# Estimating at-the-money volatility where at-the-money option is absent from the market

I am trying to estimate the intraday ATM volatility in a market where the the strike prices are relatively sparse thus the ATM option may not exist (let's say the closest strike is about 2% away from the spot price). A brute-force way to do this is to fit a volatility model, such as SABR or Heston, and compute ATM volatility using that model.I am wondering if there's any other method without fitting the model. Here's what I have tried: Suppose we have already fit a SABR model with end of day data and we have obtained the SABR parameter $\alpha, \beta, \rho, \nu$. If the strike $K$ is not too far away from the forward price $f$, we can approximate the SABR volatility using $$\sigma(K, f) = \frac{\alpha}{f^{1-\beta}}\{1 - \frac{1}{2}(1 - \beta - \rho\lambda)\ln(\frac{K}{f}) + \frac{1}{12}[(1-\beta)^2 + (2-3\rho^2)\lambda^2]\ln(\frac{K}{f})^2\}$$ where $\lambda = \frac{\nu}{\alpha}f^{1-\beta}$. Under such approximation $\sigma_{ATM} = \frac{\alpha}{f^{1-\beta}}$, hence $\lambda = \frac{\nu}{\sigma_{ATM}}$. We can then rewrite the above equation as, $$\sigma(K,f) = \sigma_{ATM}\{1 - \frac{1}{2}(1 - \beta - \frac{\rho\nu}{\sigma_{ATM}})\ln(\frac{K}{f}) + \frac{1}{12}[(1-\beta)^2 + (2-3\rho^2)\frac{\rho^2\nu^2}{\sigma_{ATM}^2}]\ln(\frac{K}{f})^2\}$$ We can directly observe $\sigma(K, f)$ from the market. If we can assume $\beta, \rho, \nu$ are constant throughout the day (or can we?), we can solve for $\sigma_{ATM}$.

However, I am not very satisfied with the above result since it has a lot of approximations and assumptions. I am wondering if there's anything I can do to improve my result or there's a better method to do this.

The simplest is to fit a parabola to the three closest points around the forward with coordinates $$(z_{-1}, \hat{\sigma}_{-1}), (z_0,\hat{\sigma}_0), (z_{1},\hat{\sigma}_1)$$. This is equivalent to a 3 points finite difference on a non uniform grid. We then have: \begin{align} \sigma_0 &= z_0 z_1 w_{-1} \hat{\sigma}_{-1} + z_{-1} z_1 w_{0} \hat{\sigma}_{0} + z_{-1} z_0w_{1} \hat{\sigma}_{1}\\ \sigma_0' &= -(z_0 + z_1) w_{-1} \hat{\sigma}_{-1} - (z_{-1}+ z_1) w_{0} \hat{\sigma}_{0} - (z_{-1}+ z_0)w_{1} \hat{\sigma}_{1}\\ \sigma_0'' &= 2 w_{-1} \hat{\sigma}_{-1} +2 w_{0} \hat{\sigma}_{0} + 2w_{1} \hat{\sigma}_{1} \end{align} with \begin{align} w_{-1} &= \frac{1}{(z_{-1}-z_{0})(z_{-1}-z_{1})}\\ w_{0} &= \frac{1}{(z_{0}-z_{-1})(z_{0}-z_{1})}\\ w_{1} &= \frac{1}{(z_{1}-z_{-1})(z_{1}-z_{0})} \end{align}
The $$\sigma_0$$ is your ATM vol. The $$z_i$$ is the moneyness or log-moneyness for the three options around ATM considered. The paper also details least-squares approaches if you want to include more points around ATM. Finally the subject is explored further in my book Applied Quantitative Finance for Equity Derivatives (Chapter 5 in general, and section 5.4.2 for SABR)