What are the axes on the longnormal plot corresponding to the Volatility Smile?

[Steve Jobs]

In BT notes for Hull chapter - Volatility smile, page 7, there are 2 longnormal plots. Just wanted to make sure...the x axis is the exercise price of the options and the y axis is the price of the option, is that correct?

 

[IRENA BLAZINCIC]

I think the two plots are probability distribution for the underlying asset price at the option expiration date.

x axis is asset price
y axis is probability density function

 

[David Harper CFA FRM]

Thank you @irenab for giving the correct answer (+1 star entered into the weekly drawing).
We probably should label these
Irenab is totally correct

• It's maybe easier to spot them as probability density (pdf) functions. They are either cumulative (CDF) or densities, but CDF functions needs to be (monotonically) increasing toward 1.0, which there are not so almost by elimination, they are densities
• It is tempting to interpret the x-axis under Hull's Fig 19.4 (on which these are based) as strike prices because the only markers on it are strike K(1) and strike K(2), but that's because these asset prices (on the lognormal plot) correspond to strike prices on the implied volatility smile. So, this confused me for a while to be candid, but the values on the x-axis are the same (the x-axis plots increasing prices) but it is a "switch" from strike prices (implied vol smile) to asset prices (density function). I've actually come to think that understanding this is the key to unlocking the intuition:
  • eg., typical equities smile is an implied volatility skew (decreasing function) where, at low strike price (left hand side of implied vol smile) say K(1), an out-of-the-money (OTM) put has a high implied volatility. Which corresponds to a heavy left tail: higher-than-lognormal probability that asset price will decrease down to K(1).
    So, on the left, low strike (implied vol smile) ~ OTM put = ITM call --> heavy left asset tail (density function)

 

[Steve Jobs]

Yes, I see, the y is probability density function, but for the x axis, I mentioned k in first reply after checking this page on slide no 10 where it's k.

 

[David Harper CFA FRM]

Thanks @Steve Jobs I honestly love to see other sources, like your link. There is often value in seeing it from a fresh perspective. But this slide 10 is very similar to Hull, and the K(1) and K(2) labels on the X-axis are exactly why this axis is confused for the strike price. But they are just corresponding values on the axis, the axis labels on the right side switches (from strike price) to asset price.

Consider the K(2), which is the higher strike price. On the vol smile chart, K2 is the strike price on the right hand side which corresponds to a lower implied volatility (left chart: y axis). On the density function, this value of K2 represents the anchor for an asset price because this is an asset distribution.

Say S(0) = $30, K1 = $20, and K2 = $40 and we have a typical equity skew (decreasing implied vol)

• On the implied vol skew, K2 = 40 refers to the lower market price observed for an OTM call or ITM put (high strike price on right hand side of x-axis as strike price). Again, high strike --> low price = low implied vol
  
• On the density function, what does this mean? It refers to the probability the asset price will be greater than 40 (high asset price on the right hand side of x axis as asset price). The low implied vol is determined by a low price which is the market saying, "the probability is less-than-given-by-lognormal that asset price will reach 40"

 

[Steve Jobs]

Got the point. When I first saw the plot, I though the PDF is for options because intuitively the relation between volatility and options price is direct, so the if the left side of the vol smile is higher, then the left side of the PDF should be also heavy, however that does explain the the middle of the PDF, thanks.

 

[Steve Jobs]

In q p2.t5.3 (4 to 6), according to the q4 and its answer: increase in demand for call options => increase in price of call options => increase in implied volatility (it has to since it's the only variable that can change, ceteris paribus, (1) is this logic correct?) => (2) would this increase in implied volatility cause increase/decrease in k as well?(the x axis of the volatility smile), (3) what about S?

I hope my questions are clear...

 

[David Harper CFA FRM]

.4 is correct, but I missed the revision to .6 which (per https://forum.bionicturtle.com/thre...ion-3-6-on-implied-vol-smile.5621/#post-15888) should read:

03.06. Option writers (market makers who take short positions) are averse to high delta call options but prefer low delta call options?
Because supply is higher for OTM call options (delta nearer to zero!) and supply is lower for ITM call options (delta nearer to 1.0), this dynamic would promote an equites-like skew: higher implied volatility on the left tail and lower implied volatility on the right tail. That is, on the right (higher strike price), OTM call options in greater supply (being preferred for their lower delta) so their price is lower (increase supply --> lower price) so the implied volatility is lower

This is true and gets to the essence, on several levels: increase in demand for call options => increase in price of call options => increase in implied volatility

• Price of options is largely a (technical) function of supply and demand; e.g., the "crashopobia" narrative starts with the presumption of higher demand for out-of-the-money puts. In this context, we can think of supply/demand as exogenous (assumption)
• Price determines implied volatility (this is the inversion of the BSM OPT). In the crashophobia narrative, higher demand for OTM puts -- > higher price --> necessarily implies higher implied volatility at lower strike prices
• In turn, with regard to implied distribution versus lognormal, this skew (higher implied vol at lower strike price) implies a heavier left tail for the asset's distribution

Re your question (2) and (3) @Steve Jobs , these questions indicate a misunderstanding, IMO: (K) is a contractual feature invariant to supply/demand and (S) is the underlying which would, in theory, be indirectly impacted via arbitrage but that's more complex than we need to get, thanks,