This question is not for homework, it is just for personal curiosity.
I am aware that we can calculate the greeks using basic calculus, and simplification. I am also aware that there is a great video by quantpie on youtube that uses a similar method to what I am trying to do.
Note: The greeks are the partial derivatives of the price of the european call option taken with respect to the parameters. I am concerned with $\Delta$ which is the partial derivative with respect to $S(0)$.
My goal: calculate $$\Delta=\partial C/\partial S(0)$$ using change of probability measure, where $$C=e^{-rT}\mathbb{E}[max(S(T)-K,0)]$$ where C is the price of the European call with the strike price K and expiration T. Also $$S(t)=S(0)e^{(r-0.5\sigma^{2})t+\sigma W(t)}$$
We know that there exists a measure Q such that C is a martingale. So we have,
$$C(0)=e^{-rt}\mathbb{E}_{Q}[max(S(t)-K,0) \mid \mathcal{F}_{0}]$$
where $\mathcal{F}_{0}$ is natural filtration with information at $t=0$. So we have
$$\frac{\partial C(0)}{\partial S(0)}=e^{-rt}\frac{\partial}{\partial S(0)}\mathbb{E}_{Q}[max(S(t)-K,0) \mid \mathcal{F}_{0}]$$ $$e^{-rt}\mathbb{E}_{Q}[\frac{\partial}{\partial S(0)} max(S(t)-K,0) \mid \mathcal{F}_{0}] \ (justification \ needed)$$ $$=e^{-rt}e^{(r-0.5\sigma^{2})t+\sigma W(t)}\mathbb{E}_{Q}[\mathbb{I}_{S(t)>K} \mid \mathcal{F}_{0}]$$ $$=e^{-rt}e^{(r-0.5\sigma^{2})t+\sigma W(t)}\mathbb{Q}[S(t)>K]$$
Note: C is martingale under $\mathbb{Q}-measure$
However, things aren't cancelling out very nicely. Ideally things would simplify to to $$\frac{\partial C(0)}{\partial S(0)}=\mathbb{Q}[S(t)>K]$$ but this is not really working out. I appreciate any help or tips!
You have
$$C_0=e^{-rT}E^Qmax(S_T-K,0)=e^{-rT}E^Qmax(S_0e^{(r-0.5\sigma^2)T+\sigma W_T}-K,0)$$
Hence
$$\frac{\partial C_0}{\partial S_0}=e^{-rT}E^Q(I(S_T-K>0)e^{(r-0.5\sigma^2)T+\sigma W_T})=e^{-rT}E^Q(I(S_T-K>0)S_T/S_0),$$
where $I$ is the indicator (more formally heaviside) function. This term can be evaluated by noting that $S_T$ is log-normal. This describes the risk neutral expectation of the stock price conditional on hitting above $K$. In the Black-Scholes model this turns out to be of the form $N(d_1)$. The derivation is a bit tedious but very standard and can be found in textbooks covering the BS model.