Thursday, February 10, 2005

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 once 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.

Anonymous said...

Very interesting, Steve!

Could you elaborate a bit more on what PI one needs to calculate for pricing derivatives? Is it $<\exp x>$, where the measure is $\exp{-(1/2)\dot{x}^2+V(x)}$, where $V(x)$ is a sum of delta functions?

Also, a more basic question. What is point of the derivatives? The pricing is done so that it is lossless to the issuer (models assumed, of course), I presume. Is it so he can raise capital for other ventures he may have in mind?
For the buyer of the derivatives, if hedging, where is the money being made? (Speculating can lead to large losses and gains...)

MFA

Carson Chow said...

You cannot tell me you are unhappy with the choice you made. You couldn't be hanging out in Eugene, sipping latte's and blogging if you were on Wall Street! I have never regretted making the academic choice, crumby backpack and all. I admire all of our friends that are successful at it but it definitely would not have been for me.

steve said...

MFA: The expression is roughly what you wrote, except the delta functions are not in the exponent, and if the path-dependent option you are computing is an arithmentical average (also called an Asian option - one of the hard cases), you have to remember that the security price is e^x(t) not x(t).

Derivatives are useful for transferring risk. Market makers provide a service (like an insurance company) and can charge a premium over what they believe to be the fair price. For example, the CFO of Pepsi may not want to deal with FX risk for all the money they make selling soft drinks in Europe. They can buy a derivative (maybe an average option!) to hedge their continuous currency exchanges from euros to dollars. Their core expertise is beverages, not FX, so they may be happy to pay a premium to be rid of that extra risk. If someone smart thinks they understand the dynamics of a particular market, and can model the risk, they can provide a useful service, making it possible for people to buy and sell risk in that market in a more and more refined way. Eventually, lots of people think they understand the risks and the market becomes deep and liquid. CMOs (collateralized mortgage obligations) and CDOs (collateralized debt obligations) are two examples that impact average people - the latter are pretty new, and let people shift around the default risk on corporate bonds. Without CMOs home mortgages would be more expensive and without CDOs companies would pay more to raise money through debt issuance.

Carson: the guys who wanted to hire me at M-S have already retired (more than once) in the intervening years, having made their "numbers". So, on that alternate path, the other me is probably sitting at a cafe in an even nicer place (Newport beach? Santa Cruz?), sipping a latte, blogging, and not having to do committee work or teach classes :-)

BTW, if I'm going to have to think about dirty, applied, complex systems (as opposed to beautiful fundamental physics), I might as well be paid for it ;-)

Carson Chow said...

Hindsight is always 20/20. The quant movement in finance could have crashed and burned. You could have bought a lottery ticket and be retired as well. Also, not everyone has retired. You yourself wrote about how the `number' can be a moving target.

Anonymous said...

Steve,

Many thanks for your clear explanations!

MFA

Anonymous said...

A foolish wondering: Why does the definiteness of the physics extension of the mathematics seem perfectly apt, while the market extension suggests there is more uncertainty in outcome than can quite be accounted for. Clever application, of course, and yet...

Anne

maxkennerly said...

Hey Steve,

How does the non-normal distribution of price movements factor in this? Doesn't the Feynman equation have an implicit assumption of some paths being more likely than anothers, such that the paths occupy at normal (i.e., "bell") distribution? Or does Feynman give no weight to the likelihood of certain paths over others?

steve said...

Carson: you are right, we could all optimize our lives further using hindsight... not a very fair comparison!

Anne: these are just models of how markets behave, and hence flawed in many ways. (See: Long Term Capital Mgmt.) But they do offer the reassuring aura of mathematics, or, better, a "spurious air of technicality"!

Max: you can change the dynamics by choice of action. The simple one everybody uses has a log normal distribution built in, which as we know underestimates rare events. I've been quite interested in what real volatility distributions look like, as you can see if you look at some earlier posts...