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Delta as nothing to do with probabilities. Take a look at the delta of a future that could be 1 or 100%. Does it mean the future would have 100% probability finishing 'in the money' (higher) ? No.

OP, It's model-dependent.

No. That 's 'At the money forward', and just for european style options. Sorry. But please, would you show us the forward for american style ones. You have all the time you need. You may need some help : "An option is at-the-money if the strike price is the same as the spot price of the...

1 year call struck at $100 Spot is $100 interest rate is 2% Dividend rate is 15% volty is 30% Delta is 0.3344 or 33.44% and this option is "At The Money" e.g strike=spot. Is this what you were looking for ? (This is a pure B&S)

You can't be serious !

I didn't want to be that rude. My point is pretty simple : - Jim Simons is famous 'cause Renaissance's returns - Empirica/Universa are known 'cause Taleb's TV shows I'm sure you see what I mean. It doesn't mean his ideas are wrong. But, as a matter of fact... And 2008 may be the best year...

It sounds very interesting, but...did Taleb make a killing during the crisis ? I mean in the real world ? Wait...I mean as a trader ? You can't be serious. Lots of recent models swap randomness with uncertainty, that means there is no distribution needed to price an option, and volatility...

Maverick74 made some very good points here as far as he didn't try to get a statistical sense of what he points out. "We can solve backwards for fair value at expiration and derive what each call and put should have traded at a specific point of time based on the expiration value of each"...

Speed is the delta of the gamma. Nobody tries to compute it with a closed form derivation. To get a sensible number, put some vegas and dvega /dspot in it.

'tsup ?

Well, for sure Laplace, Legendre, Fourier are just tools for pricing. But I never heard something about real traders making real money with fractal theory. Some examples ?

True. That's why Taylor expansions does the job most of the time.

Are you waiting for an answer like "well, you know it depends how low or high IV vs HV are and..." ? It's been said. But now, 3 pages after, would you give your opinion?

Gaps are quite rare during trading sessions.

Options and stops don't cover the same risks. One can't cover gap risk with stops, and as far as markets won't be continuous, gaps will happen. That means if your trading timeframe is about seconds, minutes, hours..., you won't need options. But if your trading timeframe is about days...

Don't you think options would be redundant if it were an existing optimal stop loss strategy ?

Because derivatives are just like geometry (take a look at Simons'background) they obey the same rules : reasoning with wrong figures to get the right result.

And what do you mean by "What" ? :D

'Cause it's cheap !

If it's the way you understand the point... My point was easy to grasp: there is no way to replicate a vanilla with CON only, and trying to replicate a vanilla with AON and CON is foolish. Maybe you can show us how to replicate a VANILLA with CON ? Do you ? Well if you couldn't You may...