There is a need to derive theta $\theta$ of an option out of standard Black-Scholes PDE. In usual notation ($P$ - price of an option, $S$ - underlying spot):
Which Greeks and volatility should be used from theoretical point of view. There are 3 choices as I see:
1) Non-adapted Black-Scholes Greeks and implied vol - most straightforward approach but lacks smile dynamics adaptation
2) Adapted Greeks (adaptation is done based on implied BS vol) and implied vol - this approach is under question. Since use of implied vol means that $\sigma(t,S_t)$ is constant for a given strike. However it contradicts the use of adaptation of the greeks.
3) Adapted Greeks and instantaneous vol. This seems to be theoretically justified approach since Greeks adaptation uses instantaneous volatility (not implied one)
Would be grateful for your feedback regarding the possibility to use approach 2.