scholarly journals Accurate and Efficient Computations of the Greeks for Options Near Expiry Using the Black-Scholes Equations

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Darae Jeong ◽  
Minhyun Yoo ◽  
Junseok Kim
Keyword(s):  
Test Problem ◽  
Truncation Error ◽  
Numerical Solutions ◽  
Option Price ◽  
Strike Price ◽  
Time Stepping ◽  
Black Scholes ◽  
Adaptive Time ◽  
Accurate Computations ◽  
Adaptive Time Stepping

We investigate the accurate computations for the Greeks using the numerical solutions of the Black-Scholes partial differential equation. In particular, we study the behaviors of the Greeks close to the maturity time and in the neighborhood around the strike price. The Black-Scholes equation is discretized using a nonuniform finite difference method. We propose a new adaptive time-stepping algorithm based on local truncation error. As a test problem for our numerical method, we consider a European cash-or-nothing call option. To show the effect of the adaptive stepping strategy, we calculate option price and its Greeks with various tolerances. Several numerical results confirm that the proposed method is fast, accurate, and practical in computing option price and the Greeks.

scholarly journals Valuation of Option Pricing with Meshless Radial Basis Functions Approximation

2020 ◽  
pp. 60-70
Author(s):  
M. O. Durojaye ◽  
J. K. Odeyemi
Keyword(s):  
Radial Basis Functions ◽  
Numerical Solutions ◽  
Interpolation Method ◽  
Option Price ◽  
Basis Functions ◽  
Time Stepping ◽  
Runge Kutta Method ◽  
Black Scholes ◽  
Radial Basis ◽  
Marching Method

This work focuses on valuation scheme of European and American options of single asset with meshless radial basis approximations. The prices are governed by Black – Scholes equations. The option price is approximated with three infinitely smooth positive definite radial basis functions (RBFs), namely, Gaussian (GA), Multiquadrics (MQ), Inverse Multiquadrics (IMQ). The RBFs were used for discretizing the space variables while Runge-Kutta method was used as a time-stepping marching method to integrate the resulting systems of differential equations. Numerical examples are shown to illustrate the strength of the method developed. The findings show that the RBFs has proven to be adaptable interpolation method because it does not depend on the locations of the approximation nodes which have overcome frequently evolving problems in computational finance such as slow convergent numerical solutions. Thus, the results allow concluding that the RBF-FD-GA and RBF-FD-MQ methods are well suited for modeling and analyzing Black and Scholes equation.

Fast Conjugate Heat Transfer Simulation of Long Transient Flexible Operations Using Adaptive Time Stepping

2018 ◽  
Vol 140 (9) ◽  
Author(s):  
R. Maffulli ◽  
L. He ◽  
P. Stein ◽  
G. Marinescu
Keyword(s):  
Heat Transfer ◽  
Conjugate Heat Transfer ◽  
Time Derivative ◽  
Computational Cost ◽  
Strong Impact ◽  
Time Step ◽  
Time Stepping ◽  
Adaptive Time ◽  
Fully Coupled ◽  
Adaptive Time Stepping

The emerging renewable energy market calls for more advanced prediction tools for turbine transient operations in fast startup/shutdown cycles. Reliable numerical analysis of such transient cycles is complicated by the disparity in time scales of the thermal responses in fluid and solid domains. Obtaining fully coupled time-accurate unsteady conjugate heat transfer (CHT) results under these conditions would require to march in both domains using the time-step dictated by the fluid domain: typically, several orders of magnitude smaller than the one required by the solid. This requirement has strong impact on the computational cost of the simulation as well as being potentially detrimental to the accuracy of the solution due to accumulation of round-off errors in the solid. A novel loosely coupled CHT methodology has been recently proposed, and successfully applied to both natural and forced convection cases that remove these requirements through a source-term based modeling (STM) approach of the physical time derivative terms in the relevant equations. The method has been shown to be numerically stable for very large time steps with adequate accuracy. The present effort is aimed at further exploiting the potential of the methodology through a new adaptive time stepping approach. The proposed method allows for automatic time-step adjustment based on estimating the magnitude of the truncation error of the time discretization. The developed automatic time stepping strategy is applied to natural convection cases under long (2000 s) transients: relevant to the prediction of turbine thermal loads during fast startups/shutdowns. The results of the method are compared with fully coupled unsteady simulations showing comparable accuracy with a significant reduction of the computational costs.

scholarly journals OPTIONS WRITTEN ON STOCKS WITH KNOWN DIVIDENDS

2004 ◽  
Vol 07 (07) ◽  
pp. 901-907
Author(s):  
ERIK EKSTRÖM ◽  
JOHAN TYSK
Keyword(s):  
Stock Price ◽  
Option Price ◽  
Option Prices ◽  
Strike Price ◽  
Call Options ◽  
Alternative Approach ◽  
Black Scholes ◽  
Market Practice ◽  
The Difference ◽  
Risk Free Rate

There are two common methods for pricing European call options on a stock with known dividends. The market practice is to use the Black–Scholes formula with the stock price reduced by the present value of the dividends. An alternative approach is to increase the strike price with the dividends compounded to expiry at the risk-free rate. These methods correspond to different stock price models and thus in general give different option prices. In the present paper we generalize these methods to time- and level-dependent volatilities and to arbitrary contract functions. We show, for convex contract functions and under very general conditions on the volatility, that the method which is market practice gives the lower option price. For call options and some other common contracts we find bounds for the difference between the two prices in the case of constant volatility.

scholarly journals An error-controlled adaptive time-stepping method for particle advancement in coupled CFD-DEM simulations

2021 ◽  
Vol 379 ◽  
pp. 203-216
Author(s):  
Hariswaran Sitaraman ◽  
Deepthi Vaidhynathan ◽  
Ray Grout ◽  
Thomas Hauser ◽  
Christine M. Hrenya ◽  
...  
Keyword(s):  
Time Stepping ◽  
Dem Simulations ◽  
Adaptive Time ◽  
Adaptive Time Stepping

An Adaptive Time Stepping Method for Transient Dynamic Response Analysis

2016 ◽  
Vol 6 (2) ◽  
pp. 152-170
Author(s):  
Jianguo Huang ◽  
Huashan Sheng
Keyword(s):  
Dynamic Response ◽  
Response Analysis ◽  
Dynamic Response Analysis ◽  
Transient Dynamic ◽  
Time Stepping ◽  
Adaptive Time ◽  
Spatial Discretisation ◽  
Transient Dynamic Response ◽  
Adaptive Time Stepping ◽  
Reduced Problem

AbstractAn efficient adaptive time stepping method is proposed for transient dynamic response analysis, which is frequently encountered in civil engineering and elsewhere. The reduced problem following the spatial discretisation can be discretised in the time by a C0-continuous discontinuous Galerkin method, and the adaptive time stepping strategy is based on optimal a posteriori error estimates developed using the energy method. Some numerical experiments demonstrate the effectiveness of our approach.

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