38 M.L. Bianchi et al.
setI−={iy|y∈R−}, remembering that 2 F 1 (a,b,c; 0 )=1. Second, in order to
obtain a fast convergence of the series (13), we split the positive imaginary line into
three subsets without intersection,
I+^1 ={iy| 0 <y≤ 0. 5 }
I+^2 ={iy| 0. 5 <y≤ 1. 5 }
I+^3 ={iy|y> 1. 5 },then we use (13) to evaluate 2 F 1 (a,b,c;z)inI+^1. Then, the first and the second
equalities of (14) together with (13) are enough to evaluate 2 F 1 (a,b,c;z)inI+^2 and
I+^3 respectively. This subdivision allows one to truncate the series (13) to the integer
N=500 and obtain the same results as Mathematica. We point out that the value of
yranges in the interval [−M,M]. This method, together with the MATLAB vector
calculus, considerably increases the speed with respect to algorithms based on the
numerical solution of the differential equation [17]. Our method is grounded only on
basic summations and multiplication. As a result the computational effort in the KR
density evaluation is comparable to that of the CGMY one. The KR characteristic
function is necessary also to price options, not only for MLE estimation. Indeed,
by using the approach of Carr and Madan [6] and the same analytic continuation as
above, risk-neutral parameters may be directly estimated from option prices, without
calibrating the underlying market model.
4 Simulation of TSαprocesses
In order to generate random variate from TSαprocesses, we will consider the gen-
eral shot noise representation of proper TSαlaws given in [21]. There are different
methods to simulate Levy processes, but most of these methods are not suitable for ́
the simulation of tempered stable processes due to the complicated structure of their
L ́evy measure. As emphasised in [21], the usual method of the inverse of the Levy ́
measure [20] is difficult to implement, even if the spectral measureRhas a simple
form. We will apply theorem 5.1 from [21] to the previously considered parametric
examples.
Proposition 1Let{Uj}and{Tj}be i.i.d. sequences of uniform random variables in
( 0 , 1 )and( 0 ,T)respectively,{Ej}and{E′j}i.i.d. sequences of exponential variables
of parameter 1 and{j}=E 1 ′+...+E′j,{Vj}an i.i.d. sequence of discrete random
variables with distribution
P(Vj=−G)=P(Vj=M)=1
2
,
a positive constant 0 <Y< 2 ( with Y= 1 ), and‖σ‖=σ(Sd−^1 )= 2 C. Further-
more,{Uj},{Ej},{E′j}and{Vj}are mutually independent. Then
Xt=d∑∞
j= 1[(
Yj
2 C)− 1 /Y
∧EjU^1 j/Y|Vj|−^1]
Vj
|Vj|I{Tj≤t}+tbT t∈[0,T], (15)