Dupire Formula with Discrete Cash Dividend

For stocks when there is cash dividend, the Dupire formula should still hold according to Bergomi. In the book "Stochastic Volatility Modeling", he says:

In the presence of cash-amount dividends, while the Dupire formula (2.3) with option prices is still valid, its version (2.19) expressing local volatilities directly as a function of implied volatilities cannot be used as is, as option prices are no longer given by the Black-Scholes formula.

The above quote can be found in the free chapter 2 in his book, the first paragraph of section 2.3.1.

I do not find this obvious though. Can you please point it out to me or give me some references to read?

In addition, why does $$\frac{\partial C}{\partial T}$$ exist, as defined by the following equation based on a rewriting of the equation 2.3 in the book, even though the stock prices can jump, making $$C$$ not continuous in $$T$$?

$$\frac{\partial C}{\partial T} = \frac{1}{2} \sigma^2(K,T; S_0)K^2 \frac{\partial^2C}{\partial K^2}-(r - q)K \frac{\partial C}{\partial K} - qC$$

Also, the $$q$$ variable in the equation makes me think the model is specific for the continuous dividend yield.

• Hi Alper, It is equation 2.3 Commented Aug 10, 2023 at 13:12
• Thx. Do you mean a rewriting of equation 2.3 or based on equation 2.3? Because equation 2.3 appears to be different in the link you have provided and as follows: $\sigma(t, S)^2=\left.2 \frac{\frac{d C}{d T}+q C+(r-q) K \frac{d C}{d K}}{K^2 \frac{d^2 C}{d K^2}}\right|_{\substack{K=S \\ T=t}}$ In any case, it would be better if you include the definitions of the terms in the equation in your post so that it can still be understood and answered in case the link is broken or changed. Commented Aug 10, 2023 at 13:32
• Thanks/Apologies, Alpher. Yes, it needs a rewrite. Commented Aug 10, 2023 at 14:11