Abstract
In an analysis of the Lie point symmetries of the equation which allows for the inclusion of transaction costs into the Black–Scholes model (J Eng Math 82:67–75, 2013) one of the symmetries was omitted. We rectify that failing and demonstrate that the correctness of the subsequent analysis is not compromised.
1 Erratum to: J Eng Math (2013) 82:67–75 DOI 10.1007/s10665-012-9595-4
The governing evolution partial differential equation for the Black–Scholes model when transaction costs are included is [1] (their Eq. 3.1)
subject to the terminal condition
where \(f (x)\) is initially unspecified and is to be determined from the analysis.
We analyse (1) for the possession of Lie point symmetries. We find that in general there are five Lie point symmetries, namely
To solve the boundary-value problem we apply the general symmetry, \(\Gamma = \sum _{i = 1} ^ 5 \alpha _i\Gamma _i \), where the \(\alpha _i \), \(i = 1,\dots , 5\), are as yet arbitrary constants, to the terminal conditions (2). We obtain the two relations
Note that we have eliminated \(\alpha _1\) in favour of \(-\alpha _5T\). The second relation may be considered as a first-order equation for \(f (x) \). If \(\alpha _5\ne 0\), its solution is, obtained using Mathematica,
in which a constant term,
has been incorporated into the constant of integration to give \(c_1\), and, if \(\alpha _5=0\), we have
where \(c_1\) is the constant of integration. The form of \(f(x)\) in (5) provides the solution (3.8) in [1], while the form in (4) will be examined elsewhere.
References
O’Hara JG, Sophocleous C, Leach PGL (2013) The application of Lie point symmetries to the resolution of certain problems in financial mathematics with a terminal condition. J. Eng. Math. 82:67–75. doi:10.10007/s10665-012-9595-4
Acknowledgments
We thank Professors Sergii Kovalenko and Olena Vaneeva of the Institute of Mathematics in National Academy of Sciences of Ukraine for drawing to our attention the missing symmetry, \(\Gamma _5\).
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The online version of the original article can be found under doi:10.1007/s10665-012-9595-4.
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O’Hara, J.G., Sophocleous, C. & Leach, P.G.L. Erratum to: The application of Lie point symmetries to the resolution of certain problems in financial mathematics with a terminal condition. J Eng Math 91, 215–216 (2015). https://doi.org/10.1007/s10665-014-9772-8
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DOI: https://doi.org/10.1007/s10665-014-9772-8