Choosing Volatility and Greeks for Black-Scholes Theta

Summary

The document raises a modeling question about computing option theta from the Black-Scholes partial differential equation when market implied volatility varies across strikes. It compares three approaches: standard Black-Scholes Greeks with implied volatility, Greeks adjusted to reflect implied volatility, and adjusted Greeks paired with instantaneous volatility. The concern is that using a fixed implied volatility for a strike may conflict with adapting the Greeks to account for the volatility smile.

The post supplies no answer, derivation, market example, or evidence that resolves the issue. It is therefore best read as a conceptual prompt about consistency between the volatility input and the Greek calculation, rather than as a prescribed theta method. The central caveat is that the appropriate treatment depends on the model assumptions behind the adapted Greeks and on how volatility changes with spot and time.

Key ideas

• The Black-Scholes PDE relates theta to option price, delta, gamma, rates, and volatility.
• Implied volatility varies across strikes, so standard Greeks may not reflect smile dynamics.
• Adjusting Greeks using implied volatility raises a consistency question when that volatility is held fixed for a strike.
• Using instantaneous volatility with adapted Greeks is proposed as a theoretically motivated alternative, but the document does not establish an answer.

Tags

# Theta from Black-Scholes PDE - is it possible to use implied volatility?


# Theta from Black-Scholes PDE - is it possible to use implied volatility?












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):

$\theta=r_dP−Sr_d\delta−\frac{1}{2}\sigma^2S^2\gamma$

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.

Thank you!

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.