In the text, we assume thatχis the characteristic function of the distribution ofXT := ln(e−rTST/S0).
This corresponds to a stock price model of the form
ST =S0erT+XT,
whereXT is the log return on a forward contract to buy the stock at the forward dateT. In some contexts, models of the form
ST =S0eYT
1Obviously, then the integral converges absolutely for allz∈Cwith Rez=R.
are used instead. HereYT is the log return on a spot contract in which one buys the stock today and sells it at dateT. Equating the stock prices leads to the relation
rT+XT =YT.
Consequently, if we are given the characteristic functionψ(u)ofYT, we can calculate the characteristic functionχ(u)ofXT as
χ(u) =E[eiuXT] =e−iurTE[eiuYT] =e−iurTψ(u).
Therefore if we know the characteristic functionψ, we at once have an expression for the characteristic functionχ(u). This can then be used to price the options as described in the text.
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Index
affine term structure, 85 asymptotic expansion, 82
of Bessel functionKν, 33
of Fourier transform of modified Lévy mea- sure, 34
autonomous coefficient function, 87 bilateral Laplace transform, 64
Bjửrk-Di Masi-Kabanov-Runggaldier model, 110
CGMY distribution, 141 Lévy density, 142 characteristic function
analytic, 111 χ2test, 103
classGτof functions, 16 class D, 125
class LD, 125
compensator of a random measure, 129 continuous in probability, 3
contract function, see payoff function convolution, 64
coupon bond, 77
cumulant generating function, 80, 111 multidimensional, 132
density plot, 99 density process, 2, 113 density, empirical, 75, 99 derivative security, 9
discount bond, see zero coupon bond discounted price, 3
discrete Fourier transform, 70 Doléans-Dade exponential, 131 Esscher transform
for 1-dim distribution, 5 for stochastic processes, 6 face value of a bond, 77
fast Fourier transform, 71 filtration, 2
forward price, 62 forward rate, 78
generalized hyperbolic distribution, 137 characteristic function, 138
density, 137
hyperbolic distribution, 138
increments independent of the past, 3 Kolmogorov distance, 101
Kolmogorov-Smirnov test, 102 Lévy density
of CGMY distribution, 142 Lévy-Khintchine formula, 3, 22
multidimensional, 130 Lévy-Khintchine triplet
of CGMY distribution, 142 Lévy-Khintchine triplet, 3 Lévy measure, 22
modified, 23
Fourier transform of, 23
of generalized hyperbolic distribution, 21–
60
of normal inverse Gaussian distribution, 40 of variance gamma distribution, 141 Lévy motion
generalized hyperbolic, 149 hyperbolic, 149
NIG, 149 Lévy process, 3
CGMY, 45, 142
generated by an infinitely divisible distribu- tion, 149
variance gamma, 140 Lévy term structure model, 77