Black–Scholes Call Theta and the Fixed-Spot Time Derivative

Summary

The document asks why a call option’s theta in the Black–Scholes model does not appear to include a term involving the time change in the underlying share price. It presents the familiar call pricing expression and a standard theta formula, then questions why derivations treat the spot price as constant when differentiating with respect to time.

The central distinction is between theta as a partial derivative of the option price with respect to calendar time while holding the current spot price fixed, and the change in option value along a path where the spot price itself moves. The latter also reflects the option’s sensitivity to spot, commonly represented by delta. The question itself provides no worked derivation or empirical evidence, and the notation and discounting in the displayed pricing expression may need checking against the usual Black–Scholes convention. The discussion concerns the definition of a model Greek, rather than a trading strategy.

Key ideas

• Black–Scholes theta is defined as a partial derivative with spot held fixed.
• A pathwise change in option value can include both the time effect and the effect of a changing spot price.
• The spot-related contribution is associated with the option’s delta.
• The document raises the conceptual question but does not provide a derivation of the theta formula.

Tags

# Back Scholes theta of the call at any time t


# Back Scholes theta of the call at any time t












I 'm trying to get the theta of a Call in the classical Black Scholes model. We have (classical result with usual notations) :

$$C_t = S_tN(d_1) - Ke^{r(T-t)}N(d_2)$$ When deriving according to time, I get, among others terms, this one : $$N(d_1)*\partial{S_t}/\partial{t}$$

But I can't find anywhere this term in any formula I found on the net ...

None of the proofs I saw of the fomula $$ \theta_t = - \frac{S \sigma}{2\sqrt{T-t}}N'(d_1) -rKe^{-r(T-t)}N(d_2) $$ are clear to me. Because I don't understand why $S$ is treated as if it was not time dependant ...

What do I miss ?

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