relocation should be used sparingly and repopulation should be delayed to exploit natural decay of the radioactive elements. These findings could provide useful recommendations to regulators in civil nuclear industry and help devise better policies for implementing emergency response and recovery measures.

Perpetual Options on Multiple Underlyings

Abstract We study three classes of perpetual option with multiple uncertainties and American-style exercise boundaries, using a partial differential equation-based approach. A combination of accurate numerical techniques and asymptotic analyses is implemented, with each approach informing and confirming the other. The first two examples we study are a put basket option and a call basket option, both involving two stochastic underlying assets, whilst the third is a (novel) class of real option linked to stochastic demand and costs (the details of the modelling for this are described in the paper). The Appendix addresses the issue of pricing American-style perpetual options involving (just) one stochastic underlying, but in which the volatility is also modelled stochastically, using the Heston (1993) framework.

Partial differential equation methods for stochastic dynamic optimization: an application to wind power generation with energy storage

A mixed financial/physical partial differential equation (PDE) can optimize the joint earnings of a single wind power generator (WPG) and a generic energy storage device (ESD). Physically, the PDE includes constraints on the ESD’s capacity, efficiency and maximum speeds of charge and discharge. There is a mean-reverting daily stochastic cycle for WPG power output. Physically, energy can only be produced or delivered at finite rates. All suppliers must commit hourly to a finite rate of delivery C, which is a continuous control variable that is changed hourly. Financially, we assume heavy ‘system balancing’ penalties in continuous time, for deviations of output rate from the commitment C. Also, the electricity spot price follows a mean-reverting stochastic cycle with a strong evening peak, when system balancing penalties also peak. Hence the economic goal of the WPG plus ESD, at each decision point, is to maximize expected net present value (NPV) of all earnings (arbitrage) minus the NPV of all expected system balancing penalties, along all financially/physically feasible future paths through state space. Given the capital costs for the various combinations of the physical parameters, the design and operating rules for a WPG plus ESD in a finite market may be jointly optimizable. This article is part of the themed issue ‘Energy management: flexibility, risk and optimization’.

SLADI: A Semi-Lagrangian Alternating-Direction Implicit Method for the Numerical Solution of Advection–Diffusion Problems with Application to Electricity Storage Valuations

In this paper, an efficient and novel methodology for numerically solving advection–diffusion problems is presented: a semi-Lagrangian approach for hyperbolic problems of advection is combined with an alternating-direction implicit method for parabolic problems involving diffusion. This is used to value a four-dimensional “storage option” (linked to storing electricity) involving three space variables and time. Efficiency is obtained by solving (only) tridiagonal systems of equations at every time step by incorporating the alternating-direction methodology. Extensive numerical experimentation indicates that the method is stable and accurate; three variants of the scheme are assessed and excellent numerical convergence can be observed. Further, a methodology for determining and results for optimal storage operation are presented.

Asymmetric synthesis of enantiomerically enriched atropisomeric amides by desymmetrisation of N,N-dialkylmesitamides

Lithiation and silylation of N,N-dialkylmesitamides using chiral lithium amide bases leads to enantiomerically enriched atropisomeric amides (up to 89% ee) by desymmetrisation of the enantiotopic methyl groups.

Detecting changes in real-time data: a user’s guide to optimal detection

The real-time detection of changes in a noisily observed signal is an important problem in applied science and engineering. The study of parametric optimal detection theory began in the 1930s, motivated by applications in production and defence. Today this theory, which aims to minimize a given measure of detection delay under accuracy constraints, finds applications in domains including radar, sonar, seismic activity, global positioning, psychological testing, quality control, communications and power systems engineering. This paper reviews developments in optimal detection theory and sequential analysis, including sequential hypothesis testing and change-point detection, in both Bayesian and classical (non-Bayesian) settings. For clarity of exposition, we work in discrete time and provide a brief discussion of the continuous time setting, including recent developments using stochastic calculus. Different measures of detection delay are presented, together with the corresponding optimal solutions. We emphasize the important role of the signal-to-noise ratio and discuss both the underlying assumptions and some typical applications for each formulation. This article is part of the themed issue ‘Energy management: flexibility, risk and optimization’.

Optimal costless extraction rate changes from a non-renewable resource

This paper considers the role of costless decisions relating to the extraction of a non-renewable resource in the presence of uncertainty. We begin by deriving a size scale of the extractable resource, above which the solution to the valuation and optimal control strategy can be described by analytic solutions; we produce solutions for a general form of operating cost function. Below this critical resource size level the valuation and optimal control strategy must be solved by numerical means; we present a robust numerical algorithm that can solve such a class of problem. We also allow for the embedding of an irreversible investment decision (abandonment) into the optimisation. Finally, we conduct experimentation for each of these two approaches (analytical and numerical), and show how they are consistent with one another when used appropriately. The extensions of this papers techniques to renewable resources are explored.