Let's say we have sold a call option, x, on a share and we have 2 other call options, y & z, with different strikes and maturities to try and achieve a portfolio that is both Gamma and Vega neutral. We just need to solve the following system of equations:

$$\begin{bmatrix} \Delta_x \\ \Gamma_x \\ \end{bmatrix} = \begin{bmatrix} \Delta_y & \Delta_z \\ \Gamma_y & \Gamma_z \end{bmatrix} % \begin{bmatrix} y \\ z \end{bmatrix} $$

However, a solution only exists if the 2x2 matrix above is invertible, i.e. $\Delta_y \dot \Gamma_z-\Delta_z\dot\Gamma_y\ne0$.

Is there any reason why the solution wouldn't exist or a name given to the situation when it happens? Or does this happen just by chance, that the determinant is 0?

Let's say we have sold a call option, x, on a share and we have 2 other call options, y & z, with different strikes and maturities to try and achieve a portfolio that is both Gamma and Vega neutral. We just need to solve the following system of equations:

$$\begin{bmatrix} V_x \\ \Gamma_x \\ \end{bmatrix} = \begin{bmatrix} V_y & V_z \\ \Gamma_y & \Gamma_z \end{bmatrix} % \begin{bmatrix} y \\ z \end{bmatrix} $$

However, a solution only exists if the 2x2 matrix above is invertible, i.e. $V_y \Gamma_z-V_z\Gamma_y\ne0$.

Is there any reason why the solution wouldn't exist or a name given to the situation when it happens? Or does this happen just by chance, that the determinant is 0?