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 traditional risk-neutral pricing thought. It allows a unified therapy of portfolio optimization, spinoff pricing, built-in possibility administration and coverage threat modeling. The lifestyles of an similar risk-neutral pricing degree isn't required. as an alternative, it ends up in pricing formulae with admire to the real-world chance degree. This yields very important modeling freedom which seems to be precious for the derivation of practical, parsimonious industry versions. the 1st a part of the publication describes the mandatory instruments from chance idea, information, stochastic calculus and the idea of stochastic differential equations with jumps. the second one half is dedicated to monetary modeling by the benchmark technique. a number of 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 incorporates quantitative analysts, postgraduate scholars and practitioners in finance, economics and coverage. It goals to be a self-contained, obtainable yet mathematically rigorous advent to quantitative finance for readers that experience an inexpensive mathematical or quantitative heritage. eventually, the e-book should still stimulate curiosity within the benchmark procedure by way of describing a few of its energy and broad applicability.

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This characteristic function uniquely identifies the distribution of the corresponding random vector. For a continuous n-dimensional random vector we have ∞ φX (θ) = −∞ ··· ∞ p exp ı −∞ θk xk fX (x1 , . . , xp ) dx1 , . . , dxp . 59) φX (θ) = exp ı µ θ − θ D θ . 2 for all θ ∈ p . Let us give another example using a p-dimensional Student t distributed random variable X = (X1 , X2 , . . , Xp ) with n > 0 degrees of freedom, zero mean vector µ = (0, . . , 0) and regular covariance matrix D. This random variable can be obtained from a multivariate Gaussian vector Y = (Y1 , Y2 , .

Xn ) = fX1 (x1 ) fX2 (x2 ) · · · fXn (xn ). 41) In addition, for n independent random variables X1 , X2 , · · · , Xn the product g1 (X1 ) g2 (X2 ) · · · gn (Xn ) involving measurable functions g1 , g2 , . . 42) whereas their sum has variance n Var n gi (Xi ) i=1 = Var(gi (Xi )). 43) i=1 Multivariate Gaussian Density Consider a random vector X = (X1 , X2 , . . , Xn ) with mean vector µ = (µ1 , µ2 , . . , µn ) = (E(X1 ), E(X2 ), . . , E(Xn )) and an n×n autocovariance matrix D = Cov(X, X) = D D ,m ,m n , ,m=1 = E ((X − µ ) (Xm − µm )) = E(X Xm ) − E(X ) E(Xm ).

The parameter µ is a location parameter. The two shape parameters for its tails are λ and α ¯ = α δ, defined so that they are invariant under scale transformations. The other parameters contribute to the scaling of the density. 27) c = 2 ¯ ⎩ δ Kλ+1 (α) otherwise. 2 Continuous Random Variables and Distributions 19 It can be shown that as λ → ± ∞ and/or α ¯ → ∞ the SGH density asymptotically approaches the Gaussian density. To illustrate certain typical SGH densities we shall describe four special cases of the SGH density in the sequel.

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