1
$\begingroup$

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%?

Thanks in advance

$\endgroup$
1
$\begingroup$

I will try to provide an answer to your questions:

  1. 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.
  1. 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.
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.