By Eckhard Platen, David Heath
A framework for monetary marketplace modeling, the benchmark strategy extends past average chance impartial pricing idea. It allows a unified remedy of portfolio optimization, spinoff pricing, built-in danger administration and assurance hazard modeling. This publication provides the required mathematical instruments, by way of an intensive creation to monetary modeling below the benchmark process, explaining a number of quantitative equipment for the reasonable pricing and hedging of derivatives.
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Extra info for A Benchmark Approach to Quantitative Finance
5. 12) u is the incomplete gamma function for u ≥ 0, a > −1, see Abramowitz & Stegun (1972) and Johnson, Kotz & Balakrishnan (1995). Non-Central Chi-Square Distribution For a non-central chi-square distributed random variable X ∼ χ2 (n, ) with n ≥ 0 degrees of freedom and non-centrality parameter > 0 its distribution function has the form ∞ FX (x) = χ2 (x; n, ) = k=0 exp − 2 k! 13) for x ≥ 0. In some sense, the non-central chi-square distribution is a weighted sum of central chi-square distributions with Poisson probabilities as weights.
7. Student t density under log scale 0 -2 log f -4 -6 -8 -4 20 15 10 -2 5 0 x a bar 2 4 Fig. 8. Normal-inverse Gaussian density under log scale Hyperbolic Density (*) Eberlein & Keller (1995) and K¨ uchler et al. (1999) proposed models, where log-returns appear to be hyperbolicly distributed. This occurs for the choice of the shape parameter λ = 1 in the SGH density. Using this parameter value the probability density function of X is fX (x) = 1 exp −¯ α 2 δ K1 (¯ α) for x ∈ , where δ2 = 1+ ¯ K1 (¯ α) c2 α .
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.