I went to see “I’m Not There”, a movie about Bob Dylan’s double. It’s a bit long, but what’s impressive in the movie, besides his music and the covers of it (and perhaps in real life according to “Don’t Look Back” as well as various biographies) is his refusal to be typecast, to be or do what his fans expect him to be. He accentuates that in his Chronicles – Volume One, which has an interesting first part but then gets too much (for me) into describing all his recording sessions.
There’s a review of the movie in The New Republic which I skimmed though and it makes the point that the movie doesn’t show the consequent sacrifices of this kind of position. It’s a risky sacrificial position to take in real life, talented or not. I know few people capable of it. He’s not a fat tail or an outlier; he’s simply not part of the distribution, and hence his present is not the expected value of his future. He’s not a martingale.
I suppose there are people like that in all fields, but I’m trying to think who. In finance Fischer Black was a little that way. In the fiction world the review mentions Pynchon, who refuses to enter the fan game at all, and wants no persona, just his books, to exist. In theater or movies, I don’t know — maybe Greta Garbo. There must be other and better examples.
On the martingale/numeraire front, I’ve been reading bits and pieces and I still feel uncomfortable with any of the expositions I’ve seen. The binomially based ones are like scratching your left ear with your right hand by by putting your arm behind your neck. The binomial proof of option valuation is economically based, with clear intuition and a straightforward proof of risk neutrality. The binomially based proofs of the martingale theorems are irritating; they try to rephrase the simple results of the binomial risk-neutral model in terms of the theorems of martingale theory, and they’re awkward and provide a very unnatural extension. But the continuous time proofs get buried in technicalities and intuition is very hard to extract.
I notice that Wilmott’s book, or at least the previous one that I have with me right now, doesn’t even attempt to cover this approach. That says something.