Monte Carlo Methods in Financial Engineering (Anglais) Relié – 7 août 2003
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Description du produit
Revue de presse
"To keep it short, let me summarize the recension in one phrase: Paul Glausserman s book is a strong buy for everybody in the financial community. ... one gets 596 pages full of valuable information on all aspects of Monte Carlo simulation. ... Altogether, I can encourage everyone interested in Monte Carlo methods in finance to read the book. It is very well written ... comes with a carefully selected bibliography (358 references) and a helpful index, thus making it really worth the buy." --Ralf Werner, OR Spectrum Operations Research Spectrum, Issue 27, 2005
"The publication of this book is an important event in computational finance. For many years, Monte Carlo methods have been successfully applied to solve diverse problems in financial mathematics. By publishing this book the author deserves much credit for a very good attempt to lift such applications to a new level. … the book may well become a major reference in the field of applications of Monte Carlo methods in financial engineering. This is because the book is well structured and well written … ." --A Zhigljavsky, Journal of the Operational Research Society, Vol. 57, 2006
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Commentaires client les plus utiles sur Amazon.com
As something of a novice to advanced Monte Carlo techniques, I find this book immensely useful. The chapter on "Generating Random Numbers" helps, even if the description of the basic uniform generators could be stronger. Given the uniform generator, its descriptions of generators for non-uniform distributions work well for me. The "Sample Path" material is where I came into this book, really, looking for more insight into generation Brownian bridges. The math certainly is not for the notation-shy, but suffices for the dedicated practitioner. The next few chapters on variance reduction, quasi-MC, discretization, and sensitivity analysis are all widely applicable - I don't have immediate use for the material, but now I know where to look when the need arises. The remaining two chapters cover specific financial applications, and I leave comment on them to other readers.
This book gave me what I wanted, and lots more besides. Much of what it offers really isn't for me, though - the financial instruments being analyzed border on abstract art. I also felt a little pain at having no background in stochastic calculus, but some determination and a willingness to skip over fine points got me through well enough. The successful reader has a working knowledge of basic calculus, linear algebra, and probability. That reader must have a real interest in MC techniques, and should care about the financial decision-making to which Glasserman applies those techniques - but, as I prove, even that isn't necessary for getting a lot of value from this text.
The next-to-last chapter discusses the difficult problem of pricing American options, which the author introduces as an `embedded optimization problem': the value of an American option is found by finding the optimal expected discounted payoff, in order to find the best time to exercise the option. When applying Monte Carlo simulation, the author restricts himself to options that can only be exercised at a finite, fixed set of opportunities, with a discrete Markov chain used to model the underlying process representing the discounted payoff from the exercise of the option at a particular time. This allows the use of dynamic programming, which the author does throughout the chapter, with the further simplification that the discounting is omitted. The author also shows how to find the optimal value by finding the best value within a parametric class, giving in the process a more tractable problem. This approach considers a parametric class of exercise regions or stopping rules. The author's discussion is somewhat too brief, but he does quote many references that the reader can easily consult.
Also discussed are random tree methods, which simulate paths of the underlying Markov chain, and which allow more control on the error as the computational effort increases. The random tree method gives two consistent estimators, one biased high and one biased low, with both converging to the true value, and attempts to find the solution to the full optimal stopping problem and estimate the true value of an American option. The author discusses briefly the numerical tests that support this method. Similar to this method are stochastic mesh methods, the difference being that stochastic mesh methods utilize information coming from all nodes in the next time step. These methods are given detailed treatment in this chapter, along with detailed discussion of their limitations and computational complexity. Regression-based methods, which estimate continuation values from simulated paths, are discussed within the framework of stochastic mesh. These methods allow the estimation of continuation values from simulated paths and consequently to price American options by Monte Carlo simulation.
Still another method that is discussed in this chapter is that of state-space partitioning, which, as the name implies, involves the partitioning of the state space of the underlying Markov chain. Monte Carlo simulation then allows the calculation of the transition probabilities and the averaged payoffs, and then these calculations are used to obtain estimates of the approximating value function. The author discusses the problems with this approach, these arising mostly in high-dimensional state spaces, as expected.
The last chapter will be of particular interest to risk managers, wherein the author applies Monte Carlo simulation to portfolio management. The measurement of market risk in his view boils down to finding a statistical model for describing the movements in individual sources of risk and correlations between multiple sources of risk, and in calculating the change in the value of the portfolio as the underlying sources of risk change. Most interesting in the discussion is the use of heavy-tailed probability distributions to model the changes in market prices and risks. A few methods for calculating VAR are discussed, which is then followed by how to use Monte Carlo simulation for estimating VAR. The author reminds the reader of the pitfalls in using probability distributions based on historical data for sampling price changes. A variance reduction technique based on the delta-gamma approximation is used to reduce the number of scenarios needed for portfolio revaluation. The author first treats the case where the risk factors are distributed according to multivariate normal distribution, and then latter the case where the distribution is heavy-tailed. The delta-gamma approximation captures some of the nonlinearity in a portfolio that contains options. This nonlinearity arises because of the dependence of the option on the price of the underlying asset. Keeping the quadratic terms in the Taylor expansion of the portfolio change yields the delta (first derivative) and gamma (second derivative) terms (the sensitivities). One then must find the distribution of a quadratic function of normal random variable, which the author does numerically via transform inversion. Particularly interesting in this discussion is the use of `exponential twisting' to obtain a dramatic reduction in variance. One then samples from the `twisted distribution' provided the `twisting parameter' is chosen intelligently. The author gives references, and discuses in slight detail, results that show the asymptotic optimality for this method.
The case for a heavy-tailed distribution if of course much more involved, since there are no moment generating functions for the quantities of interest. The author gets around this by using an `indirect' delta-gamma approximation, which involves expressing the quantities of interest in terms of a new random variable that is more convenient to work with. The author also discusses various methods for doing variance reduction in the heavy-tailed case, one of these methods again involving exponential twisting. The chapter ends with a discussion of credit risk. The main item of interest here is the calculation of the time of default, which the author discusses in terms of the default intensity and intensity-based modeling using a stochastic intensity to model the time to default.
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