# Gamma-Vega Neutral Portfolio Not Possible with Only 3 Options

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?

• @noob2 Isn't $\mathbb{P}\left[\omega \in \Omega \mid V_y \Gamma_z-V_z\Gamma_y = 0 \right] = 0$? – SmurfAcco Mar 18 at 8:31

In Black-Scholes world, we have:

$$V_y= \sigma_y \tau_y S^2 \Gamma_y$$

and similarly for $$z$$.

• Thanks! So the only way we could have the determinant being non-zero is if the time until expiration is different for both call options. – 383930283423 Mar 18 at 23:54
• Yes, assuming flat implied vol (same for all strikes and times to expiriez), then selected times to expiries must be different. – ir7 Mar 19 at 0:08