Volatility smile

[higaurav]

Hi David,

"The implied volatility decreases as the strike price increase" - I am not able to understand this concept, because as I thought that as strike price increase call option will be deep out of the money and put will be in the money (although this is relative to stock price), in both cases implied volatility should increase. At the same time, I am not able to comprehend the meaning of this statement (given in bold). Pls help.

(this was given as the explanation of one of the ans in practice exam 2007)

Rgrd,
OM

 

[David Harper CFA FRM]

Hi OM,

In fact we cannot make a general statement about what happens to implied volatility as the strike price changes, without more information.

Implied volatility is not a fundamental issue with the option pricing model (OPM). It will be difficult if you think of this only in terms of the valuation model. Rather: given a market price for the option, we solve for the implied volatility (let this equal 'ivol' below) that causes the model value to equal (=) the market value:

Find the ivol for which:
Black-Scholes Function [stock, strike, ivol?, term, riskless rate] = observed market price.

The option market could shift tomorrow and the implied volatilities will change - and the shape of the volatility smile/smirk - it is a function of market prices. This is one way to understand the smile: market prices could change and it could become any shape so it's shape it's sort of an arbitrary function of how markets are pricing options.

Since the smile (for equities) empirically tends to dip at stock = strike (that's why it's called a smile), this statement likely refers to an in-the-money call or out-of-the-money put (either because the implied vol on call & put with same strike & maturity must be the same due to put call parity) because, if there is a volatility smile (i.e., lowest implied vol near the money), then an in the money call will be decreasing in implied vol as stock price increases. However, an at-the-money call will be increasing in implied vol as strike increases.

David

 

[dennis_cmpe]

This seems related to #22 in 2006 FRM Exam:

22. With all other things being equal, a risk monitoring system that assumes constant
volatility for equity returns will understate the implied volatility for which of the following positions
by the largest amount:

a. Short position in an at-the-money call
b. Long position in an at-the-money call
c. Short position in a deep in-the-money call
d. Long position in a deep in-the-money call

Correct answer is D.

I know answer A and B are wrong because implied volatility is lower for at-the-money calls when compared to deep in-the-money calls due to the volatility smile. But I don't understand why being or long or short a deep in-the-money option would affect the volatility smile? I don't understand why answer C would not have the same implied volatility as answer D?

Here is the official explanation:

A plot of the implied volatility of an option as a function of its strike price demonstrates a pattern known as the volatility smile or volatility skew. The implied volatility decreases as the strike price increases. Thus, all else equal, a risk monitoring system which assumes constant volatility for equity returns will understate the implied volatility for a long position in a deep-in-the-money call.

 

[David Harper CFA FRM]

Dennis,

I agree with you. I don't get it. (c) and (d) have same implied volatility. Actually, I think it's doubly-flawed: a contant volatility assumption per se doesn't over- or under-state implied volatility, the constant assumption could be set higher (.e., nothing requires the constant assumption to be set per the lower ATM implied vol), so maybe i'm totally missing the question but i don't get any of it...David