Appropriate probability density function

Often one sees reference to the work of Black, Scholes, et al in attempting to assign probabilities of price movements for an option's underlying. That work makes use of the normal probability distribution. Can anyone enlighten this inquisitive soul as to the direction folks are taking in selecting other, perhaps more appropriate, probability distributions?

 

Every public-domain pricing model I'm aware of uses a lognormal probability distribution. If anyone is actually using a different probability distribution, it's proprietary.

But what people are really interested in is not so much the probability distribution. Rather, it's the implied volatility skew and how to adjust your calculations so that the skew - the actual prices (IV's) at which options at different strikes trade - fits into some consistent mathematical framework. There are a number of approaches to that, such as generating functions that will fit the existing curve, cubic spline, etc.

In other words, what high-level traders and option market making firms are doing is manipulating the change in IV strike by strike using some mathematical function that gives it some form and consistency.

I've also seen attempts to manipulate skewness and kurtosis to match price reality, but I don't believe that approach is as widely used - if it's used at all.

 

http://en.wikipedia.org/wiki/Probability_density_function

 

There's more than one way to model a cat. I have not had time to follow up on this, but as a newbie to this whole realm I am curious about the spline fitting. I have a hard time seeing how that is useful. IV is independent of time (as your comment suggests to me)? Hardly!

I have not yet read Mandelbrot's papers (suggesting an exponent change for the random process), but this is the sort of direction which appeals to my curiosity.

The idea of manipulation of skewness and/or kurtosis is exactly what appeals to me. I do not know it to be useful, but it appeals none-the-less

 

Skewness and kurtosis tell you about the distribution of the PDF. In no way does this tell you anything useful about an option. You'll find you're only comprehensible measure in modelling options is sigma squared narrowed down to the square root of sigma squared which is called standard deviation. The Mean, Median, and mode will then tell you the information your brain can comprehend. The earlier poster didn't get to the joke of lognormal distributions, which is, "Thank god a stock can't go lower than zero."

 

How about SABR as an alternative approach? It evolved as a sort of counterpoint to the local vol models, which constitute another alternative to BS. As to skewness and kurtosis, there are a couple of good books on trading vol that discuss those subjects.

Go look at papers on wilmott.com, there's all sorts of things available.

 

Instead of trying to find some tiny negligible edge by adjusting existing option models, I’d recommend to spend your energy on things that will give you greater return for your time and energy. I’d pay attention to more practical things such as that the distributions on stocks have tendencies to be leptokurtic and have negative skewness, and when selecting option strategy I’d be very cognizant of the characteristics of the skew such as how it moves and ages. I’d look at these things from a practical stand point.

 

Of what?

 

Thanks for the suggestions as they've given me new directions to dig. I like closed form solutions to problems as they satisfy some inner need. The leptokurtic nature inspired this thread. I am agnostic with regard to adjusting existing models or inventing new ones.