# Option implied risk neutral distribution vs BKM risk neutral moments

I am doing some research on the option implied risk neutral distribution and methods calculate it, and so far have come across two ways to do so.

The first way is through the Breeden-Litzenberger formula, which seems like a popular method to calculate the density via a finite differencing approach. There are many good posts both online and here describing how it is done in good detail, e.g.

How to derive the implied probability distribution from B-S volatilities?

Breeden-Litzenberger formula for risk-neutral densities

https://www.newyorkfed.org/medialibrary/media/research/staff_reports/sr677.pdf

The second way is the Bakshi, Kapadia and Madan (BKM) risk neutral moments method, where the authors' original paper is also highly cited. This post How are the BKM risk-neutral moments derived? contains the link to the original paper and the answer also provides a high level breakdown of some of the equations in the paper.

My questions are:

1. Are these two related in any way? E.g. can't we just use Breeden-Litzenberger to back out a risk neutral distribution, and with the distribution we can calculate its higher order moments (e.g. skewness/kurtosis), which is what the BKM method is trying to estimate? Do we just stick with Breeden-Litzenberger if the implied distribution is the only thing we are after (as BKM doesn't directly give you the distribution of all those moments)?

2. For the Breeden-Litzenberger method, the 'usual' way I've seen people do it is something like a) first start off with an IV smile, b) do a cubic spline interpolation/extrapolation in a way such that you don't introduce arbitrage, c) convert IV back to prices via black scholes, d) finite differencing the price function to get the density. Assuming that we already have e.g. the call prices to begin with, why don't we just perform the interpolation/extrapolation directly on the call prices, and then go straight to d)? I have access to OptionMetrics data which comes with both IV and price, why should I even bother with the IV smile at all when I already have the price?

## 2 Answers

Re 1: Yes. BL gives you an approximation of the whole density, BKM gives you an approximation of each moment, common implementations offer the first four moments but you should be able to produce more moments.

Re 2: Both ways are viable. IMO, the route via interpolating somewhere between bid/ask prices (instead of bid/ask vols) can be easier to implement / interpret. I once used a corresponding method proposed by Monnier 2013 and I was quite happy with the results.

HTH?

• 1) so technically if I have the density via BL, I don't really need the BKM for anything else, right? 2) since the aim is to make the option price vs strike price curve continuous, would a basic cubic spline be sufficient? if not, why? what is the difference between interpolating in the price-space vs interpolating in the vol-space? Commented Mar 23, 2022 at 15:55

You need to be careful, if you have a few market points (K,IV) or (K,Price) and you start interpolating points in between you very likely will generate arbitrage. It is recommended to firstly fit some parametric form, like the SABR or SVI formula and then use that formula for the necessary differentiation under the BL model. Otherwise, you will very likely notice that you will not be able to price back market instruments.