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We see in the market that a implied volatility surface is not flat. Based on this observation different models were developed to capture the structure, e.g. CEV / SABR.

A measure often used for the skew is a risk reversal, i.e.

$$\sigma_{25,c}-\sigma_{25,p}$$

and butterfly

$$\frac{\sigma_{25,c}+\sigma_{25,p}}{2}-\sigma_{ATM}$$

where $\sigma_{25,c}$ is the implied volatility of $25$ delta call.

Looking at the skew, you are interested in the slope an curvature. The mathematical objects would be for the slope of a function $f$:

$$\frac{f(x+h)-f(x)}{h}$$

and for the curvature

$$\frac{f(x+h)-2f(x)+f(x-h)}{h^2}$$

So why are the above measure (RR and BF) not constructed like this? Should they be seen as an approximation?

Moreover, why is it common to just look at a specific RR / BF, 25 for example. Wouldn't it be more reasonable to calculate these measures for every strike (delta measured) on the grid? Obviously the slope and curvature can and will change for different deltas.

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  • $\begingroup$ Hi OP, I understand that this might be a post v long ago, but where did you find the curvature formula? Any links or references? $\endgroup$
    – KaiSqDist
    Commented Apr 10 at 8:16

4 Answers 4

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You are absolutely correct that they should be seen as approximations. While it would be nice to let h go to zero in a mathematical sense this is of course impossible in real life as the options are only traded in particular intervals. While the smallest interval may be less than 25, for historical reasons traders have gotten used to using the 25 point.

Many more sophisticated models generally do use the full grid as you suggest. However, there are many complications the most important of which is values far in/out of the money can be very tough to price so various methods creating smooth surfaces are generally the most accepted methods for pricing.

Still RR and BF are good approximations and will likely continue to be used as long as humans are still trading or at least spot checking the prices.

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    $\begingroup$ Moreover, as h becomes small, small errors in the measurement of the sigma of the calls and puts are magnified. Choosing a large h reduces this noise in the calculations at the expense of approximating the average slope over a wider interval. $\endgroup$
    – jaamor
    Commented Oct 21, 2014 at 11:58
  • $\begingroup$ Thanks for your answer. However, about the grid. It seems for me not reasonable to look just at the 25 points, if there are more points available. The slope/curvature can change a lot, right. $\endgroup$
    – math
    Commented Oct 21, 2014 at 17:52
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    $\begingroup$ Yes and no. I'm guessing all but a few of the computerized pricing models used these days don't factor in the full surface. However, for near-the-money options on liquid underlyings in reasonable market conditions (ect.) the 25 point slope/curvature should be a good approximation. $\endgroup$
    – rhaskett
    Commented Oct 21, 2014 at 19:05
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I wonder if the reasons these approximations are widely used - instead of a whole set of estimates for different deltas, as proposed - have to do with liquidity and market structure.

Liquidity: A market participant willing to trade e.g. a 10 delta option for no economic reason other than skew will find, for many products, that the edge evident from a fitted IV curve no longer exists after transaction costs, or not in much size, anyway. To find enough liquidity to trade skew and skew alone in meaningful sizes, it may be necessary to trade closer to the money. But then a 25 delta rr or related estimates will be sufficient.

Market structure: the kinds of participants that use these heuristics are not the sort that are looking for nickel arbitrage opportunities from the curve. Firms making markets in options in order to do this are assuredly looking at all the strikes.

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A measure of vol skew which is used is $\frac{d\sigma}{dK}$ or $\frac{dC}{dK}$. You first need to build arbitrage free volatility curve for that. RR's and fly's are just used to get the pillar points and only in FX. The IR market directly gives pillar points i.e. $\sigma(K)$. I would suggest you read Gatheral's book if you want to know the details of volatility surface construction.

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RR and Bfly are market traded instruments in FX. They give pillar points which are then used to make the volatility skew/smile curve by interpolation. There are various methods of interpolation like, cubic spline, SVI,SABR.

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  • $\begingroup$ Thanks for your answer. There is a difference in RR as trading strategy and measure of vol skew (see: en.wikipedia.org/wiki/… ). I know that it is common in FX to provide implied vols as RR / BF. $\endgroup$
    – math
    Commented Oct 18, 2014 at 17:59

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