[...]... Engineering Stanford University Stanford, CA 94305-4026 Hakan Er Department of Business Administration Akdeniz University, Turkey Alma Garcia Department of Computer Science University of Essex, United Kingdom Kay Giesecke Department of Management Science & Engineering Stanford University Stanford, CA 94305-4026 Michael B Giles Professor of Scientific Computing Oxford University Computing Laboratory Oxford University... an examiner for the Society of Actuaries in Ireland David C Edelman is currently on the faculty of the Michael Smurfit School of Business at University College Dublin in Finance, following previous positions including Sydney University (Australia) and Columbia University (USA) David is a specialist in Quantitative and Computational Finance, Mathematical Statistics, Machine Learning, and Information Theory... finite-difference methods that work well in low-dimensional problems quickly become unmanageable in high-dimensional cases; on the other hand, standard Monte Carlo methods cannot be applied as such, due to the optimization problem that is embedded in American options Many recent papers have been devoted to finding ways of adapting the Monte Carlo method for American and Bermudan option pricing; see for instance... considerations rather than by the specific form of a payoff function By using irregular grids we gain flexibility, but there is a price to pay In the standard finite-difference method based on regular grids, simple formulas for weights can be derived from Taylor expansions Such simple rules are no longer available if we use irregular grids, and so we must look for alternatives P1: Naresh July 31, 2007... HIGH-DIMENSIONAL AMERICAN OPTIONS 17 to the classical finite-difference formulas We propose here a method based on Markov chain approximations In a discrete-time context, transformations from a continuous-state (vector autoregressive) model to a Markov chain model have been constructed, for instance by Tauchen [29] Financial models are often formulated in continuous time, typically by means of stochastic differential... measures of risk: a coherent representation of subjective risk aversion Journal of Banking and Finance, 26: 1505–1518 2 Acerbi, C (2003) Coherent representations of subjective risk aversion In Risk Measures for the XXI Century, ed G Szego Wiley, New York 3 Acerbi, C (2007) To be published in Quatitative Finance 4 Acerbi, C., Simonetti, P (2002) Portfolio Optimization with Spectral Measures of Risk... Optimization of conditional value-atrisk Journal of Risk, 2 (3) 21–41 14 Rockafellar, R.T., Uryasev, S (2001) Conditional value-at-risk for general loss distributions Journal of Banking and Finance, 26/7: 1443–1471, 2002 15 Wang, S (1996) Premium calculation by transforming the layer premium density Astin Bulletin, 26: 71–92 P1: Naresh July 31, 2007 18:59 C925X C925X˙C002 CHAPTER 2 Pricing High-Dimensional... American options, high-dimensional problems, free boundary problems, optimal stopping, variational inequalities, numerical methods, unstructured mesh, Markov chain approximation 2.1 INTRODUCTION The pricing of American options has been extensively discussed in recent years (cf Detemple [12] for a survey), and in particular much attention has been paid to the computational challenges that arise in the... adopting a noncoherent measure as a decision-making tool for asset allocation, we are choosing to face formidable (and often unsolvable) computational problems related to the minimization of risk surfaces plagued by a plethora of risk-nonsensical local minima As a matter of fact, we are persuaded that no bank in the world has actually ever performed a true VaR minimization in its portfolios, if we exclude... Carlo method for American and Bermudan option pricing; see for instance [6, 7, 16, 20, 23, 26, 28, 30, 31] A survey of Monte Carlo methods for American option pricing is provided by Glasserman [14, chap 8] As has been pointed out by Glasserman [14], a unifying framework for simulation-based approaches to American option pricing is provided by the stochastic mesh method that was developed by Broadie . the international conference on Numerical Methods for Finance held in Dublin, Ireland in June 2006 and were then submitted for publication. The refereeing. relevant numerical methods for the solution of practical problems in finance. The conference was held under the auspices of the Institute for Numerical