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## An Accurate Numerical Integrator for the Solution of Black Scholes Financial Model Equation (2015)

### Citations

5025 |
The pricing of options and corporate liabilities
- Black, Scholes
- 1973
(Show Context)
Citation Context ...r Black and Myron Scholes made a majorsbreakthrough by deriving a partial differential equation that must be satisfied by the price of any derivative security dependent on a non-dividend-paying stock =-=[2]-=-. According to [3] their work had a huge impact on how options were viewed in the financial world. In an idealized financial market, the price of a European option can besobtained as the solution of t... |

1720 |
Theory of rational option pricing
- Merton
- 1973
(Show Context)
Citation Context ...had a huge impact on how options were viewed in the financial world. In an idealized financial market, the price of a European option can besobtained as the solution of the Black-Scholes equation [4] =-=[5]-=-. However, the Black Scholes equation has beens*Corresponding author.sI. P. Akpan, J. O. Fatokuns284sderived under quite restrictive assumptions such as frictionless, liquid and complete market. In re... |

412 |
Finite difference schemes and partial differential equations, Wardsworth k Brooks/Cole
- Strikwerda
- 1989
(Show Context)
Citation Context ...schemes [21] [22]. Explicit numerical schemes have the disadvantage that restrictive conditions on the discretization parameters, time and space steps, are needed to obtainsstable, convergent schemes =-=[23]-=-. Moreover the convergence order is the only one in time.sThe Method of Lines (MOL) is a general procedure for the solution of time-dependent partial differential equations (PDEs) [24]. The basic idea... |

249 |
Numerical methods for ordinary differential systems, the initial value problem
- Lambert
- 1991
(Show Context)
Citation Context ...numerical solution to the PDE can then besused for the numerical integration. One of the salient features of the MOL is the use of existing, and generallyswell established, numerical methods for ODEs =-=[26]-=-.sThis paper is organized thus: In Section 2 we transform the black Scholes equation into a heat equation byschange in variables. In Section 3 we introduce an L-stable trapezoidal like integrator for ... |

203 |
Optimal replication of contingent claims under transaction costs”, The Review of Futures Markets
- Hodges, Neuberger
- 1989
(Show Context)
Citation Context ...e price of the option do not exist. From a binomial model, [6] derived an option price that takes into account transaction costs which approximates a BlacksScholes price but with modified volatility. =-=[14]-=- [15] computed the option price of the Black Scholes equation assthe solution of a nonlinear quasi-variational inequality. This approach has the disadvantage that the option pricesdepends on the choic... |

117 |
European option pricing with transaction costs
- Davis, Panas, et al.
- 1993
(Show Context)
Citation Context ... such as frictionless, liquid and complete market. In recent years nonlinear Black Scholes equations have been derived in order to model transaction costs arising in the hedging ofsportfolios [1] [6] =-=[7]-=- and feedback effects due to large traders [8]-[13].sIn seeking the solution of the Black Scholes equation, emphasis is always laid on derivation of formula orsequation for the price of the option of ... |

113 | liquidity, hedging and crashes - Gennotte, Leland, et al. - 1990 |

87 |
The Numerical Method of Lines: Integration of Partial Differential Equations
- Schiesser
- 1991
(Show Context)
Citation Context ...vergent schemes [23]. Moreover the convergence order is the only one in time.sThe Method of Lines (MOL) is a general procedure for the solution of time-dependent partial differential equations (PDEs) =-=[24]-=-. The basic idea of the MOL is to replace the spatial (boundary value) derivatives in thesPDEs with algebraic approximations [25]. Ones this is done, the spatial derivatives are no longer stated expli... |

86 |
Vorst T., Option replication in discrete time with transaction costs
- Boyle
- 1992
(Show Context)
Citation Context ...ions such as frictionless, liquid and complete market. In recent years nonlinear Black Scholes equations have been derived in order to model transaction costs arising in the hedging ofsportfolios [1] =-=[6]-=- [7] and feedback effects due to large traders [8]-[13].sIn seeking the solution of the Black Scholes equation, emphasis is always laid on derivation of formula orsequation for the price of the option... |

78 |
Derivative Asset Pricing with Transaction Costs,”
- Bensaid, Lense, et al.
- 1992
(Show Context)
Citation Context ...ork had a huge impact on how options were viewed in the financial world. In an idealized financial market, the price of a European option can besobtained as the solution of the Black-Scholes equation =-=[4]-=- [5]. However, the Black Scholes equation has beens*Corresponding author.sI. P. Akpan, J. O. Fatokuns284sderived under quite restrictive assumptions such as frictionless, liquid and complete market. I... |

54 | Market Manipulation, Bubbles, Corners and Short Squeezes,” - Jarrow - 1992 |

49 |
An asymptotic analysis of an optimal hedging model for option pricing with transaction costs
- Whalley, Wilmott
- 1997
(Show Context)
Citation Context ...ce of the option do not exist. From a binomial model, [6] derived an option price that takes into account transaction costs which approximates a BlacksScholes price but with modified volatility. [14] =-=[15]-=- computed the option price of the Black Scholes equation assthe solution of a nonlinear quasi-variational inequality. This approach has the disadvantage that the option pricesdepends on the choice of ... |

41 | On Feedback Effects from Hedging Derivatives,” - Platen, Schweizer - 1998 |

35 | P.: The feedback effect of hedging in illiquid markets
- Schönbucher, Wilmott
- 2000
(Show Context)
Citation Context ... In recent years nonlinear Black Scholes equations have been derived in order to model transaction costs arising in the hedging ofsportfolios [1] [6] [7] and feedback effects due to large traders [8]-=-=[13]-=-.sIn seeking the solution of the Black Scholes equation, emphasis is always laid on derivation of formula orsequation for the price of the option of interest and computation of the price of the option... |

30 | Perfect Option Hedging for a Large Trader,”
- Frey
- 1998
(Show Context)
Citation Context ...ket. In recent years nonlinear Black Scholes equations have been derived in order to model transaction costs arising in the hedging ofsportfolios [1] [6] [7] and feedback effects due to large traders =-=[8]-=--[13].sIn seeking the solution of the Black Scholes equation, emphasis is always laid on derivation of formula orsequation for the price of the option of interest and computation of the price of the o... |

30 | A.: Dynamic hedging portfolios for derivative securities in the presence of large transaction costs
- Avellaneda, Parás
- 1994
(Show Context)
Citation Context ...ion [20], and finite difference approximations [1]. The numerical discretization of the Black Scholes equations with nonlinear volatilities hassbeen performed using explicit finite difference schemes =-=[21]-=- [22]. Explicit numerical schemes have the disadvantage that restrictive conditions on the discretization parameters, time and space steps, are needed to obtainsstable, convergent schemes [23]. Moreov... |

22 |
Error estimates for the binomial approximation of American put options.
- Lamberton
- 1998
(Show Context)
Citation Context ...re, only a few results can be found on the numerical discretization of BlacksScholes equations. The numerical approaches vary from binomial approximations for American options in stochastic framework =-=[18]-=-, Monte-Carlo methods [19], finite element discretization [20], and finite difference approximations [1]. The numerical discretization of the Black Scholes equations with nonlinear volatilities hassbe... |

15 | illiquidity as a source of model risk in dynamic hedging in model risk - Frey - 2000 |

11 | Mesh adaption for the Black and Scholes equations,
- Pironneau, Hecht
- 2000
(Show Context)
Citation Context ...ation of BlacksScholes equations. The numerical approaches vary from binomial approximations for American options in stochastic framework [18], Monte-Carlo methods [19], finite element discretization =-=[20]-=-, and finite difference approximations [1]. The numerical discretization of the Black Scholes equations with nonlinear volatilities hassbeen performed using explicit finite difference schemes [21] [22... |

8 | On analytical solutions of the Black-Scholes equation
- Bohner, Zheng
(Show Context)
Citation Context ...ar quasi-variational inequality. This approach has the disadvantage that the option pricesdepends on the choice of the utility function. Seeking the analytical solution of the Black-Scholes equation, =-=[16]-=-sused the Adomain decomposition method. Adomain decomposition can provide analytical approximations to aswide class of linear and nonlinear equations without perturbation, closure approximations or di... |

8 | M.: On the numerical solution of nonlinear Black-Scholes equations
- Ankudinova, Ehrhardt
- 2008
(Show Context)
Citation Context ... 2xx Tw w xτ σ τ = ∈ ∈ s(19)sSubject tos( ) ( ) ( )1 1 2 2,0 max e e ,0 , k k w x x + − = − ∈ s(20a)s( ) 2 , 0 as , 0, 2 Tw x x στ τ → → ±∞ ∈ s(20b)sAccording to =-=[28]-=- European Call option as the solution to the Black Scholes equation on 0 ,0S t T≤ < ∞ ≤ ≤scan be approximated bys( ) ( ), ~ e as ,r T tV S t S K S− −− → ∞s(21)sEquating both hand sides of Equation (21... |

6 |
Chaotic solution for the Black-Scholes equation,
- Emamirad, Goldstein, et al.
- 2012
(Show Context)
Citation Context ... nonlinear equations without perturbation, closure approximations or discretization. Solution for the Black-Scholes equation as a semigroup on spaces of continuous functions on (0, ∞) is presented ins=-=[17]-=-.sIn the mathematical literature, only a few results can be found on the numerical discretization of BlacksScholes equations. The numerical approaches vary from binomial approximations for American op... |

3 | Applications of Monte Carlo/quasi-Monte Carlo methods in ¯nance: Option pricing
- Lai, Spanier
- 1998
(Show Context)
Citation Context ... be found on the numerical discretization of BlacksScholes equations. The numerical approaches vary from binomial approximations for American options in stochastic framework [18], Monte-Carlo methods =-=[19]-=-, finite element discretization [20], and finite difference approximations [1]. The numerical discretization of the Black Scholes equations with nonlinear volatilities hassbeen performed using explici... |

2 |
RBFs meshless method of lines for the numerical solution of timedependent nonlinear coupled partial differential equation
- Haq, Hussain, et al.
(Show Context)
Citation Context ...r the solution of time-dependent partial differential equations (PDEs) [24]. The basic idea of the MOL is to replace the spatial (boundary value) derivatives in thesPDEs with algebraic approximations =-=[25]-=-. Ones this is done, the spatial derivatives are no longer stated explicitly in terms of the spatial dependent variables. Thus, only the initial value variable, typically time in a physicalsproblem, r... |

1 |
A New Explicit Formula for the Solution of the Balck-Merton- Scholes Equation
- Goldstein, Mininni, et al.
- 2008
(Show Context)
Citation Context ...20], and finite difference approximations [1]. The numerical discretization of the Black Scholes equations with nonlinear volatilities hassbeen performed using explicit finite difference schemes [21] =-=[22]-=-. Explicit numerical schemes have the disadvantage that restrictive conditions on the discretization parameters, time and space steps, are needed to obtainsstable, convergent schemes [23]. Moreover th... |

1 |
Solving Black Scholes Equation: A Demystification. www.francoidcoppe.com/blackscholes.pdf
- Coppex
- 2009
(Show Context)
Citation Context ...sV S t S S →∞s(2b)s( ) ( ), max ,0V S T S K= −s(2c)swhere:sGeneral option priceV =s(3a)sAsset priceS =s(3b)sRisk-free interest rater =s(3c)sI. P. Akpan, J. O. Fatokuns285sVolitilityσ =s(3d)sFollowing =-=[27]-=- to transform the diffusion-advection-reaction Equation (1) into a parabolic heat PDE, we makesthe following change of variables:s( ) 2 2 T t σ τ − =s(4a)sexS K=s(4b)s( ) ( ), ,V S t Ku x τ=s(4c)sBy t... |

1 |
L-Stable Implicit Trapezoidal-Like Integrators for the Solution of Parabolic Partial Differential Equations on Manifolds
- Fatokun, Akpan
- 2013
(Show Context)
Citation Context ...retization matrix respectively;slsis the time step; and 'sdenotes differentiation with respect to time.sThe derivation of the method (28) and the analysis of the order of accuracy are as discussed in =-=[29]-=-, while thesstability properties of the method are discussed in [30].s4. Numerical ExperimentationsFor numerical experimentation the following values were used: k = 0.001, r = 0.1, σ = 0.2, K = 100, T... |

1 |
A Trapezoidal-Like Integrator for the Numerical Solution of One-Dimensional Time Dependent Schrodinger Equation
- Fatokun, Akpan
- 2014
(Show Context)
Citation Context ...fferentiation with respect to time.sThe derivation of the method (28) and the analysis of the order of accuracy are as discussed in [29], while thesstability properties of the method are discussed in =-=[30]-=-.s4. Numerical ExperimentationsFor numerical experimentation the following values were used: k = 0.001, r = 0.1, σ = 0.2, K = 100, T = 1, Δx =s0.01 and Δτ = 0.001.s5. Computation of Absolute and Relat... |