Hi,I was given the following type questions on a middle office type job interview and did not know the answers. However, as I am getting a second call from them I will be asked these types of questions again. Since the questions are practical rather than theoretical, I thought this forum would be the relevant place. Can some one please be kind enough to reply (with a bit f reasoning so I know how more than just these explicit answers)1) For a given at-the-money option, as time passes and the option gets closer to maturity, what happens to the implied volatility? increases, decreases ? why?2) For a given day, what is the behaviour of the implied vol for all the at-the-money options for different maturity dates? does it increase with the time maturity or does it decrease? why?and 3) How should I estimate the implied vol of an at-the-money option that starts in future (rather than present/past)?Many thanks in advance

QuoteOriginally posted by: LucetiosHi,I was given the following type questions on a middle office type job interview and did not know the answers. However, as I am getting a second call from them I will be asked these types of questions again. Since the questions are practical rather than theoretical, I thought this forum would be the relevant place. Can some one please be kind enough to reply (with a bit f reasoning so I know how more than just these explicit answers)1) For a given at-the-money option, as time passes and the option gets closer to maturity, what happens to the implied volatility? increases, decreases ? why?2) For a given day, what is the behaviour of the implied vol for all the at-the-money options for different maturity dates? does it increase with the time maturity or does it decrease? why?and 3) How should I estimate the implied vol of an at-the-money option that starts in future (rather than present/past)?Many thanks in advance1) For a market with backwardated volatility term struture like energy commodities (Oil, NatGas), as you get closer to maturity, the option gets more expensive i.e its implied voaltility gets higher. It is just a supply and demand phenomena. There is relatively less demand for long dated options, and more demand for short dated options, that creates the backwardation. I think the point where you can get confused is that implied vol might go up, but for a short dated ATM vega is low and going down as maturity decreases. So the sensitivity of the option to volatility is going down , as the implied volatity goes up.2) and 1) are for me essentially the same question3) forward start volatility ( never worked on this you can google it: estimation of forward start volatility)In which market are you applying to work: interest rates, fx, equities or commodities

Thanks for taking the time to answer.

Question 3: Indeed use the forward start vola. You need the implied vol of the option expiring at the future start date and the implied vol of the option expiring at the of the time interval. This is treated in chapter 9 of Taleb (Dynamic Hedging). It is actually quite simple. You use the fact that variances are additive and that the vola*sqrt(T) is the standard deviation. If you need more explanation, please let me know....

I'd say in general you'll see a lot more movement in the short term implied vol than the long term ones. In addition to general supply and demand argument, you can think about a sudden short term movement of the underlying, perhaps due to recent/upcoming events. Clearly the short term vols will move in response to that (because actual volatility is LIKELY to be higher within the short term, and those who short options may want to think about covering their short gamma positions). Now lets say the 1m vol goes up from 10 to 15 . Will the 1yr vol goes up by 5 as well? It may do, but that'll imply the expected increase actual vol will last for some time (to the 1 yr date). Is that a reasonable assumption? Sometimes yes, but very often no. So if you see a 5 vol increase in the 1m, chances are you'll see the 1yr going up by say 1 vol.

Question 3: careful, variances are additives only if there is no auto-correlation.For example, if you assume that interest rates are mean reverting, then the negative auto-correlation will make the long vol lower than the short vol but it does not mean that the forward vol is low.To visualize this, consider two succesive periods, for example now to 1-year and 1-year to 2-year.Then assume that the (log)return of the underlying is determined by two normal variable with standard deviation equal to std01 and std12.The standard dev of the return for the two years is given by:std02 = sqrt( std01^2 + std12^2 + 2 * correl *std01*std12) where corr is the correlation between the two periods.What you observe in the market is the implied vol for 0 to 1 and 0 to 2 and you would like to know the forward vol...In fact you have vol01 = std01, vol12=std12 and vol02 =std02/sqrt(2).If correl=0 then it's simple, you have: vol12 = sqrt(2*vol02^2 - vol01^2) ...

Jimmy is perfectly right. I implicitly assumed that the auto-correlation is neglible!