Volatility Smile

[[email protected]]

Dear David,

I am confusing answers about the volatility smile.
I'm working on file T5.a.Hull-Chapters-18-&-24.
I don't understand answers between questions, 18.01 and 18.05.

18.01's answer says that heavier right tail leads low prices for out-of-money calls and in-the-money puts.
However, answer of 18.05 says that less heavy right tail should lead to low prices, and therefore low volatilities for out-of-money calls.

How could heavier right tail and less heavy right tail have same conclusion about the out-of-money call's price?

 

[David Harper CFA FRM]

Hi Kuntange,

Apologies for delay. Both of these questions are from the source Hull (I say that because, to your point, frankly the language is a tad imprecise ....). Here is the meaning:

Both questions are focused on the implied volatility smile, but the explanations also include the implicit comparison to the Black-Scholes model value such that:

In 18.01(b), the higher implied volatility at higher strike price (to the right) implies a heavy right tail. The is IMPLIED BY a HIGHER market based option price and, therefore, the Black-Scholes option price (as it assumes a flat line) will be lower. Put again, the model (BSM) price will be lower than the higher market price which implies the higher implied volatility

In 18.05, same story: LIGHTER (less heavy) right tail implies a LOWER implied volatility implied by the lower market price; i.e., the BSM model price gives a higher value than the lower price observed.

So the confusion is due to the difference between BSM (model price) and observed market price. We could imagine, superimposed over each of these charts, a FLAT LINE; the flat line is the single constant volatility assumption used in the BSM. A heavier (i.e., than lognormal) distribution is then implied by an implied volatility that is higher (greater than) the flat line. And this, in turn, implies either: a higher-priced (observed) option; or equivalently, a Black-Scholes option price that would be LOWER than this observed higher price.

I hope that helps, David

 

[southeuro]

Just wanted to point out to a type in Greeks and Volatility questions set:

assume initial delta of an ATM call option with strike at $20 and the BSM model based option delta is .5.
If there's no volatility smile (flat implied vol) what is the option delta when stock price decreases to $19 (from ATM to slightly ITM)?
Answer is given as delta <0.5 (since it's moving to OTM)

it should be written as (from ATM to slightly OTM) I think since our K>S so we won't exercise the call option.

cheers,

 

[David Harper CFA FRM]

Hi @southeuro

Yes agreed, it looks like it was corrected in the source but not reverted to the PDF. See https://forum.bionicturtle.com/threads/l2-t5-08-greeks-and-volatility-smile.3387/
Thank you (+1 star),