Estimating Historical Volatility and Option Pricing Inputs

# What is the market standard for measuring historical volatility?


# What is the market standard for measuring historical volatility?












Hope to get some help with the following questions:

- 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.

- 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%?

Thanks in advance

## Answer by Pleb (score 1)

https://quant.stackexchange.com/a/60299

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.