smile

[Lun]

before asking questions, I think I should first smile : )

well, I suppose most of people here know what smile and implied volatility (IV) are

I just want to ask why people goes to study the relationship between IV and strike ? is there profit we can make from the smile ? or we can arbitrage from it ?

another question is, why it is a smile or skew (not other shapes) ? how come higher or lower strike will affect the IV in this way instead of other ways

what's the purpose to calculate IV ? there're market markers to control the price (also the liquidity) of the option (say vanilla). In other words, the IV is man-made, then is it still meaningful ?

I mean, the definition is not sth new to us, but I never think about why people go to define these things and what the usage is

Thanks for allowing me to ask so many questions : )

 

[Uncle Max]

For options volatility determines price. When traders trade options, they trade volatility. If you look at Black-Scholes formula for example, you will see that volatility is the only unknown input. Therefore, if you cannot get it right, your price will be wrong and here comes arbitrage.

You might say that we know volatility from historical data, but this is not true. Historical volatility is based on underlying security while implied volatility characterizes derivative itself. Both are related in some way, but not the same.

A lot of options are traded over the counter where market makers have no control of the price.

 

[Lun]

what's the pt to find the IV ? the trader can make money from it DIRECTLY ? or it just acts as a reference/index/measurement for options ?

we spend so much time to define things with IV, say smile/skew, vol surface, what's the application ? Practically, a trader will use these figures to make money ? or they're just some definition ?

for volatility, not IV, there're many ways to model it, then which way can give you the correct volatility, in turn, the price. As you said, wrong price can lead to arbitrage
.
.

 

[Yike Lu]

At least on exchanges, option prices are driven by market forces, and the market makers must try to balance these forces as best they can.

If there's a lot more demand than supply for a given option, the price will rise (obviously). If the price rises and the other important and variable factors governing the option price don't change much (i.e., stock price, risk free rate, time), then obviously the implied volatility must be the one that rises.

Implicitly it means either that volatility is actually high (as measured by historical volatility) or that traders EXPECT volatility to be high in the future.

Yes it's possible to more or less directly arbitrage IV, although it can be complicated unless you are just trading VIX. There are many other uses for IV - it can be a market technical indicator for examples.

Why is there a smile? Basically because Black Scholes doesn't hold exactly in the real world. I believe Stochastic Volatility is one theory which can fit the observed parameters. Another is (I forget the exact name) a higher moments extension to BS which includes skew and kurtosis.

In simple real world terms, the smile exists because people overall buy more options that are FITM and FOTM. And that's because people expect large market moves to occur more often than the Black Scholes predicts.

 

[Lun]

Thanks for your reply

what does F (of FITM, FOTM) stand for ?

 

[Yike Lu]

Far in the money and far out of the money. I guess you could also say "DITM/DOTM" d standing for deep.

 

[Lun]

I have a stupid question.

If a trader trades vanilla options, what do they trade ?
They expect the underlying to go up or down, and then trade options for leverage ?
or they purely trade for volatility ?

I haven't been a trader, I don't have a trader mind set, my question may be stupid

---------- Post added at 04:05 PM ---------- Previous post was at 03:04 PM ----------

For the smile, I want to counter prove it, see if any one can see why it doesn't work

by common sense, a vanilla option, easier to exercise, higher the price
by this logic, on a fixed day end, for call options, higher the strike, lower the option price (lower the strike, higher the option price)
if we study the relationship between IV and strike, that is, we keep the option price a constant
in other words, in order to make (man-made) the option price a constant,
for a high strike, the price is increased to the constant level, the IV is made higher
for a low strike, the price is decreased to the constant level, the IV is made lower
in other words, it is impossible to have a smile shape

any one can comment on it ?

---------- Post added at 05:47 PM ---------- Previous post was at 04:05 PM ----------

In simple real world terms, the smile exists because people overall buy more options that are FITM and FOTM. And that's because people expect large market moves to occur more often than the Black Scholes predicts.

from my understanding, smile is sth very theoretical, a graph plotted by calculated figures, is it possible to relate it to real world ?

I suppose the explanation (for the shape) is also sth theoretical, I'm not sure whether I'm right.

 

[Sanket Patel]

Dude, I think you're complicating it way too much; keep things simple. Prior to Black Monday, option markets exhibited a relatively inncocuous smile - that is, it was more or less flat. Following the crash, the markets began pricing in a jump risk premium of sorts. This premium shows up in the form of a higher IV for OTM options because people are pricing in the risk of a jump. Don't look at this in terms of price, look at it term of volatility. Relative to ATM options, OTM options tend to have a higher IV, thus being 'more' expensive.

A smile is a theoretical concept in the same manner as the return on a stock is a theoretical.

When a trader trades vanilla option, he/she trades put/calls on an exchange. When a trader trades options, he/she is technically trading vol, gamma, and theta. When you buy an option, for example, you are long vol, long gamma and short theta, so to speak. The option seller, on the other hand, is short vol, short gamma, and long theta. Short vol and short gamma because he/she will lose money as vol and gamma rise but is helped by theta. The seller is betting that the markets will remain relatively calm so that he/she gets (to keep) the premium while the buyer experiences theta decay.

And yes, options give you leverage..lots of leverage.

 

[Lun]

Then, in the real market, what's commonly traded for ?
for vol/gamma/theta or for leverager or for both ?

Maybe, the question is who trades which one ?



.

 

[Yike Lu]

I'm not a trader but I'm sure everything gets traded. Why? Because there's the chance to make money betting on any of these things: vega, gamma, theta, etc. That's why I think options are interesting - so many strategies are available. What a trader uses options for is largely dependent upon the individual trader's preferences. Do they like the inherent leverage features? Are they good at predicting volatility? Or are they more of a safe type and want to bet on time value... which accrues just as surely as the clock ticks?

The only thing that seems pointless to me is betting on delta just to bet on it (without wanting to use the inherent leverage of an option). It's usually more efficient to just buy the stock itself in these cases... tighter spreads and more liquidity.

The smile is NOT theoretical. It is an empirically observed feature of the options markets. Theories which FIT the smile include (as I mentioned above) Stochastic Volatility and Higher Moments.

And once more, as I explained above, the reason there IS a smile in the first place is because the people who trade options (and thus determine their price) estimate the probability of large market moves to be greater than are predicted under the Black Scholes model and because of this tend to more often buy options which are either deep in or out of the money. That in the end is probably the best explanation for the smile. What Sanket mentioned about Black Monday only reinforces this notion - people got more scared after the crash occurred and so started buying more options with strike prices further in/out of the money in order to hedge the probability of another market crash.

As for disproving it... good luck. It EXISTS empirically in the option markets. Trying to disprove it is like trying to prove gravity doesn't exist.

Please think critically about these things. I can say what I did about options trading in the first paragraph with conviction because it is simply logical.

 

[Sanket Patel]

Then, in the real market, what's commonly traded for ?
for vol/gamma/theta or for leverager or for both ?

Maybe, the question is who trades which one ?

.

The simple answer is that it depends. There are things to consider: are you on the buy/sell side, are you a structurer, do you trade flow, are you simply speculating, are you hedging, are you simply looking for leverage, what's you view short/long term view, are you looking for income...etc

 

[Lun]

I'm not a trader but I'm sure everything gets traded. Why? Because there's the chance to make money betting on any of these things: vega, gamma, theta, etc. That's why I think options are interesting - so many strategies are available. What a trader uses options for is largely dependent upon the individual trader's preferences. Do they like the inherent leverage features? Are they good at predicting volatility? Or are they more of a safe type and want to bet on time value... which accrues just as surely as the clock ticks?

The only thing that seems pointless to me is betting on delta just to bet on it (without wanting to use the inherent leverage of an option). It's usually more efficient to just buy the stock itself in these cases... tighter spreads and more liquidity.

The smile is NOT theoretical. It is an empirically observed feature of the options markets. Theories which FIT the smile include (as I mentioned above) Stochastic Volatility and Higher Moments.

And once more, as I explained above, the reason there IS a smile in the first place is because the people who trade options (and thus determine their price) estimate the probability of large market moves to be greater than are predicted under the Black Scholes model and because of this tend to more often buy options which are either deep in or out of the money. That in the end is probably the best explanation for the smile. What Sanket mentioned about Black Monday only reinforces this notion - people got more scared after the crash occurred and so started buying more options with strike prices further in/out of the money in order to hedge the probability of another market crash.

As for disproving it... good luck. It EXISTS empirically in the option markets. Trying to disprove it is like trying to prove gravity doesn't exist.

Please think critically about these things. I can say what I did about options trading in the first paragraph with conviction because it is simply logical.

my point of view (very personal) is that empirical is empirical. You don't hold the ball, it falls on the floor, that is empirical. However, being a quant or a physician, the duty is to use theory/model to EXPLAIN things which are empirical.

Below is what I extract from the book "Option Pricing Models and Volatility"
Just share with you all

Smiles can occur, however, because returns show greater kurtosis than stipulated under normality, so that extreme returns are more likely. This implies that deep in-the-money and deep out-of-the-money options are more expensive relative to the Black-Scholes price.

Smirks can occur because returns often show negative skewness, which again the normal distribution does not allow. This implies that large negative returns are more likely, leading to implied volatilities for in-the-money calls that are higher than implied volatilities for out-of-the-money calls.

In general, smiles and smirks are more pronounced for short-term options, and less pronounced for long-term options. This is synonymous with long-term returns being closer to normally distributed than short-term returns.

 

[Yike Lu]

Your quote is EXACTLY what I mean by "higher moments version of the Black-Scholes". Do some digging and this should be obvious. The fact that you seem to think it is new information to me makes me wonder if you know what the quote is really saying...

You said you were trying to "counter prove" the smile. It's well regarded (and in fact common sense if you happen to know a little bit about basic finance), independently of what model we use to explain option prices, that option prices are governed by the stock price, strike price, time to maturity, stock movement pattern (i.e., volatility), and the risk free rate.

If you assume the Black-Scholes model holds and use it to calculate IV, then EMPIRICALLY the exchange-traded options exhibit a volatility smile/smirk whatever. This cannot be "counter proven" or whatever it is you are trying to do. The options are empirically more expensive far in and far out of the money than they are when they are closer to the money.

Now you can question the validity of the Black-Scholes model itself, but that's a whole other story. And again, there are other models which DO explain the smile. How many times do I have to say this? Look up Stochastic Volatility... for example.

And once more... the "true" explanation is that traders expect large market moves to occur more often than the Black-Scholes model predicts and therefore purchase proportionately more options at extreme strike prices (and therefore drive up their prices).

You quantify this extra buying with whatever model you want to use.. Stochastic Volatility for example, or skew/kurtosis, or make up your own model. There is no single right answer.

 

[Lun]

Yike, thank you very much, esp your patient. I hope that I haven't made you exhausted

 

[Lun]

And once more... the "true" explanation is that traders expect large market moves to occur more often than the Black-Scholes model predicts and therefore purchase proportionately more options at extreme strike prices (and therefore drive up their prices).

In the long run, it's a normal distribution. From empirical, it's less volatile than normal, with a higher peak
In the shor run, it's "large market moves to occur more often"

long run = sum of all "short run"

if all the "short run" follows "large market moves to occur more often", how can we have a normal distribution in the long run (which is a sum of all "short run")

the only case is that some "short run" don't follow "large market moves to occur more often", then the smile will be upside down in these cases

 

[HyperVolatility]

the BSM model is obvioulsy just an approximation of the reality where everything is static, in fact, the volatility that you get out of the BS is said to be deterministic and not stochastic.

The vol smile exhist because:

1)OTM options are usually more volatile than ATM and therefore they trade at a premium to the ATM because many money managers use OTM options to hedge their portfolios.

2)the average investor buys only and therefore is more likely to hedge its positions buying protective puts

3)IV is mean-reverting

Actually trading the skew is not a great idea unless you have a clear edge which comes from a more accurate estimation of volatility itself. In other words, you must look for mismatch between the price quoted by market makers and the price calculated using your volatility forecast that you then plug into the BSM model.

you could use the Heston model to price your plain vanilla european style options but when it comes to american options you'd be better off using a binomial tree approach (even if the difference is not striking)

As I said before it is not the pricing model that makes the difference but the forecast of volatility which is important that's why you get hundreds of researches dealing with vol forecasting.

 

[financeguy]

well hang on.. maybe we're saying the same things but..
i agree that one reason why equity skews are bid for puts is because of the premium people pay for the insurance the low side otm put provides
the other thing is that equities tend to crawl up and crash down.. this implies that spot tends to be more volatile given a downward move and less volatile given an upward more - in other words, there's some correlation between spot and vol.. the higher the correlation, the steeper the smile, i.e. the larger the risk reversal.. further we know that vol doesnt remain forever constant so there also exists a volatility of volatility, which gives rise to a non-zero butterfly.. you can also draw parallels between the presence of a risk reversal and the direction in which the implied pdf of spot is skewed and between the presence of the fly and the fat tails of the implied pdf.. both concepts are consistent with one another
its probably important to understand all that in addition to the insurance / crashaphobia concept because that argument really only holds water for equity markets (can this explain why the usd/cad risk reversal is bid for usd calls?).. of course there is always some reason behind why the implied pdf of spot looks the way it does / why the spot vol correlation and vol of vol are what they are, but it can differ by asset and isnt always just one thing..

 

[HyperVolatility]

well hang on.. maybe we're saying the same things but..
i agree that one reason why equity skews are bid for puts is because of the premium people pay for the insurance the low side otm put provides
the other thing is that equities tend to crawl up and crash down.. this implies that spot tends to be more volatile given a downward move and less volatile given an upward more - in other words, there's some correlation between spot and vol.. the higher the correlation, the steeper the smile, i.e. the larger the risk reversal.. further we know that vol doesnt remain forever constant so there also exists a volatility of volatility, which gives rise to a non-zero butterfly.. you can also draw parallels between the presence of a risk reversal and the direction in which the implied pdf of spot is skewed and between the presence of the fly and the fat tails of the implied pdf.. both concepts are consistent with one another
its probably important to understand all that in addition to the insurance / crashaphobia concept because that argument really only holds water for equity markets (can this explain why the usd/cad risk reversal is bid for usd calls?).. of course there is always some reason behind why the implied pdf of spot looks the way it does / why the spot vol correlation and vol of vol are what they are, but it can differ by asset and isnt always just one thing..

I agree with much of the things you said. However, I would suggest a different interpretation of some of the statements you made:

1) "equity skews are bid for puts is because of the premium people pay for the insurance the low side otm put provides"

The premium is not the cause but the effect.It's a mere consequence of the fact that people do not want to sell downside protection and when they do they want to get an higer premium for such a sale.that's why the put are more expensive.

2) "this implies that spot tends to be more volatile given a downward move and less volatile given an upward more - in other words, there's some correlation between spot and vol"

It is not just the spot market which reacts in such a way. The increased variance that we all see in downward corrections is given by the speed at which investors sell. Panic is always faster than greed. That is why usually the volatility rises during market drops more than it does during rallies.Such a phenomenon is called leverage effect

Implied volatility is heavily affected by market structure. The bid/ask spread is a crucial factor when studying the skew since a market maker will tend to push prices higher or lower to balance his book and by doing so he will indirectly influence IV. Market making is a massive issue when it comes to options and IV skew.

 

[A modest quant]

Implied volatility or vol smile, which ceases to exist after the option expires, is obviously not the actual volatility of the underlying which exists regardless. Is this a market phenomenon? I believe it is. The paper, "Intrinsic Prices of Risk" by Truc Le, explains it all. Could someone take a look and kindly let me know what you think? Cheers