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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?

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  • $\begingroup$ @noob2 Isn't $ \mathbb{P}\left[\omega \in \Omega \mid V_y \Gamma_z-V_z\Gamma_y = 0 \right] = 0$? $\endgroup$
    – SmurfAcco
    Commented Mar 18, 2020 at 8:31

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In Black-Scholes world, we have:

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

and similarly for $z$.

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  • $\begingroup$ 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. $\endgroup$
    – 383930283423
    Commented Mar 18, 2020 at 23:54
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    $\begingroup$ Yes, assuming flat implied vol (same for all strikes and times to expiriez), then selected times to expiries must be different. $\endgroup$
    – ir7
    Commented Mar 19, 2020 at 0:08

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