By Eckhard Platen, David Heath

ISBN-10: 3540262121

ISBN-13: 9783540262121

A framework for monetary marketplace modeling, the benchmark method extends past regular possibility impartial pricing idea. It allows a unified therapy of portfolio optimization, by-product pricing, built-in threat administration and coverage chance modeling. This ebook provides the mandatory mathematical instruments, by way of an intensive advent to monetary modeling less than the benchmark method, explaining a number of quantitative equipment for the reasonable pricing and hedging of derivatives.

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Extra resources for A Benchmark Approach to Quantitative Finance

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Skewness Some random variables have probability densities with non-symmetric shapes. One way to measure their asymmetry is to compute the skewness βX of the corresponding density. The skewness of a random variable X is measured using the centralized and normalized third moment, that is βX = E X − µX σX 3 . 22) 26 1 Preliminaries from Probability Theory For a random variable X we say that its density is called positively skewed if βX > 0, negatively skewed if βX < 0 and symmetric if βX = 0. 15), the expression p1 (1 − p1 ) (2p1 − 1).

15), 1 − p1 + p21 κX = 3 . 16) a kurtosis of the form ∞ κX = i=0 4 i−λ 2 λi 1 exp{−λ} = 3 + . i! 8. 18), that it has a constant kurtosis with ∞ κX = 0 x− 1 λ 1 λ 4 λ exp{−λ x} dx = 9. 19) the constant kurtosis ∞ κX = −∞ x−µ σ 4 √ 1 1 exp − 2 2π σ x−µ σ 2 dx = 3. 35) If the kurtosis κX of a random variable X is greater than 3, then this random variable, its density and also its distribution are called leptokurtic. 36) which is larger for smaller p > 0. Thus a gamma distributed random variable is leptokurtic.

3 Moments of Random Variables 27 for p > 0. This is also indicated by inspection of its density, as displayed in Fig. 5. Finally, we mention that the density of a central Student t distributed random variable X ∼ t(n) with n > 3 degrees of freedom is symmetrically skewed, that is, ⎛ ⎞3 − n+1 ∞ 2 Γ ( n+1 x x2 2 ) ⎝ ⎠ √ dx = 0. 29) βX = 1 + n n Γ ( n2 ) π n −∞ n−2 This fact is also apparent from the shape of the density shown in Fig. 6. Kurtosis Extreme values of returns are very important in a range of financial applications.

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

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