Path integrals

I've finished lecturing on renormalization of QED, and am now covering path integrals. Feynman, following an earlier observation of Dirac, showed that the quantum amplitude for a particle to propagate from A to B is given by the sum over all possible paths connecting A to B, weighted by i times the classical action of each path:

Amplitude = Sum e^{i S[path]}

This yields a very intuitive formulation of quantum mechanics and quantum field theory. Mathematicians don't like the Feynman path integral (merely a heuristic used by physicists!). Due to its highly oscillatory integrand, little has been rigorously established about its properties or even its existence. Under better control is the related Wiener integral (an imaginary time Feynman integral or Euclidean path integral), which takes the form (S is real):

Sum e^{- S}

It didn't take long for physicists on Wall St. to realize that options pricing theory can be completely recast in path integral language. The Euclidean path integral for a free particle describes Brownian motion (a random walk). Interpret the location of the particle (in 1 spatial dimension - cake!) as the log of the price of a security, and you are off to the races! In the path integral language we value a derivative contract as the payoff averaged over all future paths. (We can only do this if the derivative can be perfectly hedged at all times, so risk preferences do not enter, but that is a subtlety.)

I derived a closed form expression using the free particle propagator and delta function potentials for the value of any possible exotic path-dependent option. At the time, this was quite novel, as such contracts were usually priced using Monte Carlo simulations. (If you look hard enough you can find an MIT Sloan school report with all the details :-) Soon after, I was offered a job in the equity derivatives group at Morgan Stanley. Being young and idealistic (dumb and naive), I decided it was better to be a postdoc at Harvard than a future multi-millionaire (although now that I think about it I did have a faculty offer at Yale by that time). A reporter from CNBC interviewed me as one of the rare "rocket scientists" who turned down Wall St. (They had no shortage of interviews with the other type :-) The camera man even shot some footage of me walking into Lyman Lab with my crummy backpack full of physics books.