Probability and Std Deviations

B

Brighton

I was comparing some probability results from Hoadley online and his Excel add-in to a similar tool from iVolatility.com, which is distributed to customers of OptionsHouse, TradeKing, Scottrade and perhaps others.

The results for 'probability at expiration' (finishing above or below) match perfectly, so that's good news. But the Standard Deviation figures from iVolatility caught my eye because calculating them using what I think is the standard method (in Excel) I get:
1SD = $10.40
a range of +/- 1SD = $21.60 to $42.40
a range of +/-2SD = $11.20 to $52.80
a range of +/-3SD = $0.80 to $63.20

The iVolatility figures are in the ballpark for 1SD but there's quite a difference at 2SDs and a huge difference at 3SDs.

Any ideas why there's such a discrepancy in SD and SD ranges? I thought this was cut and dry math, but based on the iVolatility results, they have a big error or they're using another formula that works better for this purpose. I thought they might be looking at the price history of the underlying to calculate/display results based on a non-normal distribution, but this is an underlying I made up - I did not input a ticker symbol. :confused:


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Brighton said:
The iVolatility figures are in the ballpark for 1SD but there's quite a difference at 2SDs and a huge difference at 3SDs.

Any ideas why there's such a discrepancy in SD and SD ranges?
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Normal v LogNormal Probability Distributions.
Technically should be calculated from the Forward Price.
Downside SD's should be smaller than Upside SD's ... as Downside cannot fall below zero ... no limit to Upside.
 
Thanks, James, for pointing me in the right direction. I'll have to do some digging to see if I can learn what's behind the iVolatility calculator and compare the results to the various probability distribution curves in the Hoadley Excel add-in (the ones where you can set skew and excess kurtosis; I know they're there but I haven't used them).

Just for kicks, I looked up the price distribution range the Ivolatility tool predicted for March Natural gas (and options) as of January 2, 2014. Using an underlying price of $4.30, ATM IV of 37.1%, 54 DTE and a 650 call, the calculator showed:

3 SDs at $6.52 and a probability of close above $6.50 = .15%; of touch =.37%. Hypothetically, if someone sold a $6.50 call using the tool and did a Rip Van Winkle until Feb 25th, the tool would have been accurate (the high was $6.49x) and the option would have expired worthless. Of course, Rip would have come close to losing his ass during late Feb and a few night sessions just prior to option expiration, but he was asleep.

In real life, I doubt very many people would hang on for that long, but it was an interesting exercise. I'm going to run some more scenarios ... they might be useful for the current situation in coffee.
 
Price is lognormally distributed, and return is normally distributed. If you do a bit of searching under my username, I've posted some sample calculations involving normal distributions. BTW, probability of touch is roughly twice the probability if finishing OTM.
 
panzerman said:
Price is lognormally distributed, and return is normally distributed. If you do a bit of searching under my username, I've posted some sample calculations involving normal distributions. BTW, probability of touch is roughly twice the probability if finishing OTM.
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Thanks, I used one of those formulas in a spreadsheet I'm building out.

On a related topic, I was trying the Price Probability Distribution function in Hoadley Tools where the user enters Spot, DTE, Volatility, Skew and Kurtosis and Excel draws a lognormal curve and a non-lognormal curve. Reading the Help file, I came across this.

Limitations: This function uses the Gram-Charlier expansion to adjust probabilities for skewness and kurtosis. Gram-Charlier is an approximation and works well within certain limits and combinations of skewness and kurtosis. Outside these limits negative probabilities and non-unimodal probability curves (distributions with more than one peak) can be produced. See “Edgeworth Binomial Trees” by Mark Rubinsetein (Journal of Derivatives (Spring 1998)) for more information this.

In some of the commodity data I've loaded, skew and kurtosis are outside of "certain limits" and I get negative probabilities (a curve below the X-axis). Does this mean a user should beware of the tail probabilities, or is the entire probability curve suspect?
 
Edit to add some precision to the post immediately above:

In some of the commodity data I've loaded, skew and kurtosis are outside of "certain limits" and I get negative probabilities. So far, the values are quite small - four or five decimal places each and cumulatively less than one or two percent. Is this noise? Should I beware of the probabilities just near the tails or is the entire curve suspect? In short, are there some rules of thumb on acceptable negative values that don't distort the whole curve?
 
Brighton said:
In some of the commodity data I've loaded, skew and kurtosis are outside of "certain limits" and I get negative probabilities.
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If you have a reasonably dense and liquid option chain, you can imply a probability distribution directly from the option prices. Obviously, the CAB/tick wings will look out of wack, but you should ignore them anyway. It makes for an interesting (and very educational excerise) to compare implied distribution to historical, especially if you renormalize the return to the second moment of distribution.
 
sle said:
If you have a reasonably dense and liquid option chain, you can imply a probability distribution directly from the option prices. Obviously, the CAB/tick wings will look out of wack, but you should ignore them anyway. It makes for an interesting (and very educational excerise) to compare implied distribution to historical, especially if you renormalize the return to the second moment of distribution.
More...

Implied Prob Distribution from Vertical Spread prices?
 
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