Why is Brownian motion often used in Finance?

Because you can do Brownian motion calculations in your head.  Anything more complicated and you can't do the calculations mentally.  If I mathematically combine two Brownian motions then I usually get a new Brownian motion. Brownian motion will fail very badly on the week that the financial world ends.  However, most days you go to the office and the world doesn't end at which point Brownian motion is "good enough." The difference in using drift is that it becomes harder to do the calculation in your head, and in most cases drift is irrelevant for what you are trying to calculate.  For example, your typical stock index will have a return of 10%-20% per year.  If you calculate the movements of the stock over an hour or over a day, that return is swamped by the random motions of the stock and for what you are interested in, you can ignore it.

In 1900, the French mathematician https://en.wikipedia.org/wiki/Louis_Bachelier (1870-1946) completed a doctoral thesis (supervised by the great https://en.wikipedia.org/wiki/Henri_Poincar%C3%83%C2%A9) in which he worked out a model for the variation of the prices of assets (i.e., of things that can be bought and sold many times over) like stocks and bonds.  The thesis, titled Théorie de la spéculation ("Theory of Speculation") got a positive reception from the jury and was published soon thereafter.  It was then largely forgotten for sixty years, until it was rescued from obscurity by economist https://en.wikipedia.org/wiki/Paul_Samuelson and others, who realized that Bachelier had made a seminal contribution to the problem of understanding asset prices.  Bachelier's thesis is now generally cited as the founding document of quantitative finance.  In 2006 it was issued in English translation by Princeton University Press, with a foreword by Samuelson and commentary by two British mathematicians: Bachelier's key insight was that if the asset prices showed any identifiable pattern (other than the long-term growth trend associated with macroeconomic expansion), one would expect that speculators would find it and exploit it, thereby eliminating it.  Thus, for instance, if it were possible to know now that a certain stock's price will spike in three months, speculators trading on that knowledge would act to smooth out that spike.It's worth quoting the official report on Bachelier's thesis, co-signed by Poincaré, https://en.wikipedia.org/wiki/Paul_%C3%83%C2%89mile_Appell, and https://en.wikipedia.org/wiki/Joseph_Valentin_Boussinesq: One could imagine combinations of prices on which one could bet with certainty.  The author cites some examples.  It is clear that such combinations are never produced, or that if they are produced they will not persist.  The buyer believes in a probable rise, without which he would not buy, but if he buys, there is someone who sells to him and this seller believes a fall to be probable.  From this it follows that the market taken as a whole considers the mathematical expectation of all transactions and all combinations of transactions to be null.  [emphasis added] Thus, after speculators have incorporated all available knowledge into their trades, one expects that the result will be prices showing unpredictable fluctuations, independent of their past history.  This is what is modernly called a "https://en.wikipedia.org/wiki/Random_walk" or, in certain contexts, a "https://en.wikipedia.org/wiki/Martingale_(probability_theory)".The equations that Bachelier obtained from this model correspond to what is known in physics as https://en.wikipedia.org/wiki/Brownian_motion.  Bachelier wrote this five years before Albert Einstein's celebrated paper explaining physical Brownian motion in terms of statistical mechanics.  As far as we know, Einstein never heard of Bachelier.  The mathematical theory of Brownian motion as a stochastic process was later formalized by https://en.wikipedia.org/?title=Norbert_Wiener.  Some mathematicians have therefore referred to Brownian motion as the "Bachelier-https://en.wikipedia.org/wiki/Wiener_process".Bachelier himself was well aware of (indeed, guided by) an analogy to Fourier's law for the physical diffusion of heat (see https://en.wikipedia.org/wiki/Heat_equation).  According to the official report: The manner in which M. Bachelier extracts [the https://en.wikipedia.org/wiki/Central_limit_theorem] is very original and all the more interesting as his reasoning could extend with some changes to the theory of [the https://en.wikipedia.org/wiki/Propagation_of_uncertainty].  He develops it in a chapter whose title seems a little strange, because he calls it 'Radiation of Probability'. It is in effect a comparison with the analytic theory of heat propagation to which the author has had recourse.  A little reflection shows that the analogy is real and the comparison legitimate.  The reasoning of Fourier is applicable almost without change to this problem, so different from that for which it was created. Bachelier's insights were refined and systematized in the 1970s by Black, Scholes, and Merton (see https://en.wikipedia.org/wiki/Black%C3%A2%C2%80%C2%93Scholes_model), in the context of solving the https://en.wikipedia.org/wiki/Option_(finance) pricing problem in finance.  This won Scholes and Merton the Nobel Prize in economics for 1997 (by which time Black was dead).  Black and Scholes begin their 1973 paper by explaining that If options are correctly priced in the market, it should not be possible to make sure profits by creating portfolios of long and short positions in options and their underlying stocks.  Using this principle, a theoretical valuation formula for options is derived. Note that such a model is based on the assumption that all economically relevant information that could affect the prices has in fact been incorporated by speculators into the prices themselves.  This has been called the "https://en.wikipedia.org/wiki/Efficient-market_hypothesis".  In economics, an "efficient" outcome is one in which all gains from trade have been realized, so that everyone is as well off as they can be, without having to make someone else worse off.  In this particular context, it means that speculators have found all possible gains that can be made from playing the market, so that what's left is a Brownian motion of the asset prices (about the long-term trend determined by "fundamentals", such as macroeconomic growth, rather than by speculation).To what extent this is true in the real world is a matter of longstanding and intense academic debate.  An often told joke in this context is that two economics professors are walking down the street, when one notices a $100 bill lying on the ground.  As the first economist is about to pick it up, the other (who evidently believes in efficient markets) says: "Leave it.  If it were a real $100 bill someone else would've picked it up already."The efficient market hypothesis (and therefore the Brownian motion models) seems to work well in the short term.  In the long term, however, there appear large fluctuations which are difficult to reconcile with the market being efficient.  This connects to the hard problem of explaining theoretically the largest fluctuations of the https://en.wikipedia.org/wiki/Business_cycle, including the recurrence of financial crises and macroeconomic recessions.Debate on this subject was highlighted by the 2013 Nobel Prize in economics, awarded to Fama, Hansen, and Shiller (see the relevant "http://www.nobelprize.org/nobel_prizes/economic-sciences/laureates/2013/advanced.html" on the Nobel Foundation's website).  Fama is one of the most notable and effective defenders of the efficient market hypothesis.  Shiller coined the term "irrational exuberance" to describe how crowd psychology can lead to bubbles in the prices of certain assets, which eventually burst.  Hansen is somewhere in the middle of those views, and has developed mathematical tools to study the problem based on data.  Thus, the Nobel committee hedged its position on the question of the efficiency of asset markets (as any sensible financial analyst would've advised them to do in the face of uncertainty).One obvious inefficiency of asset markets is "insider trading", when someone knows something that the rest of the world doesn't know yet, and uses that knowledge to make a profitable trade in the market.  However, in a big, sophisticated, and robust asset market, one would expect opportunities for insider trading generally to be one-off and very short-term.  A deeper problem is whether the risk ratings of various securities are generally reliable, as they should be in an efficient market.  That such ratings should've led many investors astray in the run-up to the financial crisis of 2007-08 suggests to me that the market might sometimes depart significantly from efficiency.

All excellent answers, especially from Alejandro. Here are some additional mathematical observations and why they're relevant to the debate about asset prices: 1) In the limit, and assuming small steps, Brownian motion and random walk are the same (the term "random walk" was first used in 1905, a few years after Bachelier published his thesis).  Most of the mathematics of modern finance is built on this limiting case. 2) When Bachelier did his work he assumed that in the short term, (hours, days, weeks, but not years) asset prices moved randomly and the probability distribution was 'normal' or Gaussian. 3) Everyone is correct in pointing out that these assumptions (eg: random walk and Gaussian probability distribution) simplified the statistical mathematics - which is one reason why they became so popular. However, these simplifying assumptions were being questioned as far back as the 1950's and certainly in recent years have been demonstrated to not match reality. As Alejandro points out, certainly in the longer run (years, to decades) asset returns are not random.  There is a roughly 5-7 year up/down business cycle and over 15 to 20 years a definite upward drift as pointed out by the questioner. But even more fundamental, in the short term, although asset prices do appear to be random (certainly on the scale of hours or days), their probability distribution is definitely NOT Gaussian. This has been amply pointed out by Mandelbrot in several publications.  In fact the probability distribution is much closer to a power law distribution, the same type of probability distribution that describes the frequency vs magnitude of earthquakes or the distribution of wealth vs population. Why is this important? Because according to traditional economics and finance orthodoxy (based on Gaussian distributions), events such as the one day 20%+ drop in the S&P 500 on October 19, 1987, or the 2007 - 2009 market meltdown (and several other large market swings), were statistically impossible. If you assume a power law distribution however, these same events are improbable, but possible, similar to a magnitude 9 earthquake for example. Finance academia has been slow to accept this uncomfortable reality for several reasons: 1) The mathematics of power law distributions has proven difficult to fit into the existing 50 year body of work that the current finance academic structure has been built on. 2) It's potentially career limiting for grad students to attempt to break away from this orthodoxy.  If someone succeeds there's probably a Nobel prize in it, but the chances of failing are high. EDIT: I forgot to add one more key point. Also pointed out by several others, the Gaussian distribution assumption is a "good enough" cheap and cheerful approximation most of the time, but its most serious flaw is it underestimates risk. So long as the market is puttering along in calm waters this isn't a problem. But as Long Term Capital Management painfully discovered in 1998, when the financial waters start to get very stormy, risk assumptions based on the Gaussian distribution no longer work. Long Term Capital Management was founded in 1994 on the assumption that one could use the sophisticated mathematics of modern finance (Gaussian based), and powerful computers to create arbitrage and hedging strategies. Nobelists Myron Scholes and Robert Merton, academics who were significant contributors to mathematically-based financial theory, were among their founding principals. Since market inefficiencies are typically small they used massive leverage to amplify their returns; in 1988 they had $4.7B in equity and $124B in debt, giving assets of $129B all used in arbitrage positions.  Obviously, with a huge debt/equity ratio of 25:1, they were closely monitoring risk based on Schole's and Merton's math. They did very  well for 4 years, reaching a 40% annual return.  Then around March 1998 things suddenly went very wrong. They had been betting on the Asian financial crisis at the time.  Through a complex confluence of unpredictable events their positions suddenly turned on them (the financial equivalent of one of those 'statistically impossible' magnitude 9 earthquakes) and they were unable to get out quickly enough and were eventually forced to liquidate at very unfavorable prices.  With a potential $125B default hanging in the balance, they had to be bailed out by the federal reserve to avoid a market collapse.  They lost most of their equity but struggled on for several years.  In 2009 the fund shut down. Here's the value of $1,000 invested in LTCM from founding to the end of 1998. Source:  Jay Henry; Own Work; Public Domain

The simple answer is that they don't really anymore. It was originally used because it is an assumption that provides great simplification (you could approximate an option price through BM just in excel formulae), and it is not a ridiculous assumption to make. It evolved from the PDE school of pricing, Monte Carlo methods provide a solution to the BS equation. This is why it was used. The efficient market hypothesis is of little real world significance to BS. In reality a liquid market exists in vanilla derivatives, and so the pricing of options is not a problem that requires much computation in reality. Most option traders will not even quote prices when dealong in house, as such, they will simply quote an implied volatility level. Pricing of vanilla options is rarely considered in any theoretical detail these days, unless it is done as a calibration step for more complex products. It should be noted that BM and BS are both wrong, and do not give correct prices for the majority of derivatives out there. The actual pricing of derivatives now only really sits with exotic derivative desks, and any monte Carlo pricing that is done will at a minimum use a local volatility model, and almost always use stochastic volatility. These do not yield a Brownian motion.

Brownian motion is used commonly in finance ( and many other non financial areas) to allow one to build and explore models based on randomness/patternless/ independent events. While brownian motion may be known as one of the corner stones in quantum physics, ultimately it is a TOOL that allows us to model based on uncertainty. It is not that the economics/ financial mathematics fraternities completely belive that financial economics and finance is based completely on uncertainty, but from a scientific methodology it is useful to explore this perspective. I'm much the way fund managers use benchmarks, this too can be seen as a benchmark of sorts. "In the worst/best case let's just assume that stock prices are completely independent observations, let's see what we get with this model and compare that to our other models based on other theories that we can contrast to this uncertainty assumption" Brownian motion is a well developed concept which allows whatever studies based on it to be easily reviewed, understood and further developed.

Because modulo technicalities, all continuous time martingales are time changed Brownian motion.

The movement of asset prices is non-stationary, albeit there are periods when it approximates Brownian Motion.  Asset prices reflect the complex interactions between economic forces, market dynamics and investors' sentiments, all of which are influenced by news, which is inherently unpredictable and which is therefore very difficult to model.    Efforts expended by many in trying to model markets in this simplistic Brownian Motion manner played very well in the hands of those of us who chose other paths.

I agree with Alejandro, but 1) Brownian motion is used because one can do the calculus with it ( there are a lot of processes that are martingales!!!) 2) It is principaly used in exotic option pricing where in fact the models are principally used to interpol and project complex products on simpler one's and the fact that historic data doesn not repsect normal laws is then a a lesser problem