Volatility and dividend risk in perpetual American options

American options are financial instruments that can be exercised at any time before expiration. In this paper we study the problem of pricing this kind of derivatives within a framework in which some of the properties --volatility and dividend policy-- of the underlaying stock can change at a random instant of time, but in such a way that we can forecast their final values. Under this assumption we can model actual market conditions because some of the most relevant facts that may potentially affect a firm will entail sharp predictable effects. We will analyse the consequences of this potential risk on perpetual American derivatives, a topic connected with a wide class of recurrent problems in physics: holders of American options must look for the fair price and the optimal exercise strategy at once, a typical question of free absorbing boundaries. We present explicit solutions to the most common contract specifications and derive analytical expressions concerning the mean and higher moments of the exercise time.Comment: 21 pages, 5 figures, iopart, submitted for publication; deep revision, two new appendice

Back to basics: historical option pricing revisited

We reconsider the problem of option pricing using historical probability distributions. We first discuss how the risk-minimisation scheme proposed recently is an adequate starting point under the realistic assumption that price increments are uncorrelated (but not necessarily independent) and of arbitrary probability density. We discuss in particular how, in the Gaussian limit, the Black-Scholes results are recovered, including the fact that the average return of the underlying stock disappears from the price (and the hedging strategy). We compare this theory to real option prices and find these reflect in a surprisingly accurate way the subtle statistical features of the underlying asset fluctuations.Comment: 14 pages, 2 .ps figures. Proceedings, to appear in Proc. Roy. So

Segregation of a Two Species Granular Flow

A cellular automaton model is presented for the segregation of a granular flow. The flow consists of particles of two different sizes, which in the specific industrial problem presented by Elkem are lumps of coal. It is known from experiments that these particles show different mobilities under different circumstances. This effect is incorporated in the current model via the inclusion of a 'hydrostatic' pressure term

The escape problem under stochastic volatility: the Heston model

We solve the escape problem for the Heston random diffusion model. We obtain exact expressions for the survival probability (which ammounts to solving the complete escape problem) as well as for the mean exit time. We also average the volatility in order to work out the problem for the return alone regardless volatility. We look over these results in terms of the dimensionless normal level of volatility --a ratio of the three parameters that appear in the Heston model-- and analyze their form in several assymptotic limits. Thus, for instance, we show that the mean exit time grows quadratically with large spans while for small spans the growth is systematically slower depending on the value of the normal level. We compare our results with those of the Wiener process and show that the assumption of stochastic volatility, in an apparent paradoxical way, increases survival and prolongs the escape time.Comment: 29 pages, 12 figure

Stability of central finite difference schemes for the Heston PDE

This paper deals with stability in the numerical solution of the prominent Heston partial differential equation from mathematical finance. We study the well-known central second-order finite difference discretization, which leads to large semi-discrete systems with non-normal matrices A. By employing the logarithmic spectral norm we prove practical, rigorous stability bounds. Our theoretical stability results are illustrated by ample numerical experiments

Numerical performance of penalty method for American option pricing

This paper is devoted to studying the numerical performance of a power penalty method for a linear parabolic complementarity problem arising from American option valuation. The penalized problem is a nonlinear parabolic partial differential equation (PDE). A fitted finite volume method and an implicit time-stepping scheme are used for, respectively, the spatial and time discretizations of the PDE. The rate of convergence of the penalty methods with respect to the penalty parameters is investigated both theoretically and numerically. The numerical robustness and computational effectiveness of the penalty method with respect to the market parameters are also studied and compared with those from an existing popular method, project successive over relaxation.Department of Applied Mathematic

Rendering an Account: An Open-State Archive in Postgraduate Supervision

The paper begins with a brief account of the transformation of research degree studies under the pressures of global capitalism and neo-liberal governmentality. A parallel transformation is occurring in the conduct of research through the use of information and communication technologies. Yet the potential of ICTs to shape practices of surveillance or to produce new student-supervisor relations and enhance the processes of developing the dissertation has received almost no critical attention. As doctoral supervisor and student, we then describe the features and uses of a web-based open state archive of the student's work-in-progress, developed by the student and accessible to his supervisor. Our intention was to encourage more open conversations between data and theorising, student and supervisor, and ultimately between the student and professional community. However, we recognise that relations of accountability, as these have developed within a contemporary "audit revolution" (Power, 1994, 1997) in universities, create particular "lines of visibility" (Munro, 1996). Thus while the open-state archive may help to redefine in less managerial terms notions of quality, transparency, flexibility and accountability, it might also make possible greater supervisory surveillance. How should we think about the panoptical potential of this archive? We argue that the diverse kinds of interactional patterns and pedagogical intervention it encourages help to create shifting subjectivities. Moreover, the archive itself is multiple, in bringing together an array of diverse materials that can be read in various ways, by following multiple paths. It therefore constitutes a collage, which we identify as a mode of cognition and of accounting distinct from but related to argument and narrative. As a more "open" text (Iser, 1978) it has an indeterminacy which may render it less open to abuse for the technologies of managerial accountability

Oscillation mark formation in continuous casting

A mathematical model based on lubrication was used to study the formation of notches steel that is cast from a vertically oscillating mould. The analysis is a continuation of a problem that was presented at Heriot-Watt in 1988 and Oxford in 1989

Modification terms to the Black-Scholes model in a realistic hedging strategy with discrete temporal steps

Option pricing models generally require the assumption that stock prices are described by continuous-time stochastic processes. Although the time-continuous trading is easy to conceive theoretically, it is practically impossible to execute in real markets. One reason is because real markets are not perfectly liquid and purchase or sell any amount of an asset would change the asset price drastically. A realistic hedging strategy needs to consider trading that happens at discrete instants of time. This paper focuses on the impact and effect due to temporal discretisation on the pricing partial differential equation (PDE) for European options. Two different aspects of temporal discretisation are considered and used to derive the modification or correction source terms to the continuous pricing PDE. First the finite difference discretisation of the standard Black-Scholes PDE and its modification due to discrete trading. Second the discrete trading leads to a discrete time re-balancing strategy that only cancels risks on average by using a discrete analogy of the stochastic process of the underlying asset. In both cases high order terms in the Taylor series expansion are used and the respective correction source terms are derived