Friday, May 14, 2010

“Fuzzy” option spreads

I was skimming through Kees Van Deemter’s book Not Exactly: In Praise of Vagueness (Oxford University Press, 2010) when I came across the following diagram:

It displays our intuitive answer to the question: When is a person tall? We know that all heights below a certain value definitely don’t qualify as tall and all those above another value unequivocally count as tall. That leaves a grey area. Van Deemter suggests that the following function might be suitable (where v stands for degree of truth):

0 if x < 150
1 if x > 190
(x-150)/(190-150) otherwise

We have three intervals—two horizontal lines--0 below 150 cm and 1 above 190 cm—with a diagonal line in between. Welcome to the world of fuzzy membership functions.

If we shift the binary functions 0 and 1 to defined negative and positive values and think in terms of financial instruments, what do we have? Well, yes, a bull call option spread.

(compliments of OIC)

There’s the maximum profit, the maximum loss, and all that fuzzy stuff in between.

And the point, you ask? Actually, I’m not sure there is any. But isn’t it interesting?

1 comment:

  1. Ha! Very interesting... got me thinking about an economics paper that claimed to improve upon the "rational agent" assumption. Basically, the dominant paradigm in designing economic models is to assume that people are utility maximizers (in an absolute sense). The authors argued that people seek a minimum level of utility (food, job, etc.) and after that minimum they seek to maximize their utility relative to other people.

    Or put more plainly, after we meet our basic needs we want higher social status. Obvious, yes, but this is pretty good for economists. They go on to claim that this explains stock market momentum and bubbles, et cetera, but their proof involved industrial strength maths so I can't verify it.

    Anyway, perhaps the fuzzy height function could be improved using their "insight"... the height of the person judging the other person's tallness would be a factor in the equation, although the minimum and maximum would still hold, as the judge would be well aware of average height and average opinion, regardless of their own tallness or shortness.