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# FRM 2006 PI q22 - Volatility smile vs. constant volatility

#### fullofquestions

##### New Member
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

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.

Reference: Options, Futures, and Other Derivatives, Hull, 2006.

The answer does not explain how a risk system with an assumed constant volatility understates the implied volatility. Any suggestions?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
HI FoQ,

This came up last year: http://www.bionicturtle.com/forum/viewthread/812/#1980

I think it's doubly-flawed:

1. the long/short have the same implied volatility, so there is no difference btwn (c) and (d)
true, the long is sitting on a positive intrinsic value and short has negative the same, but higher implied vol is same for both, exposure for both...so I see no difference btwn (c) and (d)

2. there is a presumption that constant vol is "anchored" ATM or something like that; constant vol could be set higher than the skew ... so even if choice (C) gave "deep in the money put" so that (D) made sense...to your point, there is still an imprecision in making an assumption about where the constant vol is set (!) .. saying constant volatility is not even (IMO) as good as saying "if the system assumes lognormal distribution"

...this question makes no sense to me

(the equity does have skew, so if choice (C) instead were "short position in a deep-in-the-money *put* ... then (D) becomes defensible: for equities, we do expect a skew/smirk instead of a smile where implied volatility is *higher* -- think left side of skew chart -- for ITM call/OTM put and *lower* for OTM call/ITM put)

David