Hull volatility smiles

[Arka Bose]

percentage impact of non constant volatility on option prices are more pronounced as time to maturity increases-This is what Hull's text says. Can anyone explain me why impact on prices become more pronounced even though the volatility smile itself becomes less pronounced when the maturity term increases?

 

[QuantMan2318]

Hi there @arkabose , that's a brilliant question
I think you mean the increase in the implied volatility itself in absolute amounts, as the time to maturity increases though the smile is less pronounced? If thats so,

1. I think it is based on the expectation of the Market that as the time increases, there is greater chance for risks and hence volatility. I think, in addition the BSM increases the option price and hence the premium if the time to maturity increases, which is again a reflection of the impact on prices that show higher volatility

2. Hull states that the delta of an option with non constant volatility is given by: dc/dS+(dc/d imp.vol)*(d imp.vol/dS), it seems that the term d imp. vol/dS becomes an increasing function of the Asset prices and thus increases implied volatility w.r.t Asset prices, particularly as time increases, because of the first point and the fact that implied volatility changes w.r.t K/S

3. The impact of the jumps, as we may observe, the jumps cause the opposite effect of a smile and increases the implied volatility during the middle ranges (ATM) and reduces the same when the strike prices are at the extremes. Thus we may conclude that as time to maturity progresses, there is greater chance for options to close at their extremes because of larger movements in Asset prices, as we expect the extreme asset price movements, we may conclude that the implied volatility falls at the tails and thus, this causes a reduced effect on the volatility smile as the time to maturity increases

4. Finally, we may state that at the uncertain longer time periods, we can only guesstimate that the implied volatility will be more throughout the entire strike price ranges and not say with conviction the movement of the implied volatility over the ranges

Again, I wish to state that all this is my interpretation, more advanced practitioners can verify if I am correct

 

[Arka Bose]

Hi @QuantMan2318 thanks for the reply,

Why are you saying 'The impact of the jumps, as we may observe, the jumps cause the opposite effect of a smile'??

Hull says that 'The percentage impact of jumps on both prices and the implied volatility becomes less pronounced as the maturity of the option is increased.
He added via footnote that 'for sufficiently long dated options, jumps tend to get 'averaged out' - this contributes to lower implied volatility ----( and i agree and is intuitive). In that case, does you statement go with it?

Moreover, I found out a stack exchange page from where i actually do get myself relief (at least got some intuition) you may check it out too.
http://quant.stackexchange.com/questions/18701/implied-volatility-and-nonconstant-volatility

 

[Arka Bose]

Moreover, this equation dc/dS+(dc/d imp.vol)*(d imp.vol/dS) given by Hull, dont you think there will be an '=' sign rather than '+' sign? I believe he is trying to break the delta of the option into two pieces just to show the positive relationship between implied volatility and stock price. (Since volatility is decreasing function of K/S ratio)

 

[QuantMan2318]

Hi @arkabose

You are probably right w.r.t the second one, I thought about it as well. If he was using the Chain rule, then the + sign gets replaced by =, but I was not too sure if thats what he meant and quoted him exactly. Then, again, your knowledge of Mathematics involved exceeds mine. The point is that there is a new term that incorporates the implied volatility w.r.t stock prices and is an increasing function of Stock price

What I meant earlier w.r.t jumps ( where the jumps are known ) was that jumps have a volatility frown as opposed to a smile, therefore the implied volatility is certainly less for the extremes and then again, there is this beautiful statement by Hull

"The percentage impact of non constant volatility on prices is more pronounced but its effect on implied volatility is less pronounced as maturity increases
The percentage impact of jumps on prices and implied volatility is less pronounced as maturity increases

Therefore, he concludes that the "net effect of ALL this is that the volatility smile becomes less pronounced as maturity rises" we have to see both together, the inverse shape of the chart in case of the jump coupled with all the reasons above. You may notice that the author of the stack exchange post agrees with my line of reasoning in his second post

 

[Arka Bose]

Aha, I see it now, actually i didn't go through the very last part of the chapter where the relationship between jump and implied volatility was shown. Thanks a lot,. But does a jump always makes a frown? I am confused about that

PS: I am no maths guy

 

[Dr. Jayanthi Sankaran]

Hi @arkabose,

Yes, when there is a jump or jumps in the underlying stock price process, corresponding to say a bimodal distribution, the implied volatility vs moneyness is no longer a smile but a frown...

Thanks!