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Let $\sigma(F,K)$ be the SABR implied vol. In the shifted model, the formula essentially becomes $\sigma(F+x,K+x)$ (you have to shift the strike as well). So to answer your question in the ATM vol calibration you take $K=F$ in order to have $F+x=K+x$. There is no need to "reconcile" anything as it is just a model. Once you have your model, you have to ...
$X_t$ being a stochastic process, one cannot use ordinary calculus to express the differential of a (sufficiently well-behaved) function $f$ of $t$ and $X_t$. Instead one should turn to Itô's lemma, one of the key results of stochastic calculus, which stipulates (assuming $X_t$ is here a continuous, square integrable stochastic process) $$df(t,X_t) = ... 7 This is an interesting and not so easy question. Here's my 2 cents: First, you should distinguish between mathematical models for the dynamics of an underlying asset (Black-Scholes, Merton, Heston etc.) and numerical methods designed to calculate financial instruments' prices under given modelling assumptions (lattices, Fourier inversion techniques etc.). ... 4 Let$$q (S) := \frac{d\mathbb {Q}(S_T \leq S)}{dS}  denote the probability density function of the stock price at time $T>0$ under the risk-neutral measure. By definition, the price of a European call then writes \begin{align} C (K,T) &= P (0,T) E_0^{\mathbb {Q}}[(S_T-K)^+] \\ &= P (0,T) \int_K^\infty (S - K) q (S) dS \end{align} with \$P ...