By Eckhard Platen, David Heath

ISBN-10: 3540262121

ISBN-13: 9783540262121

A framework for monetary marketplace modeling, the benchmark procedure extends past regular danger impartial pricing concept. It allows a unified remedy of portfolio optimization, by-product pricing, built-in hazard administration and assurance threat modeling. This booklet offers the mandatory mathematical instruments, through a radical advent to monetary modeling lower than the benchmark procedure, explaining a variety of quantitative equipment for the reasonable pricing and hedging of derivatives.

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Extra info for A Benchmark Approach to Quantitative Finance

Example text

1) the joint distributions FX1 ,X2 ,··· ,Xn satisfy lim FX1 ,X2 ,··· ,Xn (x1 , x2 , . . 36) for i ∈ {1, 2, . . , n} and fixed xj , j ∈ {1, 2, . . , i − 1, i + 1, . . , n}. ,xn →+∞ FX1 ,X2 ,··· ,Xn (x1 , x2 , . . , xn ) = 1. 37) 46 1 Preliminaries from Probability Theory In addition, FX1 ,X2 ,··· ,Xn is non-decreasing and continuous from the right in xi for i ∈ {1, 2, . . , n}. For any permutation {i1 , i2 , . . , in } of the set {1, 2, . . , n} we have FXi1 ,Xi2 ,··· ,Xin (xi1 , xi2 , . .

16) Z n turns out to be a central Student t, or in short a Student t, distributed with n degrees of freedom. 17) for x ∈ . We write X ∼ t(n) if the random variable X has a Student t distribution with n degrees of freedom. In Fig. 6 we plot the density of the Student t distribution for n = 4 degrees of freedom. As will be shown later, this distribution seems to model log-returns of indices extremely well. It is interesting to express the Student t distribution function Ft(n) (x) in terms of rational and trigonometric functions for small integers n, see Shaw (2005).

20) and E (X1 − E(X1 X2 ))2 = Var(X1 ) − (Cov(X1 , X2 ))2 . 21) Here E(X1 | X2 ) denotes the conditional expectation of X1 given the information generated by X2 . We emphasize that the above constructions use jointly Gaussian distributed random variables. Now consider two independent N (0, 1) standard Gaussian random variables Y1 and Y2 . From these we construct X1 = |Y2 | sgn(Y1 ) and X2 = Y2 . Using these definitions it can be shown that X1 ∼ N (0, 1) and X2 ∼ N (0, 1) with Cov(X1 , X2 ) = E(X1 Y2 ) − E(X1 ) E(Y2 ) = E(Y2 |Y2 | sgn(Y1 )) = E(Y2 |Y2 |) E(sgn(Y1 )) = 0, but X1 and X2 are dependent random variables.

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


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