scholarly journals Reconstruction of the Time-Dependent Volatility Function Using the Black–Scholes Model

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Yuzi Jin ◽  
Jian Wang ◽  
Sangkwon Kim ◽  
Youngjin Heo ◽  
Changwoo Yoo ◽  
...  
Keyword(s):  
Cost Function ◽  
Piecewise Linear ◽  
Reconstruction Algorithm ◽  
Descent Method ◽  
Time Dependent ◽  
Steepest Descent Method ◽  
Market Values ◽  
Black Scholes ◽  
Volatility Function ◽  
The Cost

We propose a simple and robust numerical algorithm to estimate a time-dependent volatility function from a set of market observations, using the Black–Scholes (BS) model. We employ a fully implicit finite difference method to solve the BS equation numerically. To define the time-dependent volatility function, we define a cost function that is the sum of the squared errors between the market values and the theoretical values obtained by the BS model using the time-dependent volatility function. To minimize the cost function, we employ the steepest descent method. However, in general, volatility functions for minimizing the cost function are nonunique. To resolve this problem, we propose a predictor-corrector technique. As the first step, we construct the volatility function as a constant. Then, in the next step, our algorithm follows the prediction step and correction step at half-backward time level. The constructed volatility function is continuous and piecewise linear with respect to the time variable. We demonstrate the ability of the proposed algorithm to reconstruct time-dependent volatility functions using manufactured volatility functions. We also present some numerical results for real market data using the proposed volatility function reconstruction algorithm.

scholarly journals Polyhedral DC Decomposition and DCA Optimization of Piecewise Linear Functions

Algorithms ◽  
2020 ◽  
Vol 13 (7) ◽  
pp. 166 ◽  
Author(s):  
Andreas Griewank ◽  
Andrea Walther
Keyword(s):  
Piecewise Linear ◽  
Linear Representation ◽  
Local Minimizer ◽  
Descent Method ◽  
Linear Functions ◽  
Steepest Descent Method ◽  
Piecewise Linear Functions ◽  
Piecewise Linearization ◽  
Algorithmic Differentiation ◽  
Concave Part

For piecewise linear functions f : R n ↦ R we show how their abs-linear representation can be extended to yield simultaneously their decomposition into a convex f ˇ and a concave part f ^ , including a pair of generalized gradients g ˇ ∈ R n ∋ g ^ . The latter satisfy strict chain rules and can be computed in the reverse mode of algorithmic differentiation, at a small multiple of the cost of evaluating f itself. It is shown how f ˇ and f ^ can be expressed as a single maximum and a single minimum of affine functions, respectively. The two subgradients g ˇ and − g ^ are then used to drive DCA algorithms, where the (convex) inner problem can be solved in finitely many steps, e.g., by a Simplex variant or the true steepest descent method. Using a reflection technique to update the gradients of the concave part, one can ensure finite convergence to a local minimizer of f, provided the Linear Independence Kink Qualification holds. For piecewise smooth objectives the approach can be used as an inner method for successive piecewise linearization.

RELATIONSHIP BETWEEN A FUZZY LOGIC AND A STEEPEST DESCENT APPROACH TO OPTIMIZE A FEEDFORWARD ARTIFICIAL NEURAL NETWORK CONFIGURATION

1994 ◽  
Vol 05 (04) ◽  
pp. 299-312
Author(s):  
ROBERT N. SHARPE ◽  
MO-YUEN CHOW
Keyword(s):  
Neural Network ◽  
Fuzzy Logic ◽  
Mathematical Models ◽  
Cost Function ◽  
Evaluation Criteria ◽  
Descent Method ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Network Configuration ◽  
Training Time

The neural network designer must take into consideration many factors when selecting an appropriate network configuration. The performance of a given network configuration is influenced by many different factors such as: accuracy, training time, sensitivity, and the number of neurons used in the implementation. Using a cost function based on the four criteria mentioned previously, the various network paradigms can be evaluated relative to one another. If the mathematical models of the evaluation criteria as functions of the network configuration are known, then traditional techniques (such as the steepest descent method) could be used to determine the optimal network configuration. The difficulty in selecting an appropriate network configuration is due to the difficulty involved in determining the mathematical models of the evaluation criteria. This difficulty can be avoided by using fuzzy logic techniques to perform the network optimization as opposed to the traditional techniques. Fuzzy logic avoids the need of a detailed mathematical description of the relationship between the network performance and the network configuration, by using heuristic reasoning and linguistic variables. A comparison will be made between the fuzzy logic approach and the steepest descent method for the optimization of the cost function. The fuzzy optimization procedure could be applied to other systems where there is a priori information about their characteristics.

scholarly journals Proliferation of non-linear excitations in the piecewise-linear perceptron

2021 ◽  
Vol 10 (1) ◽  
Author(s):  
Antonio Sclocchi ◽  
Pierfrancesco Urbani
Keyword(s):  
Cost Function ◽  
Optimization Problem ◽  
Hard Spheres ◽  
Piecewise Linear ◽  
Power Laws ◽  
Universality Class ◽  
Local Minima ◽  
Convex Optimization Problem ◽  
Non Linear ◽  
The Cost

We investigate the properties of local minima of the energy landscape of a continuous non-convex optimization problem, the spherical perceptron with piecewise linear cost function and show that they are critical, marginally stable and displaying a set of pseudogaps, singularities and non-linear excitations whose properties appear to be in the same universality class of jammed packings of hard spheres. The piecewise linear perceptron problem appears as an evolution of the purely linear perceptron optimization problem that has been recently investigated in [1]. Its cost function contains two non-analytic points where the derivative has a jump. Correspondingly, in the non-convex/glassy phase, these two points give rise to four pseudogaps in the force distribution and this induces four power laws in the gap distribution as well. In addition one can define an extended notion of isostaticity and show that local minima appear again to be isostatic in this phase. We believe that our results generalize naturally to more complex cases with a proliferation of non-linear excitations as the number of non-analytic points in the cost function is increased.

Optimizing Piecewise Linear Isolator for Steady State Vibration

2003 ◽  
Author(s):  
A. Narimani ◽  
M. F. Golnaraghi ◽  
R. N. Jazar
Keyword(s):  
Linear System ◽  
Cost Function ◽  
Frequency Response ◽  
Averaging Method ◽  
Piecewise Linear ◽  
Relative Displacement ◽  
Experimental Frequency ◽  
Piecewise Linear System ◽  
Transmitted Force ◽  
The Cost

In this work we have studied the effect of stoppers in quarter car model. While in rough roads these stoppers prevent the system from excessive displacement around the resonance frequency. Although stoppers prevent the undesired motion they increase the transmitted force that is undesirable in suspension systems. In order to optimize between the relative displacement and transmitted force an analytical model is considered. The piecewise linear system is the model of presented nonlinearity which cannot be considered small. Therefore standard perturbation methods are not able to provide an analytical solution. For this case we have adopted an averaging method which leads to frequency response of the piecewise linear system at resonance. In order to confirm the results obtained by averaging method an experimental device is fabricated and frequency response of the system is measured. The experimental frequency response is in very good agreement with analytical approach and numerical simulations. Sensitivity analysis methods are used to minimize the cost function of maximum relative displacement in frequency domain. The range of the parameters that minimizes the cost function where certain amount of clearance exists in the system are obtained.

Interpolated Multichannel Singular Spectrum Analysis (I-MSSA): a reconstruction method that honors true trace coordinates

Geophysics ◽  
2020 ◽  
pp. 1-65
Author(s):  
Fernanda Carozzi ◽  
Mauricio D. Sacchi
Keyword(s):  
Spectrum Analysis ◽  
Singular Spectrum Analysis ◽  
Reconstruction Algorithm ◽  
Descent Method ◽  
Steepest Descent Method ◽  
Reconstruction Method ◽  
Gridded Data ◽  
Bilinear Interpolation ◽  
Singular Spectrum ◽  
Reconstructed Volume

The Multichannel Singular Spectrum Analysis (MSSA) reconstruction algorithm denoises and reconstructs seismic traces on a regular grid. We present a modified version of MSSA that can cope with denoising and reconstruction of traces with irregular coordinates. The proposed method, Interpolated Multichannel Singular Spectrum Analysis (I-MSSA), connects off-the-grid observations to the desired gridded data via a non-invertible bilinear interpolation operator. The algorithm consists of two steps. In the first step, we use the steepest descent method to estimate the gridded data that honors off-the-grid observations. The second step guarantees convergence to a solution by applying the MSSA filter to the gridded data. The final solution is the reconstructed volume that honors off-the-grid observations. We apply the algorithm to synthetic and field data. We also provide an application where 3D prestack data corresponding to an orthogonal survey is fully reconstructed using cross-spread gathers. We use I-MSSA to reconstruct each subset individually. The output is a complete seismic volume described in a regular CMP grid.

Time-Dependent Barrier Options and Boundary Crossing Probabilities

2003 ◽  
Vol 10 (2) ◽  
pp. 325-334
Author(s):  
A. Novikov ◽  
V. Frishling ◽  
N. Kordzakhia
Keyword(s):  
Piecewise Linear ◽  
Time Dependent ◽  
Piecewise Linear Approximation ◽  
Boundary Crossing ◽  
Barrier Options ◽  
Standard Brownian Motion ◽  
Linear Boundary ◽  
Fair Price ◽  
Black Scholes ◽  
Crossing Probabilities

Abstract The problem of pricing of time-dependent barrier options is considered in the case when interest rate and volatility are given functions in Black–Scholes framework. The calculation of the fair price reduces to the calculation of non-linear boundary crossing probabilities for a standard Brownian motion. The proposed method is based on a piecewise-linear approximation for the boundary and repeated integration. The numerical example provided draws attention to the performance of suggested method in comparison to some alternatives.

Solution of inverse coefficient problem for fractional anomalous diffusion equation

2012 ◽  
Vol 9 (2) ◽  
pp. 65-70
Author(s):  
E.V. Karachurina ◽  
S.Yu. Lukashchuk
Keyword(s):  
Anomalous Diffusion ◽  
Fractional Derivatives ◽  
Descent Method ◽  
Extremum Problem ◽  
Steepest Descent Method ◽  
Coefficient Problem ◽  
Caputo Fractional Derivatives ◽  
Inverse Coefficient Problem ◽  
Residual Functional ◽  
Inverse Coefficient

An inverse coefficient problem is considered for time-fractional anomalous diffusion equations with the Riemann-Liouville and Caputo fractional derivatives. A numerical algorithm is proposed for identification of anomalous diffusivity which is considered as a function of concentration. The algorithm is based on transformation of inverse coefficient problem to extremum problem for the residual functional. The steepest descent method is used for numerical solving of this extremum problem. Necessary expressions for calculating gradient of residual functional are presented. The efficiency of the proposed algorithm is illustrated by several test examples.

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