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)

$$\begin{aligned} 0 = u_t + \frac{1}{2}\tilde{\sigma } ^ 2x ^ 2u_{xx} + b\sigma ^ 2x ^3u_{xx} ^ 2 + r (xu_x -u) \end{aligned}$$
(1)

subject to the terminal condition

$$\begin{aligned} u (T,\,x) = f (x), \end{aligned}$$
(2)

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

$$\begin{aligned} \begin{aligned}&\Gamma _1 = \partial _t, \quad \Gamma _2 = x\partial _u, \quad \Gamma _3 = e^{rt}\partial _u, \quad \Gamma _4 = x\partial _x + u\partial _u, \\&\Gamma _5 = t\partial _t + rtx\partial _x + \left\{ \frac{\tilde{\sigma }^2x}{8b\sigma ^2}\left( 2+2rt+\tilde{\sigma }^2t-2\log x\right) + (rt-1)u\right\} \partial _u. \end{aligned} \end{aligned}$$
(3)

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

$$\begin{aligned}&\alpha _1+\alpha _5 T = 0 \\&\text{ and }\\&\alpha _2x + \alpha _3 e^{rT}+\alpha _5\frac{\tilde{\sigma }^2x}{8b\sigma ^2}\left( 2+2rT+\tilde{\sigma }^2T-2\log x\right) + (\alpha _4+\alpha _5(rT-1))f = (\alpha _4+\alpha _5 rT)xf'. \end{aligned}$$

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,

$$\begin{aligned} f (x)&= -\frac{\alpha _3 \text{ e }^{r T}}{\alpha _4+\alpha _5 (-1+r T)}+\frac{\left( 8 \alpha _2 b \sigma ^2+ \tilde{\sigma }^2 \left( 2 \alpha _4+\alpha _5 \left( 2+4 r T+ \tilde{\sigma }^2 T\right) \right) \right) x}{8 \alpha _5 b \sigma ^2}\nonumber \\&+\, c_1 x^{1- \alpha _5/(\alpha _4+\alpha _5 r T)}-\frac{\tilde{\sigma }^2 x \log x}{4 b \sigma ^2}, \end{aligned}$$
(4)

in which a constant term,

$$\begin{aligned} (\alpha _4+\alpha _5 rT)^{1- \alpha _5/(\alpha _4+\alpha _5 r T)}, \end{aligned}$$

has been incorporated into the constant of integration to give \(c_1\), and, if \(\alpha _5=0\), we have

$$\begin{aligned} f(x)=c_1x-\frac{\alpha _3}{\alpha _4}e^{rT}+\frac{\alpha _2}{\alpha _4}x\log x \end{aligned}$$
(5)

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.