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 pathdependent 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 multimillionaire (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.
Subscribe to:
Post Comments (Atom)
Blog Archive

▼
2005
(253)

▼
02
(29)
 Riskadjusted return
 LTCM (and the *real* smart guys)
 Housing and Inflation
 The shadow of LTCM
 Null energy condition
 Smart money reaches $1 trillion
 First FX domino?
 "Easy Al" Greenspan
 Finance for beginners
 Out on the tail
 Summers firing
 Asperger watch
 Book recommendation
 Evolutionary timescales
 Housing revisited
 Place your bets...
 Liquidity feedback
 Fed symposium on China exchange rate
 A tale of two geeks
 Path integrals
 Carly Fiorina no longer HP CEO
 Household debt
 Science funding
 ChinaJapan relations
 Cosmic accounting
 The myth of fingerprints
 Coming soon: the $4K car from China
 Housing bubble
 Virtual meetings overdue

▼
02
(29)
Labels
 physics (326)
 genetics (263)
 finance (257)
 globalization (248)
 brainpower (237)
 technology (180)
 american society (173)
 China (162)
 economics (158)
 psychometrics (153)
 genomics (152)
 photos (152)
 science (151)
 innovation (135)
 psychology (133)
 travel (125)
 credit crisis (115)
 universities (110)
 higher education (107)
 human capital (105)
 biology (103)
 ai (100)
 iq (96)
 startups (90)
 genetic engineering (89)
 cognitive science (85)
 credit crunch (78)
 machine learning (78)
 careers (76)
 evolution (71)
 gilded age (71)
 elitism (70)
 income inequality (69)
 autobiographical (68)
 statistics (68)
 books (61)
 quantum mechanics (59)
 caltech (57)
 genius (57)
 social science (57)
 political correctness (53)
 politics (53)
 talks (50)
 bgi (48)
 bounded rationality (48)
 history of science (48)
 mma (48)
 sci fi (48)
 geopolitics (47)
 kids (47)
 education (46)
 cdo (45)
 silicon valley (45)
 academia (43)
 derivatives (42)
 mathematics (41)
 podcasts (41)
 harvard (39)
 behavioral economics (38)
 intellectual history (37)
 bubbles (36)
 history (36)
 mortgages (36)
 realpolitik (36)
 MSU (35)
 literature (35)
 jiujitsu (34)
 computing (32)
 expert prediction (32)
 film (32)
 hedge funds (32)
 bjj (30)
 foo camp (30)
 physical training (30)
 ufc (30)
 quants (29)
 efficient markets (28)
 google (28)
 many worlds (28)
 black holes (27)
 economic history (27)
 neuroscience (27)
 sports (27)
 von Neumann (27)
 nuclear weapons (26)
 entrepreneurs (25)
 feynman (25)
 movies (25)
 subprime (25)
 housing (24)
 obama (24)
 singularity (24)
 taiwan (24)
 race relations (23)
 berkeley (22)
 ultimate fighting (22)
 athletics (21)
 music (21)
 wall street (21)
 affirmative action (20)
 cds (20)
 scifoo (20)
 biotech (19)
 security (19)
 gender (18)
 goldman sachs (18)
 internet (18)
 blogging (17)
 meritocracy (17)
 quantum field theory (17)
 treasury bailout (17)
 university of oregon (17)
 venture capital (17)
 conferences (16)
 freeman dyson (16)
 smpy (16)
 autism (15)
 cosmology (15)
 cryptography (15)
 japan (15)
 oppenheimer (15)
 personality (15)
 algorithms (14)
 happiness (14)
 height (14)
 new yorker (14)
 hedonic treadmill (13)
 probability (13)
 wwii (13)
 aspergers (12)
 fitness (12)
 india (12)
 malcolm gladwell (12)
 net worth (12)
 nobel prize (12)
 social networks (12)
 television (12)
 chess (11)
 christmas (11)
 dna (11)
 geeks (11)
 les grandes ecoles (11)
 neanderthals (11)
 string theory (11)
 blade runner (10)
 entropy (10)
 football (10)
 government (10)
 italy (10)
 mutants (10)
 nerds (10)
 olympics (10)
 privacy (10)
 pseudoscience (10)
 Einstein (9)
 ability (9)
 complexity (9)
 dating (9)
 eugene (9)
 flynn effect (9)
 crossfit (8)
 determinism (8)
 encryption (8)
 harvard society of fellows (8)
 keynes (8)
 nsa (8)
 pca (8)
 philip k. dick (8)
 philosophy of mind (8)
 pop culture (8)
 real estate (8)
 research (8)
 robot genius (8)
 usain bolt (8)
 video (8)
 Go (7)
 aig (7)
 alpha (7)
 art (7)
 ashkenazim (7)
 basketball (7)
 data mining (7)
 energy (7)
 free will (7)
 game theory (7)
 games (7)
 hugh everett (7)
 james salter (7)
 manhattan (7)
 poker (7)
 qcd (7)
 russia (7)
 turing test (7)
 war (7)
 Fermi problems (6)
 alan turing (6)
 anthropic principle (6)
 bayes (6)
 class (6)
 climate change (6)
 environmentalism (6)
 france (6)
 fx (6)
 nassim taleb (6)
 noam chomsky (6)
 prostitution (6)
 success (6)
 tail risk (6)
 teaching (6)
 volatility (6)
 academia sinica (5)
 bobby fischer (5)
 democracy (5)
 econtalk (5)
 luck (5)
 paris (5)
 patents (5)
 renaissance technologies (5)
 sad but true (5)
 software development (5)
 warren buffet (5)
 100m (4)
 Iran (4)
 Poincare (4)
 bill gates (4)
 borges (4)
 cambridge uk (4)
 charles darwin (4)
 cold war (4)
 creativity (4)
 fake alpha (4)
 feminism (4)
 global warming (4)
 godel (4)
 hormones (4)
 humor (4)
 inequality (4)
 intellectual property (4)
 iraq war (4)
 kasparov (4)
 kerviel (4)
 markets (4)
 microsoft (4)
 mixed martial arts (4)
 monsters (4)
 moore's law (4)
 nonlinearity (4)
 perimeter institute (4)
 solar energy (4)
 soros (4)
 trento (4)
 vietnam war (4)
 200m (3)
 babies (3)
 brain drain (3)
 censorship (3)
 charlie munger (3)
 chet baker (3)
 correlation (3)
 demographics (3)
 ecosystems (3)
 equity risk premium (3)
 facebook (3)
 fannie (3)
 fst (3)
 information theory (3)
 intellectual ventures (3)
 jim simons (3)
 judo (3)
 language (3)
 lee kwan yew (3)
 lewontin fallacy (3)
 lhc (3)
 magic (3)
 michael lewis (3)
 nathan myhrvold (3)
 neal stephenson (3)
 new york times (3)
 olympiads (3)
 path integrals (3)
 quantum computers (3)
 rationality (3)
 risk preference (3)
 search (3)
 sec (3)
 sivs (3)
 society generale (3)
 thailand (3)
 alibaba (2)
 assortative mating (2)
 bear stearns (2)
 bruce springsteen (2)
 charles babbage (2)
 cheng ting hsu (2)
 cloning (2)
 david mamet (2)
 digital books (2)
 donald mackenzie (2)
 drones (2)
 eliot spitzer (2)
 empire (2)
 exchange rates (2)
 freddie (2)
 gaussian copula (2)
 heinlein (2)
 industrial revolution (2)
 james watson (2)
 ltcm (2)
 mating (2)
 mba (2)
 mccain (2)
 monkeys (2)
 national character (2)
 nicholas metropolis (2)
 no holds barred (2)
 offices (2)
 oligarchs (2)
 palin (2)
 population structure (2)
 prisoner's dilemma (2)
 simulation (2)
 skidelsky (2)
 socgen (2)
 sprints (2)
 supercomputers (2)
 systemic risk (2)
 variance (2)
 virtual reality (2)
 abx (1)
 anathem (1)
 andrew lo (1)
 antikythera mechanism (1)
 athens (1)
 atlas shrugged (1)
 ayn rand (1)
 bay area (1)
 beats (1)
 book search (1)
 bunnie huang (1)
 car dealers (1)
 carlos slim (1)
 catastrophe bonds (1)
 cdos (1)
 ces 2008 (1)
 chance (1)
 children (1)
 cochranharpending (1)
 cpi (1)
 david x. li (1)
 dick cavett (1)
 dolomites (1)
 drugs (1)
 dune (1)
 eharmony (1)
 epidemics (1)
 escorts (1)
 faces (1)
 fads (1)
 favorite posts (1)
 fiber optic cable (1)
 francis crick (1)
 frauds (1)
 gary brecher (1)
 gizmos (1)
 greece (1)
 greenspan (1)
 hypocrisy (1)
 igon value (1)
 iit (1)
 inflation (1)
 information asymmetry (1)
 iphone (1)
 jack kerouac (1)
 jaynes (1)
 jfk (1)
 john dolan (1)
 john kerry (1)
 john paulson (1)
 john searle (1)
 john tierney (1)
 jonathan littell (1)
 las vegas (1)
 lawyers (1)
 lehman auction (1)
 les bienveillantes (1)
 lowell wood (1)
 lse (1)
 mcgeorge bundy (1)
 mexico (1)
 michael jackson (1)
 mickey rourke (1)
 migration (1)
 mit (1)
 money:tech (1)
 myron scholes (1)
 netwon institute (1)
 networks (1)
 newton institute (1)
 nfl (1)
 oliver stone (1)
 phil gramm (1)
 philanthropy (1)
 philip greenspun (1)
 portfolio theory (1)
 power laws (1)
 pyschology (1)
 randomness (1)
 recession (1)
 sales (1)
 singapore (1)
 skype (1)
 standard deviation (1)
 star wars (1)
 starship troopers (1)
 students today (1)
 teleportation (1)
 tierney lab blog (1)
 tomonaga (1)
 twitter (1)
 tyler cowen (1)
 ussr (1)
 venice (1)
 violence (1)
 virtual meetings (1)
 war nerd (1)
 wealth effect (1)
8 comments:
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
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.
MFA: The expression is roughly what you wrote, except the delta functions are not in the exponent, and if the pathdependent 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 MS 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 ;)
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.
Steve,
Many thanks for your clear explanations!
MFA
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
Hey Steve,
How does the nonnormal 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?
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...
Post a Comment