I want to clarify first that, I am not arguing against the essentially Black-Scholes formula with the time-dependent volatility per se, you wrote in your previous post, as evidenced by my ready agreement before. Our difference is how to take the partial derivative of the variance integral [$]U[$] with respect to time, specifically whether it should be [$]U_{t,1}[$] (line 1) or [$]U_{t,2}[$] (line 2).
You said:
The formulas I wrote for [$]C(t,T,S_t)[$] are standard. If you agree with them, then apparently we are arguing about how to take a partial derivative.
So, yes, we are indeed arguing about how to take a partial derivative.
I understand your reasoning regarding the deterministic volatility case. I thought of the same thing before I came to my current and different conclusion. I think the confusing factor is that we are now talking about the deterministic volatility as an end to itself. Yes, I used the deterministic volatility, but only to make a simplified computation as an extreme and degenerate example of the stochastic volatility case, not as an end to itself. My ultimate goal is to investigate the proper stochastic volatility, not the deterministic volatility. At this stage, this degenerate case seems to add unnecessary confusion. In this respect, the exponential example will not clarify but only highlight our difference, because my answer will be different from yours. I have already written out the two different partial derivatives, namely [$]U_{t,1}[$] and [$]U_{t,2}[$] for this exponential volatility function, in my last post. You want to pick [$]U_{t,2}[$], and I want to pick [$]U_{t,1}[$]. We already know our difference, particularly for this example. We now need to resolve it.
With the above preamble, let us back up and look at a proper stochastic volatility model instead of its degenerate deterministic volatility model.
I understand you do not agree with setting the volatility arbitrarily, specifically to [$]\beta_0[$], at [$]t_1[$] and [$]t_2[$] for the deterministic volatility. Now, for the proper stochastic volatility case, e.g., a Heston model with a positive volatility of volatility, do you agree we can? Specifically, we value two calls with the same stock price [$]S_0[$] and the same volatility [$]\beta_0[$] at two distinct valuation times [$]t_1[$] and [$]t_2[$]. We denote the two call values as [$]C(t_1,\beta_0)[$] and [$]C(t_2,\beta_0)[$].
Do you agree we can do that?