Is the Relationship between Theta and Vega Linear or Non-Linear?

The relationship between Theta and Vega is non-linear. Both Greeks are related to the option's extrinsic value and time to expiration, but their interaction is complex.

For example, as time to expiration decreases, Theta increases exponentially, while Vega decreases. This non-linear relationship is governed by the Black-Scholes-Merton partial differential equation.