Choosing Volatility Inputs and Time Units for the Black Model
Summary
This document discusses which price series should be used to estimate volatility for the Black model and whether volatility needs to be annualized. The responses emphasize matching the volatility estimate to the instrument used for hedging: a hedge based on futures calls for futures-derived volatility, while a hedge using the underlying calls for its volatility. These series can differ in practice because rates, dividends, liquidity, and, for commodities, storage costs and convenience yields affect futures behavior.
One response notes that under simplified assumptions, spot and futures volatility may coincide because futures prices are linked to spot. That equivalence can break when those assumptions fail. The text also explains that the standard Black formula expects annualized volatility when used in its usual form. Other time units can work if the formula and all inputs are adjusted consistently. These are general modeling guidelines, not empirical comparisons; the appropriate estimate depends on the hedge instrument and market conditions.
Key ideas
- Estimate volatility from the instrument used to hedge the position.
- Spot and futures volatility may differ when rates or other pricing factors are stochastic.
- Commodity futures can reflect storage costs and convenience yield as well as spot-price risk.
- The standard Black formula uses annualized volatility when applied in its usual form.
- Alternative time units are acceptable if all model inputs are scaled consistently.
Tags
Full text
# Black model - volatility estimation
# Black model - volatility estimation
In the Black (1976) model:
- We should use the settlement prices of the underlying futures contract in order to estimate the volatility, right? Or can we also use the spot prices? Because the behaviors of these series of prices are quite different.
- Should the volatility always be annualized?
## Answer by Rustam (score 1)
https://quant.stackexchange.com/a/8727
In standard Black model
1) We have to use the volatility of underlying. It is similar for futures and stocks, but it is not the same! Futures prices differ from stock prices not only by discounting, but also by dividends, for example.
2) Volatility should be annualized if you want to use formula "as is" without amendments.
## Answer by Christian Fries (score 1)
https://quant.stackexchange.com/a/8766
You should use the volatility of your hedge instrument. If you do hedging with the underlying you use the underlyings vol. If you do hedging with futures then derive the vol form there...
## Answer by KAT (score 0)
https://quant.stackexchange.com/a/8717
In theory, since the futures price is $F(t,T)=S(t)e^{r(T-t)}$, the only risk source is coming from S(t), so the volatilities of the two series should be the same.
The thing is that the theory doesn't consider the fact that the interest rates could be stochastic, or that the future market is more liquid and so incorporates more easily information. Also if the underlying of the future contract is a commodity then you also have the volatility coming from storage cost and convenience yield. So indeed the two series could represent different volatilities.
And, the one you should use is of course the volatility obtained from the futures prices.
In what regards weather you should use annualised or monthly volatilities, it shouldn't matter, in theory, as long as you fix the time unit and you are consistent with it throughout the calculations. It does come handy though to use the year as a unit time.
The reason why it doesn't matter what time unit you chose resides in the scaling properties of the BM (in case you are interested).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.