# Historical volatility calculation to price options with the Black-Scholes formula

I'm looking for a reference algorithm for calculating historical volatility to price options. I know there are several volatility calculation models that use the time series of the underlying's returns. Is there a reference method for determining the number of days to consider for the calculation?

• Use a time period having the same order of magnitude as your option's tenor. Please do not try and trade around the price you find. It will be fairly different from what sophisticated market participants work with (i.e. in trading with them you would lose money). – Brian B Jan 10 '20 at 19:29
• There is no agreement as to how many days to use to compute historical volatitliy. Some years ago S. Figlewski suggested that you should use as many days of history as the maturity of the option you consider. For example for a 3 month option you should use 3 months of daily data. This is just a heuristic which he suggested based on some empirical tests. – noob2 Jan 10 '20 at 19:51

To answer strictly: no there's no reference method for determining the number of days.

### Delving a bit deeper

Always remember the profit from option trading does not come from the price at which you trade but from the mismatch between implied volatility and realised volatility.

Moreover, you can reap that profit only setting up the replicating portfolio, that is constantly delta-hedging the position.

Whatever the way you use to estimate the volatility "generated" by the movements of the underlying between the inception of the contract and the expiry of the option, you should hedge using that estimate when computing the delta-hedge amount of underlying.

### Delving way deeper

Check on the market and see an option (either call or put) with maturity $$T$$ and (irrelevant) strike $$K$$ trading at some (irrelevant) price $$O(T,K)$$. You invert the Black-Formula and obtain the so called implied volatility $$\sigma_{Blk}(K,T)=25\%$$.

Say you have a very good model for the instantaneous volatility of the underlying process $$\sigma_{mdl}(t)$$. It can be stochastic, state-dependent, whatever. Given your model, compute the following quantity (also called root-mean-squared-volatility)

$$\hat{\sigma}(T) = \sqrt{\frac{1}{T}\int_0^T\sigma^2(u)du}$$

Turns out $$\hat{\sigma}(T)=50\%$$ from your model. Now you should:

1. Buy the option on the market
2. Delta-hedge the option using $$\hat{\sigma}(T)$$
3. Continue rebalancing your hedge portfolio after each $$dt$$ computing the Delta always using $$\hat{\sigma}(T-dt)$$
4. If your model is correct, you would end up in profit irregardless if the option expires in the money or not.

What does it mean "the model is correct"? It means that the underlying had a realised volatility of $$50\%$$ instead of $$25\%$$.

How much profit? The difference

$$Profit = O(T,K,\sigma_{Blk}=50\%) - O(T,K,\sigma_{Blk}=25\%)$$

How much is $$dt$$? Theory says "infinitesimal amount of time".

What happens if $$dt$$ longer than "infinitesimal amount of time"? Your $$Profit$$ in formula up there is no more going to be a single number, but it turns out to have a distribution (centered on the original value). This distribution is going to be wider and wider the higher $$dt$$ is, therefore making your profit more and more random (and possibly negative).

• nice answer. Could you pls elaborate a little bit on the last paragraph about the length of $dt$ turning the profit into distribution rather than a single number? – Jan Stuller Jun 10 '20 at 9:39
• For a dt larger than an infinitesimal amount of time, your portfolio will not be constantly delta neutral and therefore the adjustments will be made potentially after the market moves a bit. As a consequence the final result will depend on how the market moved in relation to your hedging strategy. That's why the profit will be a distribution around the single number of the constantly delta neutral strategy. – David Duarte Jun 10 '20 at 13:22
• Totally agree with @DavidDuarte comment (thanks David). To add a little bit: in the theoretical setting "there is no lucky path" (as long as you have continuous paths) that's why the theoretical value of the option is correct no matter what happens during the life of the option. You can picture it easily: hedge the option just once! you see that your profit at expiry will depend on when you hedged and what happened after. I can point you to chapter 4 of Volatility and Correlation by Riccardo Rebonato, where this entire story is explained perfectly (IMHO). – LePiddu Jun 12 '20 at 9:48

You should review the difference between implied volatility and realised volatility. Historical volatility is the realised volatility that happen in the past but an option price will have to be determined by the view of the market about volatility in the future for a given period. Normally you do the inverse, ie the option price is given by the market but you can work out the implied volatility from that price.