# What is the market standard for measuring historical volatility?

Hope to get some help with the following questions:

1. Can someone explain what is the industry standard to calculate stock options historical volatility? I am using this estimator https://portfolioslab.com/yang-zhang with 52 weekly prices times square 52, but glad to learn more accurate methods.

2. Which rates to use when pricing an option? For example TSLA I would use fed rate of 0.25% and dividend rate of 0.00%?

• 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.