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In this paper the Black Scholes differential equation is transformed into a parabolic heat equation by appropriate change in variables. The transformed equation is semi-discretized by the Method of Lines (MOL). The evolving system of ordinary differential equations (ODEs) is integrated numerically by an L-stable trapezoidal-like integrator. Results show accuracy of relative maximum error of order 10
^{–10}.

Financial derivative, in particular options, became very popular financial contracts in the last few decades. Options can be used to hedge assets and portfolios in order to control the risk due to movements in the share price. A European call (put) option provides the right share price to buy or sell a fixed number of assets at the fixed exercised price E, at the expiry time t_{0} [

In seeking the solution of the Black Scholes equation, emphasis is always laid on derivation of formula or equation for the price of the option of interest and computation of the price of the option. This calls for the usage of numerical methods because explicit theoretical solutions for the price of the option do not exist. From a binomial model, [

In the mathematical literature, only a few results can be found on the numerical discretization of Black Scholes equations. The numerical approaches vary from binomial approximations for American options in stochastic framework [

The Method of Lines (MOL) is a general procedure for the solution of time-dependent partial differential equations (PDEs) [

This paper is organized thus: In Section 2 we transform the black Scholes equation into a heat equation by change in variables. In Section 3 we introduce an L-stable trapezoidal like integrator for the numerical integration of the transformed Black Scholes equation. Section 4 is devoted to the numerical test of the method on the transformed Black Scholes equation. Section 5 explains the computation of the errors and relative errors of the method while results are discussed in Section 6.

Given the Black Scholes equation:

Subject to:

where:

Following [

By taking appropriate partial derivatives of

Substituting

Defining

And substituting for k in Equation (6) we obtain

By defining

Then, the partial derivatives of

Making

Substituting Equations (13), (14) and (15) into Equation (8) we obtain

By letting the coefficients of

Under the condition that

and

the Black Scholes equation given in Equation (1) is transformed into the parabolic heat equation PDE

Subject to

According to [

Equating both hand sides of Equation (21) yields

Substituting for S from Equation (4) in Equation (22) gives

Substituting

By equating the RHS of Equations (2b) and (24)

Substituting

Substituting Equation (26) for

Remark

By appropriate change in variables, Equation (1) is transformed into Equation (19) which is a parabolic heat equation to be discretized by the MOL. Equation (27) is the derived approximate theoretical solution to the transformed Black Scholes equation.

The trapezoidal-like integrator

where

The derivation of the method (28) and the analysis of the order of accuracy are as discussed in [

For numerical experimentation the following values were used: k = 0.001, r = 0.1, σ = 0.2, K = 100, T = 1, Δx = 0.01 and Δτ = 0.001.

In this section we explain how the absolute errors and relative errors of the methods shown in

t | New Scheme | Theoretical Solution | Errors |
---|---|---|---|

0.0 | 0.0100000917 | 0.0100000917 | 0 |

0.001 | 0.01000359219 | 0.0100035927 | 5.1 × 10^{−10 } |

0.002 | 0.01000709380 | 0.0100070936 | 2.0 × 10^{−10} |

0.003 | 0.01001059654 | 0.0100105966 | 6.0 × 10^{−11} |

0.004 | 0.01001410040 | 0.0100141006 | 2.0 × 10^{−10} |

0.005 | 0.01001760539 | 0.0100176056 | 2.1 × 10^{−10} |

0.006 | 0.01002111150 | 0.0100211126 | 1.10 × 10^{−9} |

0.007 | 0.01002461875 | 0.0100246196 | 8.5 × 10^{−10} |

0.008 | 0.01002812713 | 0.0100281285 | 1.37 × 10^{−9} |

0.009 | 0.01003163663 | 0.0100316373 | 6.7 × 10^{−10} |

0.010 | 0.01003514726 | 0.0100351481 | 4.4 × 10^{−10} |

0.011 | 0.01003865902 | 0.0100386598 | 7.8 × 10^{−10} |

0.012 | 0.01004217190 | 0.0100421733 | 1.4 × 10^{−9} |

0.013 | 0.01004568591 | 0.0100456878 | 1.89 × 10^{−9} |

0.014 | 0.01004920106 | 0.0100492031 | 2.04 × 10^{−9} |

0.015 | 0.01005271734 | 0.0100527192 | 1.86 × 10^{−9} |

0.016 | 0.01005623474 | 0.0100562362 | 1.46 × 10^{−9} |

0.017 | 0.01005975327 | 0.0100597549 | 1.63 × 10^{−9} |

0.018 | 0.01006327294 | 0.0100632755 | 2.56 × 10^{−9} |

0.019 | 0.01006679374 | 0.0100667959 | 2.16 × 10^{−9} |

0.020 | 0.01007031567 | 0.0100703180 | 2.33 × 10^{−9} |

t | New scheme | Theoretical solution | Errors | Relative errors |
---|---|---|---|---|

0.000 | 0.0100000917 | 0.0100000917 | 0 | 0 |

0.001 | 0.01000359219 | 0.0100035927 | 5.1 × 10^{−10 } | 5.049486987 × 10^{−10 } |

0.002 | 0.01000709380 | 0.0100070936 | 2.0 × 10^{−10} | 1.980184111 × 10^{−10 } |

0.003 | 0.01001059654 | 0.0100105966 | 6.0 × 10^{−11} | 5.940531731 × 10^{−11 } |

0.004 | 0.01001410040 | 0.0100141006 | 2.0 × 10^{−10} | 1.980170374 × 10^{−10 } |

0.005 | 0.01001760539 | 0.0100176056 | 2.1 × 10^{−10} | 2.079171677 × 10^{−10 } |

0.006 | 0.01002111150 | 0.0100211126 | 1.10 × 10^{−9} | 1.089086145 × 10^{−9 } |

0.007 | 0.01002461875 | 0.0100246196 | 8.5 × 10^{−10} | 8.415636443 × 10^{−10 } |

0.008 | 0.01002812713 | 0.0100281285 | 1.37 × 10^{−9} | 1.356397869 × 10^{−9 } |

0.009 | 0.01003163663 | 0.0100316373 | 6.7 × 10^{−10} | 6.633455582 × 10^{−10 } |

0.010 | 0.01003514726 | 0.0100351481 | 4.4 × 10^{−10} | 4.356284045 × 10^{−10 } |

0.011 | 0.01003865902 | 0.0100386598 | 7.8 × 10^{−10} | 7.722476682 × 10^{−10 } |

0.012 | 0.01004217190 | 0.0100421733 | 1.4 × 10^{−9} | 1.386080737 × 10^{−9 } |

0.013 | 0.01004568591 | 0.0100456878 | 1.89 × 10^{−9} | 1.871202484 × 10^{−9 } |

0.014 | 0.01004920106 | 0.0100492031 | 2.04 × 10^{−9} | 2.019703589 × 10^{−9 } |

0.015 | 0.01005271734 | 0.0100527192 | 1.86 × 10^{−9} | 1.841488038 × 10^{−9 } |

0.016 | 0.01005623474 | 0.0100562362 | 1.46 × 10^{−9} | 1.445464072 × 10^{−9 } |

0.017 | 0.01005975327 | 0.0100597549 | 1.63 × 10^{−9} | 1.613765910 × 10^{−9 } |

0.018 | 0.01006327294 | 0.0100632755 | 2.56 × 10^{−9} | 2.534494681 × 10^{−9 } |

0.019 | 0.01006679374 | 0.0100667959 | 2.16 × 10^{−9} | 2.138472434 × 10^{−9 } |

0.020 | 0.01007031567 | 0.0100703180 | 2.33 × 10^{−9} | 2.306770092 × 10^{−9 } |

The absolute errors of the scheme were computed by the use of the formula:

where the numerical solution at the grid point

Relative errors of the method were computed by use of the formula:

where the numerical solution at the grid point

On the implementation of the L-stable trapezoidal-like integrator for the solution of transformed Black Scholes equation after discretizing with MOL, the errors and relative errors of the scheme were computed as shown in

Iyakino P.Akpan,Johnson O.Fatokun, (2015) An Accurate Numerical Integrator for the Solution of Black Scholes Financial Model Equation. American Journal of Computational Mathematics,05,283-290. doi: 10.4236/ajcm.2015.53026