Spectral calibration of exponential Lévy Models [2]

The calibration of financial models has become rather important topic in recent years mainly because of the need to price increasingly complex options in a consistent way. The choice of the underlying model is crucial for the good performance of any calibration procedure. Recent empirical evidences suggest that more complex models taking into account such phenomenons as jumps in the stock prices, smiles in implied volatilities and so on should be considered. Among most popular such models are Levy ones which are on the one hand able to produce complex behavior of the stock time series including jumps, heavy tails and on other hand remain tractable with respect to option pricing. The work on calibration methods for financial models based on Lévy processes has mainly focused on certain parametrisations of the underlying Lévy process with the notable exception of Cont and Tankov (2004). Since the characteristic triplet of a Lévy process is a priori an infinite-dimensional object, the parametric approach is always exposed to the problem of misspecification, in particular when there is no inherent economic foundation of the parameters and they are only used to generate different shapes of possible jump distributions. In this work we propose and test a non-parametric calibration algorithm which is based on the inversion of the explicit pricing formula via Fourier transforms and a regularisation in the spectral domain.Using the Fast Fourier Transformation, the procedure is fast, easy to implement and yields good results in simulations in view of the severe ill-posedness of the underlying inverse problem

Extraction of soma support.

<p>The figure illustrates the detection of the cross section of the soma from a representative image taken from the bottom of a confocal image stack. (<b>A</b>) Denoised image. (<b>B</b>) Region obtained by applying the thresholding strategy within the region identified by the 2D mask as described in the text. (<b>C</b>) Detected support region obtained after applying a combination of filling and erosion operators to the image from Panel B.</p

Shearlet-based surface detection of a soma.

<p>The figure shows two views of the surface morphology reconstruction of a soma obtained by processing a confocal image stack with the shearlet surface detection routine.</p

Image preprocessing.

<p>(<b>A</b>) MIP image of a representative neuron from a confocal image stack. (<b>B</b>) Shearlet-based denoised version of the same image. (<b>C</b>) Smoothed version of the denoised image, where the smoothing is obtained by convolving the image with a Gaussian kernel of size 3 × 3 and standard deviation <i>σ</i> = .6</p

Proposed algorithm for 2D soma detection.

<p>Proposed algorithm for 2D soma detection.</p

3D soma detection.

<p>(<b>A—B</b>) Visualization of two representative image stacks of neuronal cultures. Note that the stacks have a very limited extension along the <i>z</i> direction where less that 10 <i>μm</i> are available, corresponding to about 25–30 optical slices. (<b>C—D</b>) The images illustrate, in red color, the detection of the somas from the confocal image stacks shown above.</p

Performance of soma segmentation.

<p>2D segmentation and soma detection of representative MIP images. The Performance metrics for the segmentation of the somas contained in these images are reported in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0121886#pone.0121886.t002" target="_blank">Table 2</a>.</p

Illustration of 2D soma detection algorithm.

<p>(<b>A</b>) Denoised image, obtained using a shearlet-based denoising routine on the MIP of the image stack. Image size = 512 × 512 pixels (1 pixel = 0.28 × 0.28<i>μm</i>). (<b>B</b>) Segmented binary image. (<b>C</b>) Directional Ratio plot; the values range between 1, in red color (corresponding to more isotropic regions), and 0, in blue color (corresponding to more anisotripc regions); note that the Directional Ratio is only computed inside the segmented region, i.e., inside the red region in Panel B. (<b>D</b>) Detection of initial soma region, obtained by applying a threshold to the values of the Directional Ratio in Panel C. (<b>E</b>) Soma detection, obtained by applying the level set method with the initialization curve determined by the boundary of the initial soma region in Panel D. (<b>F</b>) Separation of contiguous somas; two regions from Panel E are recognized as too large and hence divided using the level set method.</p

Validation of soma count.

<p>Performance on soma detection and separation of contiguous somas.</p><p>Validation of soma count.</p

Separation of clustered somas.

<p>The figure illustrates the application of the multiscale Directional Ratio in combination with the level set method to separate contiguous somas on the MIP of a confocal image of a neuronoal network culture. (<b>A</b>) Segmented image (detail). (<b>B</b>) Directional Ratio plot using directional filters of length 20 pixels (note that the diameter of a soma is about 40 pixels). (<b>C</b>) The blue region shows the points where the Directional Ratio exceeds the threshold 0.9, identifying the more isotropic region. (<b>D</b>) Directional Ratio plot using directional filters of length 40 pixels; to note that the larger values of the Directional Ratio are now concentrated within a smaller set inside the blob-like regions. (<b>E</b>) The blue region shows the points where the Directional Ratio in panel D exceeds the threshold 0.9, identifying the more isotropic region; note that contiguous somas are now split into two regions. (<b>F</b>) Soma detection obtained from the application of the level set method, using the initialization curves determined by the boundary of the initial soma region in (<b>E</b>).</p