Estimating At-the-Money Call Premiums from Historical Volatility
Summary
The document offers a simple way to estimate premiums for near at-the-money calls in a covered-call simulation. It presents an at-the-money Bachelier approximation in which the premium depends on the underlying price, volatility, and square root of time to expiry. For volatility, it suggests estimating the dispersion of periodic simple returns from historical prices, with an option to simplify by treating the average return as zero. A cruder alternative uses the average absolute periodic return and average interval length.
The approximation is explicitly limited: it is not exact, has an error associated with higher powers of volatility and time, and omits a smaller interest-rate effect. Historical volatility is a practical substitute for forward-looking volatility, but the answer cautions that it is not always the best estimate. Black–Scholes is mentioned as a more accurate model, though obtaining historical implied volatility data may be difficult. The discussion does not provide a full covered-call simulation framework or address changing volatility, dividends, or other option-pricing inputs.
Key ideas
- A simple at-the-money call premium approximation scales with spot price, volatility, and the square root of time to expiry.
- Historical periodic returns can be used to estimate volatility for a basic simulation.
- Assuming zero average return or using mean absolute returns offers simpler but cruder estimates.
- The approximation omits interest-rate effects and has an error that grows with volatility and time.
- Historical volatility may differ from forward-looking volatility, while a Black–Scholes approach requires suitable volatility inputs.
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# Simple model for option premium (for covered call simulation)?
# Simple model for option premium (for covered call simulation)?
Given a historical distribution of weekly prices and price changes for a stock, how can I estimate the the option premium for a nearly at-the-money (ATM) option, say with an expiration date 3 months in the future? We could also have the stock's historical beta and the current option premium if desired. To keep things simple assume the volatility of the stock is constant.
I am writing a little Monte-Carlo sim for a buddy in Excel and would like to simulate a strategy of writing ATM covered calls, letting the option expire if OTM and re-writing. I will assume exercise at the end of the option period if the call is ITM.
I have never modeled options before so another title for this question might be "Options Modeling 101". For starters I would just like to improve upon my current model, which is constant premiums. Thanks!
## Answer by Brian B (score 10, accepted)
https://quant.stackexchange.com/a/585
The main component of that option premium is (forward-looking) volatility $\sigma$. The very simplest formula you could use for ATM options is the Bachelier model \begin{equation} \text{Call}_T = \sigma S \sqrt{\frac{T}{2\pi}} \end{equation} where the time to expiration is $T$ and $S$ is the current underlying price. This formula is "wrong" strictly speaking, but only by a factor of $\sigma^3T^{\frac32}$ which in your case will be around 5%. You'll also be ignoring a somewhat smaller error due to nonzero interest rates.
To obtain $\sigma$ you can work with your available historical data to get a historical volatility. Historical volatility is not always the very best choice but it is far better here than your current constant price assumption, and it is very simple to calculate:
\begin{equation} \sigma_{\text{Hist}} = \sqrt{\frac1{N-1}\sum_{i=1}^N{(r_i-\bar{r})^2}} \end{equation}
where the $r_i$ are the periodic returns
\begin{equation} r_i = \frac{\frac{S_{i+1}}{S_i}-1}{\Delta t_i} \end{equation}
taken of the underlying $S_i$ at times $t_i$, $\Delta t_i=(t_{i+1}-t_i)$ and $\bar{r}$ is their mean. (The Black-Scholes model would have used log returns instead.)
If you are happy with a crude estimate, you may assume $\bar{r}$ is zero rather than bothering to calculate it. And for a very crude estimate of historical volatility, you can instead use
\begin{equation} \sigma_\text{Inaccurate} = \text{Mean}\left[|r_i|\right] \frac1{\sqrt{\text{Mean}\left[\Delta t_i\right]}} \end{equation}
For maximum accuracy, you would of course want to use the Black-Scholes model. But frankly you would have an easier time finding the requisite historical option prices than you would finding a historical time series of forward-looking (implied) Black-Scholes volatilities.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.