Mandelbrot's basic argument here is that stock price movements don't have the log-normal distribution assumed by the conventional theory underpinning the Black-Scholes-Merton option pricing formulae, but rather a power-law distribution, as the evidence appears to bear out. This means that much of modern finance theory is based on assumptions that are demonstrably wrong. As Mandelbrot (and/or his co-author Richard Hudson) puts in, an a passage I found rather endearing -
If this were astronomy, the argument would have ended long ago. Imagine observatories suddenly finding a new planet where, the standard theory says, none should be. And then another, and another and another. Astronomers, after checking their instruments, would not ignore the data; they would question their understanding of celestial mechanics and a new and fruitful episode in astronomy would dawn. But it does not work that way in economics, even though the equivalent of countless new planetary systems have been recorded.
Fair comment on the reluctance of economists to change their paradigms (although it does seem as though power-law distributions are now well established in the statistical finance field). But I do think he's being rather generous in his assessment of astronomers...
I interviewed Mandelbrot about his financial work a few years ago. He's probably the most egotistical person I've ever interviewed, but I guess he's entitled to be. And I'll always treasure his explanation of why his work is like a beautiful woman.