Black-Scholes formula with deterministic discrete dividend (Musiela approach)

For deterministic discrete dividend, there are two approach

• Musiela approach, works when every dividend are paid at maturity of the option.


• Hull approach, works when every dividend are paid immediately after ex-dividend date.

I spend 1 day to understand the Musiela approach, but I can not understand his formula. In his book "Martingal Method for Financial Modelling 2nd Edit" $3.2.2, his first approach firstly define quantity :

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  Timeline $0 < T_1
  




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  Value of all posterior-t dividend compounded to Maturity time : $$ I_t = \sum^m_{i=1} q_i e^{r(T-T_i)} \mathbf{1}_{[0,T_i]}(t) $$ Note that $I_t$ decrease in time $t$ and piecewise constant. At each time $T_i$, $I_t$ drop down $q_i$

  
  


• Value of all anterior-t dividend compound to time $t$. $$ D_t = \sum^m_{i=1} q_i e^{r(t-T_i)} \mathbf{1}_{[T_i,T]}(t) $$ Here, $D_t$ increase in time $t$. At each time $T_i$, $D_t$ jump up $q_i$


• He define the capital gain process $$ G_t = S_t + D_t $$

Note that $$ D_T=I_0 \hspace{1cm} G_0=S_0 \hspace{1cm} G_T=S_T+D_T=S_T+I_0 $$
And all jump in price process $S_t$ are separated to $D_t$, he can model $G_t$ by the geometric brownian as usual, i.e under risk-neutral measure $$ \frac{dG_t}{G_t} = rdt + \sigma dW_t $$ Now, he can give the B&S formula for European Call option at time zero $$ C_0 = e^{-rT}\mathbb{E}[(S_T-K)^+] = e^{-rT}\mathbb{E}[(G_T-(K+I_0))^+] $$ Since the modelled process is $G_t$, this price at time $0$ is easily found by Black-Scholes calculation routine. $$ C_0 = S_0 \mathcal{N}(d_+) - e^{-rT}K \mathcal{N}(d_{-}) $$ with $$ d\pm = \frac{ \text{ln}\frac{S_0}{K+I_0} + (r\pm\frac{\sigma^2}{2})T } {\sigma \sqrt{T}} $$ For this price formula at time $0$, I can understand it. An then I tried to compute for an arbitrary time $t$ $$ C_t = e^{-r(T-t)}\mathbb{E}[(S_T-K)^+|\mathcal{F}_t] = e^{-r(T-t)}\mathbb{E}[(G_T-(K+I_0))^+|\mathcal{F}_t] $$ Again, the calculation routine of Black-Scholes should give $$ C_t = G_t \mathcal{N}(d_+) - e^{-r(T-t)} (K+I_0) \mathcal{N}(d_{-}) $$ with $d\pm$ should be $$ d\pm = \frac{ \text{ln}\frac{G_t}{K+I_0} + (r\pm\frac{\sigma^2}{2})(T-t) } {\sigma \sqrt{T-t}} $$ But in the Musiela's book, he give the different result without detail proof. His result is $$ C_t = S_t \mathcal{N}(\hat{d}_+) - e^{-r(T-t)} (K+I_t) \mathcal{N}(\hat{d}_{-}) $$ with $$ \hat{d}\pm = \frac{ \text{ln}\frac{S_t}{K+I_t} + (r\pm\frac{\sigma^2}{2})(T-t) } {\sigma \sqrt{T-t}} $$ So the annoying differences are

• He have strike term as $K+I_t$, I have $K+I_0$



• He have random process as $S_t$, I have $G_t$

Can anyone help please. I've spent to much time without success.

PS : one more question. There are maybe something that I am missing. The fact that he use $D_t$ to model the dividend, but in the result, he use $I_t$, that seems strange.