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

116 CHAPTER 3. NUMERICAL IMPLEMENTATION
where w k are the weights corresponding to the chosen integration rule. For a given strike K
such that x 0 ≤ log K ≤ x M−1 , the option price can now be computed by linear interpolation
(other interpolation methods can also be used):
Ĉ Q √
(T, K) = C A
BS (T, K) + log K − x n K ˆ˜z T (x nK +1) + x n K +1 − log K
ˆ˜z T (x nK ), (3.31)
x nK +1 − x nK x nK +1 − x nK
√
where n K := sup{n : x n ≤ log K} and C A
BS
(T, K) denotes the Black Scholes price of a call
option with time to maturity T , strike K and volatility √ A.
Error control The above method of evaluating option prices contains three types of numerical
errors: truncation and discretization errors appear when the integral is replaced by a finite sum
using Equation (1.28), and interpolation error appears in Equation (3.31). Supposing that the
grid in Fourier space is centered, that is, u 0 = L/2 and ∆ = L/(M − 1) for some constant L,
Section 1.4 allows to obtain the following bounds for the first two types of error.
Since we have supposed that A > 0, Equation (1.30) provides a uniform (with respect to Q)
bound for the truncation error:
|ε T | ≤
16e−T
AL2
πT AL 3 .
Proposition 1.13 does not give uniform bounds for the discretization (sampling) error, however,
since the calibrated Lévy measures are usually similar to the Lévy measures of the prior
process (see Section 3.6), the order of magnitude of the discretization error can be estimated
by computing the bounds of Proposition 1.13 for the prior process. The exact form of the error
depends on the integration rule used; for example for Simpson’s rule one has
( L 4 )
log L
|ε D | = O
M 4 .
A uniform bound for the interpolation error depends on the interpolation method that one
is using. For linear interpolation using Equation (3.31) one easily obtains
|ε I | ≤ d2
8
max ˜z′′ T ,
and the second derivative of the time value is uniformly bounded under the hypothesis A > 0