Intuition of the Vol Smile

[TheBigShort]

Having traded the skew countless of times, today I realized, I do not have a sound understanding of why it exists(apart from the hedgers vs liquidity providers aspect).

Trading Scenario:
I will be trading the IV vs RV spread using the options from SPX 1 month chain.

Let's say we have extremely high skew in SPX 1 month options where the 95% moneyness vs 100% moneyness is 10%. With ATM vol being 15 and 90% moneyness being 25. We buy both options.

If the SPX realizes 15% vol over the life of the options, do we not realize 15% vol on both options no matter where the underlying trades? So why do we have the skew?
Thanks in advance

 

[Jgills]

The skew is pricing in more than just expectation of realized vol, there is also a vol of vol component and higher order things to consider.

As for the realized vol you see in your gamma pnl, that is very spot dependant when you have fixed strike options. If the underlying moves far enough away from your 1m options, the impact to your pnl from delta hedging becomes small, unless you are constantly adjusting strikes to always be long atm. In the case of your 90% PUT, in 1m space that's a very low delta option with very little premium, so what your e paying for can't really be described as a 25 vol vs what the underlying realized (in regards to thinking of what implied to realized spread is) with spot here since your delta and gamma will soon become zero.

Hope this is helpful

 

[TheBigShort]

The skew is pricing in more than just expectation of realized vol, there is also a vol of vol component and higher order things to consider.

As for the realized vol you see in your gamma pnl, that is very spot dependant when you have fixed strike options. If the underlying moves far enough away from your 1m options, the impact to your pnl from delta hedging becomes small, unless you are constantly adjusting strikes to always be long atm. In the case of your 90% PUT, in 1m space that's a very low delta option with very little premium, so what your e paying for can't really be described as a 25 vol vs what the underlying realized with spot here since your delta and gamma will soon become zero.

Hope this is helpful

Hey Jgillis, I am still a bit confused. If we can both agree on our expected final PnL (if delta hedged until expiration) is vega(RV - IV), what inputs are you suggesting I use for RV/IV for an OTM option? And I still don't see why RV should be any different depending on where the strike is.

Also even if we put on a straddle, the gamma will also trend towards 0, yet I don't think we should change our vega(RV-IV) formula.
Further explanation would be great

 

[Jgills]

I'm not saying rv will be different depending on strike, but the pnl assumption of vega (rv-iv) implicitly has the underlying near the strike. I say this because if the underlying is not near the strike you will not have nearly enough gamma in your position to make the realized vol matter since there is no delta, nor changes in delta to be hedging.

Edit* if you want to think of your pnl strictly in vega(rv-iv) then you need to trade things such as vol swaps or var swaps. take var swaps as the easier example, the replicating portfolio includes all strikes, not one single strike (or two wide strikes) in the example your e providing.

 

[TheBigShort]

thanks for the response.

then what is our expected PnL of an OTM option? I am having a hard time seeing it being different then vega(rv-iv). As long as there is gamma in the option (which there is all the way up until expiration). That "should" be our expected pay-off.

 

[Jgills]

thanks for the response.

then what is our expected PnL of an OTM option? I am having a hard time seeing it being different then vega(rv-iv). As long as there is gamma in the option (which there is all the way up until expiration). That "should" be our expected pay-off.

I wouldn't say there is gamma up until expiration of the 90% PUT. I'm on vacation so my access to models right now is poor, but think through this example.

You buy 1m 90% otm put today and it's 10d.

On day 2 the market goes up 1.5% and your delta goes to 0.05, so you sell some your delta.
Days 2-15 the market doesn't move and your option decays to 0 delta.
Day 16 is up 3%. You have no delta against this option, so you have no delta to sell.
Days 17 until expiry is back down and up 3% each day. Unfortunately you still have no delta (or gamma) with your option.

The market realized a lot in that example, but due to the path depdancy of your fixed strike options you are not getting to profit off of any of the realized moves because your option has little to no gamma and spot is too far away from strike.

Edit* It didnt matter if you initially purchased your option on a 10 vol or 15 vol because once the realized vol came you had no exposure to it. I'm heading to bed, if your e still confused and no one else gets back send me a pm and we can get into it further another time. Good luck.

This is what I was trying to get at when looking at options in the wings.

 

[Kevin Schmit]

With ATM vol being 15 and 90% moneyness [vol] being 25. We buy both options.

The two "vols" are actually the same if you think of vol as being the standardized second moment of the risk-neutral distribution implied by option prices. The apparent difference is due to risk-neutral implied vol (2nd moment) being expressed in solved-for BS log-normal equivalent point densities. The skew is a measure of the difference (divergence) between the shape of the log-normal density plot and the implied risk-neutral density plot. You could just as easily express the skew in terms of asymmetric KS or JS divergence between the two distros.

If the SPX realizes 15% vol over the life of the options, do we not realize 15% vol on both options

Pretty close, other than the fact that the vol is realized over the price of the underlying not the options. Fit the RND using Breeden-Litzenberger or Dual-Delta and solve for the normalized 2nd moment (vol) numerically -- it will be much closer to the BS ATM "vol" than the BS OTM "vol."

 

[GRULSTMRNN]

I feel nobody so far actually answered your question.

As I am quite lazy here a quote I agree most with :

The volatility smile is the result of market forces knowing form experience that out of the money option pay out more often that what would be expected by a normal (Gaussian) distribution. For years Quants speculated why the market drove the out of the money options higher that the price of the Black-Scholes model.

The best theory speculates that the smile is because the distribution of returns of stock prices are not a normal distribution having large jumps in price that occur too often. The distribution is much closer to a power law distribution or Pareto distribution.

The problem is that the math becomes messy without the normal distribution so the most recent models that include smile uses a jump diffusion process ( a normal distribution with random jumps).

P. S. I market made caps, floors, swaptions in FI before trading prop equity options for several banks for over 12 years.

Having traded the skew countless of times, today I realized, I do not have a sound understanding of why it exists(apart from the hedgers vs liquidity providers aspect).

Trading Scenario:
I will be trading the IV vs RV spread using the options from SPX 1 month chain.

Let's say we have extremely high skew in SPX 1 month options where the 95% moneyness vs 100% moneyness is 10%. With ATM vol being 15 and 90% moneyness being 25. We buy both options.

If the SPX realizes 15% vol over the life of the options, do we not realize 15% vol on both options no matter where the underlying trades? So why do we have the skew?
Thanks in advance

 

[newwurldmn]

Having traded the skew countless of times, today I realized, I do not have a sound understanding of why it exists(apart from the hedgers vs liquidity providers aspect).

Trading Scenario:
I will be trading the IV vs RV spread using the options from SPX 1 month chain.

Let's say we have extremely high skew in SPX 1 month options where the 95% moneyness vs 100% moneyness is 10%. With ATM vol being 15 and 90% moneyness being 25. We buy both options.

If the SPX realizes 15% vol over the life of the options, do we not realize 15% vol on both options no matter where the underlying trades? So why do we have the skew?
Thanks in advance

In plain English, the skew represents the expectation of realized vol along the path dependency of the option. In equities You expect more realized vol on the downside where you will have more gamma on the puts (being closer to the money in thar case) vs the calls in this stock path. The higher price for those puts (because of this plausible scenario) translates to higher implied vol now.

Commodities have a smirk because you expect more vol if the commodity changes price in either direction.

You sometimes see that single stocks have inverted skew. This is especially true in takeover spec names where the expectation of a big up move means you will be realizing more vol where your gamma is higher if you are long those calls.

 

[GRULSTMRNN]

You probably meant that the skew represents the deviation of expected realized vol from the one implied by the underlying model distribution. It's the differential that causes the skew not the absolute value.

But in terms that even kids understand, skew originates from the fact that extreme events happen more often than implied by the underlying market agreed distribution used in pricing the option.

In plain English, the skew represents the expectation of realized vol along the path dependency of the option. In equities You expect more realized vol on the downside where you will have more gamma on the puts (being closer to the money in thar case) vs the calls in this stock path. The higher price for those puts (because of this plausible scenario) translates to higher implied vol now.

Commodities have a smirk because you expect more vol if the commodity changes price in either direction.

You sometimes see that single stocks have inverted skew. This is especially true in takeover spec names where the expectation of a big up move means you will be realizing more vol where your gamma is higher if you are long those calls.