A Benchmark Approach to Quantitative Finance by Eckhard Platen

By Eckhard Platen

The benchmark procedure presents a common framework for monetary marketplace modeling, which extends past the normal risk-neutral pricing idea. It allows a unified remedy of portfolio optimization, spinoff pricing, built-in possibility administration and assurance chance modeling. The life of an identical risk-neutral pricing degree isn't really required. as a substitute, it results in pricing formulae with recognize to the real-world likelihood degree. This yields very important modeling freedom which seems to be helpful for the derivation of practical, parsimonious industry versions. the 1st a part of the ebook describes the mandatory instruments from likelihood concept, statistics, stochastic calculus and the idea of stochastic differential equations with jumps. the second one half is dedicated to monetary modeling by the benchmark procedure. numerous quantitative tools for the real-world pricing and hedging of derivatives are defined. the overall framework is used to supply an figuring out of the character of stochastic volatility. The publication 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, obtainable yet mathematically rigorous creation to quantitative finance for readers that experience an inexpensive mathematical or quantitative history. ultimately, the ebook should still stimulate curiosity within the benchmark procedure by means of describing a few of its energy and broad applicability.

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9 we graph the hyperbolic density in a logarithmic scale. Variance Gamma Density (*) Madan & Seneta (1990) proposed that log-returns are distributed with a normal-variance gamma mixture distribution. This case is obtained when√the shape parameters are such that λ > 0 and α ¯ = 0, that is, δ = 0 and α = c2 λ . 31) for x ∈ . The variance gamma density is a three parameter density. The parameter λ is the shape parameter with smaller λ implying larger tail heaviness. Furthermore, when λ → ∞ the variance gamma density asymptotically approaches the Gaussian density.

48) E(X r ) = r α Γ (p) for α > 0, p > 0 and r > −p. 49) for r > −p, which does not depend on α. Properties of Moments General properties of moments can be used to gain an understanding of, and insight into, many of the problems that arise in quantitative finance. 5), inherits the additivity property. 50) for any two random variables X1 , X2 and any two real numbers a, b, provided the expectations are finite. 51) E(X1 ) ≤ E(X2 ). 52) holds for any convex function g : → , which is a function satisfying the relation g(λ x + (1 − λ) y) ≤ λ g(x) + (1 − λ) g(y) for all x, y ∈ and λ ∈ [0, 1].

The derivatives of the Laplace transform 36 1 Preliminaries from Probability Theory with respect to λ can be used to find the moments. 45), by the equations k1 = α1 = µX , k2 = m2 , k3 = m3 , k4 = m4 − 3m22 . 76) for λ ∈ . This Laplace transform can be used to obtain expectations for asset prices under the standard market model, which is the lognormal or Black-Scholes model. Under this model returns are normalized increments of exponentials of Gaussian random variables or, equivalently, the log-returns are Gaussian.

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