I will try to provide an answer to your questions:
- In retrospect, I do not believe that there is any industry standard for calculating historical volatility (I could be wrong on that part). As long as you have a consistent (and unbiased) estimator of variance (quadratic variation) you're set. As described in the original article of the Yang-Zhang estimator, the authors argue that the estimator is unbiased and consistent under a modified Merton jump-diffusion model. One way of procuring more accurate variance estimates is simply to use a higher frequency (eg. daily data) under the same variance estimator.
- If you're pricing stock options on eg. TSLA, you can recover interest rates and dividend yield via the put-call parity:
- Get option-data for $m$ maturities over the same stock (often 5 or 6 maturities are fine):
$$T_j \in \mathcal{T} = \{T_1,\ldots,T_m\}, \qquad j=1,\ldots,m.$$
- For each maturity $T_j$, gather $n$ call options with strikes around ATM (you can use 10 strikes around ATM),
\begin{align*}
K_i \in \mathcal{K} &=\{K_1, \ldots, K_n\}, \qquad i=1,\ldots,n.
\end{align*}
- From these call options, we can compute the corresponding put options using put-call
parity:
\begin{equation}
P(S_0, K_i, T_j) = C(S_0, K_i, T_j) - S_0 e^{-qT_j} + K_ie^{-r\cdot T_j} \qquad
\text{for} \; \; T_j\in \mathcal{T} \quad \text{and} \quad K_i \in \mathcal{K}
\end{equation}
and choose $r_t$ and $q$ for all $T_j \in \mathcal{T}$ for
$j=1,\ldots m$ and $T_j > T_{j-1}$ such that the put prices derived from
the put-call parity matches the observed market prices. This can be done by minimizing the sum of squared error between the put prices found from parity and the market prices:
\begin{equation}
\min_{r_{T_j},q} \sum_{j=1}^{m} \sum_{i=1}^{n} \left(Put_{PC}(K_i,T_j,r_{T_j},q) -
Put_{Market}(K_i,T_j)\right)^2.
\end{equation}
The method nets you with deterministic interest rate (different for each maturity $T_j$), $r_{T_1},\ldots,r_{T_m}$and constant a constant dividend yield $q$, which can be used to price options on the same stock. I've allowed for a slight abuse of notation in the put-call parity formula, since $r$ does not depend on time for convenience.
