By Eckhard Platen
The benchmark method offers a normal framework for monetary industry modeling, which extends past the traditional risk-neutral pricing concept. It allows a unified remedy of portfolio optimization, spinoff pricing, built-in probability administration and assurance threat modeling. The lifestyles of an similar risk-neutral pricing degree isn't really required. as a substitute, it ends up in pricing formulae with admire to the real-world chance degree. This yields vital modeling freedom which seems to be useful for the derivation of life like, parsimonious marketplace types. the 1st a part of the e-book describes the mandatory instruments from chance conception, records, stochastic calculus and the speculation of stochastic differential equations with jumps. the second one half is dedicated to monetary modeling by the benchmark method. numerous quantitative equipment for the real-world pricing and hedging of derivatives are defined. the overall framework is used to supply an realizing of the character of stochastic volatility. The e-book is meant for a large viewers that comes with quantitative analysts, postgraduate scholars and practitioners in finance, economics and assurance. It goals to be a self-contained, available yet mathematically rigorous creation to quantitative finance for readers that experience an inexpensive mathematical or quantitative history. eventually, the ebook should still stimulate curiosity within the benchmark strategy via describing a few of its strength and broad applicability.
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Extra resources for A Benchmark Approach to Quantitative Finance
In some cases the kurtosis of a ﬁtted model may not even be ﬁnite. 1 summarizes the moments for several distributions discussed previously. 8 1 -3 Fig. 1. 1. 8 9 3 λ2 0 2 0 1 2 p− 2 3 2 √ n 0 2 3 3 p+2 p (n+4) n n−2 n−4 Finally, let us mention that sometimes the notion of excess kurtosis κX of a random variable X is used. This is simply the diﬀerence between the kurtosis κX and the value 3 for the Gaussian kurtosis, that is κX = κX − 3. 39) Higher Order Moments In general, a new random variable is obtained when we transform or combine random variables by functions or arithmetic operations.
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. As a consequence, X1 and X2 are not jointly Gaussian distributed and Cov(|X1 |, |X2 |) = E(|Y2 |2 ) − (E(|Y2 |))2 > 0.
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).
A Benchmark Approach to Quantitative Finance by Eckhard Platen