Using butterfly spread to imply shape of probability distribution of Nifty ( or any other underlying asset) and a surprise

I spent sometime digging through option data for some insight about the nifty level. Although the search led to an anticlimax(see the last section) I think the method itself could be used to determine the distribution of nifty level

Butterfly Spread : 

A butterfly spread is a portfolio of three call options (c1,c2,c3)

one long position in call option with strike K1(c1), two short positions in call at strike K2(c2) and one long position in call option with strike K3(C3)

K1+K3 = 2K2 , K2 is midpoint Strike

Pay off : max(S-K1,0)+max(S-K3,0)-2*max(S-K2,0)

Cost of setting up the portfolio : c1+c3-2c2

 A butterfly spread always has a positive pay off and the maximum pay off is fixed at K2-K1. 

Implying probability distribution:

There are a lot of methods to imply probability distribution  of the underlying but there are a lot of assumptions that one has to make about risk free rate and so on. However, Butterfly spread is a better bet because

1) Shape of Payoff is fixed i.e. it is the area under the triangle is fixed with fixed base and fixed height.  

2)It is a good approximation delta function concentrated at K2 i.e. it is not too off to think that the entire mass is concentrated at K2(this is an appoximation but a good one.  If we had options at continuous strikes this would indeed get very close to delta function i.e. a spike at a point).

3) The price of the portfolio(c1+c3-2*c2) . As we move  mid point strike K2 along the X axis this price gives us the measure of probability that the market assigns to event that the price is K2 for various values of K2

So if we fix the maturity and move  mid point strikes then we get cost of setting up this portfolio(c1+c3-2c2) for various mid point strikes K2. Now pay off is fixed and maturity is fixed. The the only thing that changes while moving the triangle along the X axis is the probability that the price level will be between K1 and K3 or the probability that the price will be approximately K2. So the cost of setting up this portfolio is proportional to probability of the price being K2. 

Using the Nifty option chain data(fancy for option price at various strikes) for July 28th maturity following chart can be derived. 

Maximum probability happens to be around 8550 levels which is not surprising given that maturity is just a week away and hence the option market predicts that the price of the Nifty on 28th July will be close to the current levels.

Anti Climax

Now lets look at what the market has to say about August maturity option. I have kind of converted the prices to probabilities knowing that prices are proportional to probabilities:

So one month down the line the mean still seems to be 8550 levels. But what is more surprising is that other than spreading out the whole distribution i.e. making tails fatter the market is not saying anything about the future. 

What this means is that we are less certain about a one month horizon than a 7 day horizon(which is why the graph is spread out and is fatter) but there is no other insight about the level that option market can provide. I think everyone is on the same boat with respect to their understanding of Nifty levels in future i.e. no one knows. At least that is what the data tells!