Processus de Lévy en Finance - Laboratoire de Probabilités et ...

118 CHAPTER 3. NUMERICAL IMPLEMENTATION
and formulas (3.28–3.29) yield:
where ψ A k
∂ ˜ζ T (u k )
∂q m
= T eiu krT +T ψ k {
e
x m+iu k x m
− 1 − (1 + iu k )(e xm − 1) }
iu k (1 + iu k )
= T (e xm − 1) eiu krT +T ψ k
+ T e xm eiukrT (e T ψ k
− e T ψA k )
1 + iu k iu k (1 + iu k )
− T e xm eiu krT (e T ψ k
− e T ψA k )
iu k (1 + iu k )
e iu kx m
+ T e xm eiu krT +T ψk A (e iu kx m
− 1)
,
iu k (1 + iu k )
= − A
2 u k(u k − i). Suppose that the grid in strike space is such that x 0 = m 0 d for
some m 0 ∈ Z. Then for every j ∈ {0, . . . , M − 1},
M−1
∑
DFT j [f k e iu M−1−kx m
] = e iu M−1x m
Introducing additional notation:
k=0
e −2πik(j−m−m 0)/M f k
H k = eiu krT +ψ k T
1 + iu k
G k = eiu krT +ψ A k T
iu k (1 + iu k ) ,
we obtain the final formula for computing the gradient:
∂ˆ˜z T (x j )
∂q m
= ∆ 2π e−ix ju M−1
T (e xm − 1)DFT j
[
w k e −ix 0∆k H M−1−k
]
= e iu M−1x m
DFT (j−m0 −m) mod M [f k ].
+ T e xm (e iu M−1x m ˆ˜z T (x (j−m0 −m) mod M ) − ˆ˜z T (x j ))
+ ∆ ]
2π ei(xm−x j)u M−1
T e xm DFT (j−m0 −m) mod M
[w k e −ix0∆k G M−1−k
− ∆ [
]
2π e−ix ju M−1
T e xm DFT j w k e −ix0∆k G M−1−k . (3.32)
The time values ˆ˜z T have already been computed in all points of the grid when evaluating option
prices, and the Fourier transforms of G k do not need to be reevaluated at each step of the
algorithm because G k do not depend on q i . Therefore, compared to evaluating the functional
alone, to compute the gradient of Ĵ α one only needs one additional fast Fourier transform per
maturity date. The complexity of evaluating the gradient using the above formula is thus only
about 1.5 times higher than that of evaluating the functional itself (because evaluating option
prices requires two Fourier transforms per maturity date). If the gradient was to be evaluated
numerically, the complexity would typically be M times higher. The analytic formula (3.32)
therefore allows to reduce the overall running time of the calibration algorithm from several
hours to less than a minute on a standard PC.