Spectral calibration of exponential Lévy Models [1]

Denis Belomestny and Markus Reiβ

parameters σ = 0.1, λ = 5, λ- = 4, λ+ = 8,p = 1/3 and apply the non-
parametric calibration procedure given the noisy observation of N = 50
European options with maturity T = 0.25, interest rate r = 0.06 and noise
levels δj = O(xj)/10. The strike prices giving rise to the design points (xj)
have been obtained by a random sample which yields more option prices at
the money than further in or out of the money.

In this example we use a standard procedure based on cubic smoothing
splines with cross validation for step (a) of the algorithm. The spectral cut-
off frequency U is selected in a data-driven way by looking for the values
where the estimates stabilize when U increases. As a postprocessing step
the estimated Levy density is corrected to ensure that it is non-negative.
A more precise description of the implementation of the entire procedure is
given in Belomestny and Reiβ (2006).

In Figure 1 (left) the simulated observations (Oj) are depicted as a
function of the corresponding log-forward moneyness (xj ) on the horizontal
axis. The calibrated Levy density ^ is shown in Figure 1 (right) together
with the true density ν from (20). The parameters were estimated as σ =
0.131, λ = 4.983, γ = 0.424(γ = 0.423). We observe that the calibration
recovers the main features of the Levy triplet like the magnitude of the
volatility and jump intensity or the mode and the skewness of the jump
density.

Simulations show that for twice as many data points (N = 100) and ob-
servations with half as much noise (δj = O(xj )/20) the calibration results
are already very satisfactory. Usually, the quality of the estimators depends
slightly more on the noise level (δj ) than on the number of observations
N , that is the distance of observations ∆. The double-exponential jump
density in the Kou model is difficult to estimate because of its nondiffer-
entiability at zero. For smoother jump densities, as in the Merton (1976)
model, even better estimates are obtained. In Belomestny and Reiβ (2006)
further calibration results are presented.

4. Risk bounds

4.1. The main results

We shall use throughout the notation A . B if A is bounded by a constant
multiple of B, independently of the parameters involved, that is, in the
Landau notation A = O (B). Equally A & B means B < A and A ~ B
stands for A . B and A & B simultaneously.

To assess the quality of the estimators, we quantify their risks under a
smoothness condition of order s on the transformed jump density μ.

Definition 1. For s ∈ N and R, σ_max > 0 let G_s (R, σ_max) denote the set
of all Levy triplets T = (σ² ,γ,μ), satisfying the martingale condition and
Assumption 1 with C2 6 R, such that μ is s-times (weakly) differentiable
and

^σ ∈ [^{0, σ}max ]^,    ∣^γ Ij ʌ ∈ [^0, R ]^,   ∩mθx ^kμ^ )∣l-b ²(R) ⁶ R^,    kμ^(s) l∣L∞(R) ^{6 R}.

06k6s

<<   8   9   10   11   12   13   14   >>

More intriguing information