Download PDF by Eckhard Platen, David Heath: A Benchmark Approach to Quantitative Finance

By Eckhard Platen, David Heath

ISBN-10: 3540262121

ISBN-13: 9783540262121

A framework for monetary industry modeling, the benchmark method extends past general possibility impartial pricing thought. It allows a unified remedy of portfolio optimization, by-product pricing, built-in danger administration and coverage hazard modeling. This publication provides the required mathematical instruments, via a radical advent to monetary modeling below the benchmark procedure, explaining a variety of quantitative tools for the reasonable pricing and hedging of derivatives.

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3) When a continuous random variable has a probability density fX , then the corresponding expression for its mean is ∞ µX = E(X) = −∞ x fX (x) dx. 4) One may say that the product fX (x) dx approximates the probability that X takes its value in the interval (x, x + dx). 1). 4) can both be equivalently expressed as an integral with respect to the measure P . That is, we can write X(ω) dP (ω). 4) actually exist, that is, they are finite and well defined for each subset of Ω. 12 at the end of this chapter.

1. More precisely, the function g should be Borel measurable. This is the case when g is, for instance, continuous or piecewise continuous. For more details on these issues the reader is referred to Shiryaev (1984). 41) i∈N when X is discrete, or ∞ E(g(X)) = −∞ when X is continuous with a density function fX . It should be noted that these expectations may be undefined or infinite. Typical functions of a random variable X are the polynomials g(x) = xp or g(x) = (x − µX )p for integers p ≥ 1. 43) or the pth central moment respectively.

57) for all a > 0. 58) for all a > 0. For two random variables X1 and X2 the Cauchy-Schwartz inequality provides the estimate |E(|X1 X2 |)| ≤ E ((X1 )2 ) E ((X2 )2 ). 59) Further properties of moments can be found at the end of the following section. Conditional Expectations The notion of conditional expectation is central to many of the ideas that arise in probability theory and stochastic calculus. 6) . If we know that some event A has occurred we may be able to improve on this estimate. For instance, suppose that the event A = {ω ∈ Ω : X(ω) ∈ [a, b]} has occurred.

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A Benchmark Approach to Quantitative Finance by Eckhard Platen, David Heath

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