Huw Rhys

The Timing of Real Option Exercise: Some Recent Developments

ABSTRACT This paper briefly reviews one recent development in real options modeling. This development aims to extend traditional real options models. We show that previous models are incomplete in that only the optimal action threshold is derived. The new literature aims to provide, further information on optimal option exercise. By introducing the first passage time approach, empirically testable propositions on the probability of option exercise and the expected waiting time before an option is exercised are provided.

On the ‘Steady State’ Properties of Financial Ratios

Abstract This paper uses the methods of continuous time stochastic calculus to investigate the ‘steady state’ properties of financial ratios. Basing our analysis on previous work in the area, we show that, if a financial ratio can be characterised as a diffusion process which possesses an asymptotic equilibrium, then the Fokker-Kolmogorov-Planck forward equation may be used to ‘retrieve’ its probability density. The approach is ‘flexible’ enough to incorporate a wide variety of density functions, many of which have not been investigated in the literature. We demonstrate the procedures which may be used to derive both the cross-sectional and time series tests implied by these distributions. The paper also includes a section dealing with the methods which may be used for parameter estimation, once the underlying distribution has been determined.

Non‐linear equity valuation

Abstract We incorporate a real option component into the Ohlson (1995) equity valuation model and then use this augmented model to make assessments about the form and nature of the systematic biases that are likely to arise when empirical work is based on linear models of the relationship between the market value of equity and its determining variables. We also demonstrate how one can expand equity valuation models in terms of an infinite series of ‘orthogonal’ polynomials and thereby determine the relative contribution which the linear and non‐linear components of the relationship between equity value and its determining variables make to overall equity value. This procedure shows that non‐linearities in equity valuation can be large and significant, particularly for firms with low earnings‐to‐book ratios or where the undeflated book value of equity is comparatively small. Moreover, it is highly unlikely the simple linear models that characterise this area of accounting research can form the basis of meaningful statistical tests of the relationship between equity value and its determining variables.

Student's distribution and the value of real options

Publisher Summary This chapter presents a closed-form solution for the value of the option to invest in a capital project when the option to do so has a finite life. In doing so, it is assumed that the capital projects net present value evolves in terms of the student distributions that exhibit the fat tail properties characterizing at least some of the empirical distributions of research and development. However, most analytical work conducted in this area assumes that the option to undertake an investment project has an infinite life. Unfortunately, many real-life investment opportunities are not infinitely lived but expire and become worthless at a known point in time. This analysis provides an explicit closed-form solution for the valuation of finite-lived derivative securities of this kind. Closed-form solutions are however, notoriously difficult to come by. Yet, despite the difficulties associated with obtaining closed-form solutions, there is one variation to the analysis that is worthy of further investigation. This stems from the fact that many asset prices appear to evolve in terms of distributions that exhibit not only fat tails but also significant skewness.

The Friedman rule and inflation targeting

We use concepts from the financial economics discipline – and in particular the methods of continuous time finance – to develop a monetarist model under which the rate of inflation evolves in terms of a first-order mean reversion process based on a ‘white noise’ error structure. The Fokker–Planck (i.e. the Chapman–Kolmogorov) equation is then invoked to retrieve the steady-state (i.e. unconditional) probability distribution for the rate of inflation. Monthly data for the UK Consumer Price Index (CPI) covering the period from 1988 until 2012 are then used to estimate the parameters of the probability distribution for the UK inflation rate. The parameter estimates are compatible with the hypothesis that the UK inflation rate evolves in terms of a slightly skewed and highly leptokurtic probability distribution that encompasses non-convergent higher moments. We then determine the Hamilton–Jacobi–Bellman fundamental equation of optimality corresponding to a monetary policy loss function defined in terms of the squared difference between the targeted rate of inflation and the actual inflation rate. Optimising and then solving the Hamilton–Jacobi–Bellman equation shows that the optimal control for the rate of increase in the money supply will be a linear function of the difference between the current rate of inflation and the targeted inflation rate. The conditions under which the optimal control will lead to the Friedman rule are then determined. These conditions are used in conjunction with the Fokker–Planck equation and the mean reversion process describing the evolution of the inflation rate to determine the probability distribution for the inflation rate under the Friedman rule. This shows that whilst the empirically determined probability distribution for the UK inflation rate meets some of the conditions required for the application of the Friedman rule, it does not meet them all.

Stochastic Depreciation and Optimal Consumption-Investment Decisions

Capital accumulation is viewed as the net outcome of accumulation and wasting. Accumulation is modelled by a Gaussian process whilst wasting is generated by a continuous time and continuous state-space pseudo-Poisson process. These assumptions imply that an economic agents preferences, consumption, capital accumulation and wastage are governed by a form of Hamilton-Jacobi-Bellman equation containing an infinite number of terms. We demonstrate the optimisation procedures applicable to consumption and the expected wastage of capital and in the process show that isoelastic preferences are consistent with the model. Copyright Blackwell Publishers Ltd 2002.