What does it mean that stocks are “memoryless”?
"It reminds me of the Efficient Market Hypothesis, except that just states in its weakest form that the current market price accounts for all information embedded in previous market prices. In other words, people buying today at 42 know it was selling for 40 yesterday, and the patterns and such. To say that stock is memoryless strikes me as not quite right -- to the extent that stocks are valued based on earnings, much of what we infer about future earnings relies on past and present earnings. One obvious counterexample to this ""memoryless"" claim is bankruptcy. If a stock files bankruptcy, and there isn't enough money to pay senior debt, your shares are worth 0 in perpetuity." "I think that ""memoryless"" in this context of a given stock's performance is not a term of art. IMO, it's an anecdotal concept or cliche used to make a point about holding a stock. Sometimes people get stuck... they buy a stock or fund at 50, it goes down to 30, then hold onto it so they can ""get back to even"". By holding the loser stock for emotional reasons, the person potentially misses out on gains elsewhere." "This is an interesting question that may actually be better suited to Quant.SE. First of all, stock prices are random variables, or, to be more precise, stochastic processes (a time-ordered string of random variables).  The alternative to being stochastic is being deterministic, and I doubt you believe that stock prices are deterministic (meaning, they are fully knowable in advance).  The fact that real world events drive the randomness has no bearing on whether or not it is random.  So, to start, I think you have confused the technical definition of random with a colloquial concept. Now, the heart of the question is whether stock prices are memoryless.  Ultimately, this is an empirical question that has been addressed in many academic studies.  The conclusion of most of this research is that stock prices are ""almost"" memoryless, in the sense that the distribution of future stock prices displays very little dependence upon past realizations, although a few persistent anomalies remain.  One of the most robust deviations from memorylessness is the increase in the volatility of a stock following large declines.  Another is persistence in volatility.  In general, in fact, the volatility is far more predictable than the mean of stock price changes.  Hence ""memorylessness"" is a far stronger assumption than the efficient markets hypothesis. The bottom line, however, is that the deviations from memorylessness are relatively small.  As such, despite its limitations, it is a decent working assumption in some contexts." "With my current, limited knowledge (see end), I understand it the following way: Are share prices really described as ""memoryless""?  Yes.  Is there a   technical meaning of the term? What does it really mean? The meaning comes from Markov Models: Think of the behavior of the stock market over time as a Markov Chain, i.e. a probabilistic model with states and probabilistic transitions. A state is the current price of all stocks of the market, a transition is a step in time. Memoryless means that transitions that the stock market might make can be modelled by a relation from one state to another, i.e. it only depends on the current state. The model is a Markov Chain, as opposed to a more general Stochastic Process where the next state depends on more than the current state. So in a Markov Chain, all the history of one stock is ""encoded"" already in its current price (more precisely in all stock's prices).  The memorylessness of stocks is the main statement of the Efficient Market Theory (EMT). If a company's circumstances don't change, then a drop in its share price is going to be followed by a rise later.  So if the EMT holds, your statement above is not necessarily true. I personally belief the EMT is a good approximation - only large corporations (e.g. Renaissance Technologies) have enough ressources (hundreds of mathematicians, billions of $) to be able to leverage tiny non-random movements that stem from a not completely random, mostly chaotic market.  The prices can of course change when the company's circumstances change, but they aren't ""memoryless"" either. A company's future state is influenced by its past. In the EMT, a stock's future state is only influenced by its past as much as is encoded in its current price (more precisely, the complete market's current state). Whether that price was reached by a drop or a rise makes no difference.  The above is my believe, but I'm by far no finance expert. I am working professionally with probabilistic models, but have only read one book on finance: Kommer's ""Souverän investieren mit Indexfonds und ETFs"". It's supposed to contain many statements of Malkiel's ""A Random Walk Down Wall Street""." "@jidugger mostly got it right. It basically mean that past performance of a stock, or a basket of stocks, are not at all useful when trying to predict its future. There is no proven correlation between past and future performance. If there was such a correlation, that was ""proven"" or known, then investors would quickly exploit this correlation by buying or selling this stock, thus nullifying the prediction. It doesn't mean the specific individuals cannot predict the future stock market - hell, if I set up 2^100 different robots, where every robots gives a different series of answers to the 100 questions ""how will stock X do Y days from now"" (for 1<=Y<=100), then one of those robots would be perfectly correct. The problem is that an outside observer has no way of knowing which of the predictor robots is right. To say that stock is memoryless strikes me as not quite right -- to   the extent that stocks are valued based on earnings, much of what we   infer about future earnings relies on past and present earnings. To put it another way - you have $1000 now, and need to decide whether to invest in a particular stock, or a stock index. The ""memoryless"" property means that no matter how many earning reports you view ... by the time you see them, the stock price already accounts for them, so they're not useful to you. If the earning reports are positive, the stock is already ""too high"" because people bought it before you did. So on average, you can't use this information to predict the stock's future performance, and are better off investing in an index fund (unless you desire extra risk that doesn't come with more profitability)." "It means price movements in the past do not affect price movements in the future. Think of the situation of a coin, if you flip it once, and then you flip it a second time, the results are independent of each other. If the first time, you flipped a HEAD, it does not mean that the coin will remember it, and produce a TAIL the second time. This is the meaning of ""memoryless"". FYI, stock markets are clearly not memoryless. It is just an assumption for academic purposes."
"Are you talking long-term institutional or retail investors? Long-term *retail* investors look for *orderly markets*, the antithesis of HFT business models, which have a direct correlation between market volatility and profits. To a lesser extent, some ""dumb money""/""muppet"" institutionals do as well. HFT firms tout they supply liquidity into markets, when in fact the opposite is true.  Yes, HFTs supply liquidity, *but only when the liquidity's benefits runs in their direction*. That is, they are applying the part of the liquidity definition that mentions ""high trading activity"", and conveniently ignoring the part that simultaneously requires ""*easily* buying *or* selling an asset"".  If HFT's are the new exchange floor, then they need to be formalized as such, *and become bound to market maker responsibilities*. If they are actually supplying liquidity, like real Designated Market Makers in the NYSE for example, they become responsible to supply a specified liquidity for specified ticker symbols in exchange for their informational advantage on those tickers. The indisputable fact is that HFT cannot exist at their current profit levels without the information advantage they gain with preferential access to tick-by-tick data unavailable to investors who cannot afford the exchange fees ($1M per exchange 10 years ago, more now). Restrict the entire market, including HFTs, to only second-by-second price data without the tick-by-tick depth, and they won't do so well.  Don't get me wrong, I'm not knocking HFTs *per se*; I think they are a marvelous development, so long as they really do ""supply liquidity"". Right now, they aren't doing so, and especially in an orderly manner. If you want retail investors to keep out of the water as they are doing now, by all means let HFT (and regulatory capture, and a whole host of other financial service industry ills) run as they are. There are arguments to be made about ""only let the professionals play the market"", where there is no role for retail and anyone who doesn't know how to play the long-term investment game needs to get out of the kitchen. But if you are making such an argument, come out and say so." Yes, the net effect is zero. If you own zero shares by Nov 30, for example, and don't buy any more shares by 12/31, the year is done, and nothing left to account for. "Actually, this is a pretty good analogy to certain types of stocks, specifically tech and other ""fad"" stocks. Around the turn of the century, there were a lot of ""Bobs"" buying tech stocks (like they would baseball cards), for tech stocks' sakes. That's what drove the internet and tech stock bubbles of high valuations. At other times, the tech stocks are bought and sold mainly by ""Steve's"" for business reasons such as likely (not merely possible) future appreciation, and command a much lower valuation." "Hmm, that doesn't quite answer my question -- sorry if I was unclear. Let me try rephrasing:  Here's what I'm asking: is the Black-Scholes equation derived using martingale pricing methods (e.g. Girsanov's theorem, the Martingale Representation Theorem, and so on) *equivalent* to the equation derived using the method in the 1973 paper, e.g. constructing a risk-less portfolio and eliminating the drift term? I was led to believe that both of these approaches *lead to the same result*.  I agree that there's an enormous difference between ""equilibrium"" and ""arbitrage free"" *term structure models*; I'm just not convinced that this is a meaningful difference in the case where market completeness holds, e.g. European equity options.   As you correctly said, it's generally impossible to parameterise a Vasicek model to get it to match the observed term structure, while this is something that can be readily done with a Hull-White / Ho-Lee / Libor Market Model. Hence, the prices you get from a Vasicek will obviously be different from what you get using a Hull-White (since you won't be able to match the initial term structure). But such a difference doesn't hold with a Black-Scholes equation derived using martingale vs. the original method.  Sorry if this sounds nitpicky; I just want to make sure I'm understanding these concepts correctly.  EDIT: Added italics." Value is the key word here. Traders should ideally trade on the perceived future value of a company.  Changes in the perceived future value is what leads them to buy and sell shares. That said, if a company were to have some catastrophe happen (say it and all of its employees and property disappeared)  and somehow every shareholder agreed to not sell, the companies market capitalization would remain unmoved even though the value of the company is gone. So theoretically yes, but it is unlikely. &gt; One is whether prices are correlated to each other for long periods of time as a preliminary study suggested (which would go against efficient markets hypothesis, since you could use that info to game the market) or if that result is illusory and the long term returns are close to a standard normal distribution which would follow the effiecient markets hypo.  The fascinating thing about this is that the returns themselves show no correlation at all (at any time scale), but the *absolute* returns do.  i.e. following a sharp rise/drop in price, you can predict that a sharp rise/drop is likely to follow, you just can't say in which direction.  And this effect carries over for long periods.  Given that by the central limit theorem the sum of identically distributed random variables converges to a gaussian, leads one to think that short term returns *ought* to be gaussian also.  However, they're not.  Evidently there is something very subtle going on. How I understand it is: supply/demand affect price of stock   negatively/positively, respectively. Correct. Volume is the amount of buying/selling activity in these stocks (more   volume = more fluctuation, right?). Sort of.  Higher volume means higher liquidity.  That is, a stock that is traded more is easier to trade.  It doesn't necessarily mean more fluctuation and in the real world, it often means that these are well-understood stocks with a high amount of analyst coverage.  This tends towards these stocks not being as volatile as smaller stocks with less liquidity. Company revenue (and profit) will help an investor predict company   growth. That is one factor in a stock price.  There are certain stocks that you would buy without them making a profit because their future revenue looks potentially explosive.  However, these stocks are very risky and are bubble-prone. If you're starting out in the share market, it's generally a good idea to invest in index funds (I am not a broker, my advice should not be taken as financial advice).  These funds aggregate risk by holding a lot of different companies.  Also, statistics have shown that over time, buying and holding index funds long term tends to dramatically outperform other investment strategies, particularly for people with low amounts of capital. "I'm not familiar with the Dupire model; I'll have to take a look at (it sounds cool though).  I think that all arbitrage-free models are ""incomplete"" in the sense that they don't say, ""This is a price that doesn't imply any arbitrage opportunities *anywhere*,"" but instead say, ""This is a price that doesn't imply any arbitrage opportunities within a specific set of securities.""  What set you're using will vary from one model to another, and I'd say (although other people might reasonably disagree) that taking a volatility structure as given is as much no-arbitrage as taking a term structure as given.  As a side note, I'd say that what (theoretically) distinguishes an equilibrium model is that you're supposed to *know* the parameters, not guess at them or observe them from the real world.  By that definition, a really complete CAPM or Black-Scholes would explain how to derive the correct beta or volatility from fundamental analysis.  Also, I've upvoted you elsewhere for some really good comments you made about intrinsic value." Yeah, lots of value in companies is much less tangible than it used to be. But then i think we put too much faith in tangible value. EG a coal mining company my buddy invested in cos it had so much tangible value; had precisely zero tangible value when it collapsed tho "There's the question whether knowledge about unannounced products is actually ""material"" if everyone (the public) knows that something new will be released. If you work at Apple on the development of the iPhone 8, that's not material. If you worked at Apple and you knew that they stopped developing new phones, that would be very, very, very material information.  The important thing as far as the stock market is concerned is what sales look like, and that's not something you know as a product developer." The last column in the source data is volume (the number of stocks that was exchanged during the day), and it also has a value of zero for that day, meaning that nobody bought or sold the stocks on that day. And since the prices are prices of transactions (the first and the last one on a particular day, and the ones with the highest/lowest price), the prices cannot be established, and are irrelevant as there was not a single transaction on that day. Only the close price is assumed equal to its previous day counterpart because this is the most important value serving as a basis to determine the daily price change (and we assume no change in this case). Continuous-line charts also use this single value. Bar and candle charts usually display a blank space for a day where no trade occurred. Nominal. What you say is true, but I'm guessing it would be too complicated to modelate. Plus, a shareholder of a very large company would not necessarily experience said loss if he/she sells the stock in small chunks at a time. "In general, I think you're conflating a lot of ideas.   The stock market is not like a supermarket.  With the exception of a direct issue, you're not buying your shares from the company or from the New York Stock Exchange you're buying from an owner of stock, Joe, Sally, a pension fund, a hedge fund, etc; it's not sitting on a shelf at the stock market. When you buy an Apple stock you don't own $10 of Apple, you own 1/5,480,000,000th of Apple because Apple has 5,480,000,000 shares outstanding.  When a the board gets together to vote on and approve a dividend the approved dividend is then divided by 5.48 billion to determine how much each owner receives.  The company doesn't pay dividends out to owners from a pot of money it received from new owners; it sold iPhones at a profit and is sending a portion of that profit to the owners of the company.   ""When you buy stock, it is claimed that you own a small portion of the   company. This statement has no backing, as you cannot exchange your   stock for the company's assets."" The statement does have backing.  It's backed by the US Judicial system.  But there's a difference between owning a company and owning the assets of the company.  You own 1/5,480,000,000 of the company and the company owns the company's assets.  Nevermind how disruptive it would be if any shareholder could unilaterally decide to sell a company's buildings or other assets. This is not a ponzi scheme because when you buy or sell your Apple stock, it has no impact on Apple, you're simply transacting with another random shareholder (barring a share-repurchase or direct issue).  Apple doesn't receive the proceeds of your private transaction, you do. As far as value goes, yes the stock market provides tons of value and is a staple of capitalism.  The stock market provides an avenue of financing for companies.  Rather than taking a loan, a company's board can choose to relinquish some control and take on additional owners who will share in the spoils of the enterprise.   Additionally, the exchanges deliver value via an unbelievable level of liquidity.  You don't have to go seek out Joe or Sally when you want to sell your Apple stock.  You don't need to put your shares on Craigslist in the hope of finding a buyer.  You don't have to negotiate a price with someone who knows you want to sell.  You just place an order at an exchange and you're aligned with a buyer.   Also understand that anything can move up or down in value without any money actually changing hands.  Say you get your hands on a pair of shoes (or whatever), they're hot on the market, very rare and sought after.  You think you can sell them for $1,000. On tonight's news it turns out that the leather is actually from humans and the CEO of the company is being indicted, the company is falling apart, etc.  Your shoes just went from $1,000 to $0 with no money changing hands (or from $1,000 to $100,000 depending on how cynical you are)." "It's not either or.  Much of the time the value of the stock has some tangible relation to the financial prospects of the company. The value of Ford and GM stock rose when they were selling a lot of cars, and collapsed when their cars became unpopular. Other companies (Enron for example) frankly 'cook the books' to make it appear they are prospering, when they are actually drowning in debt and non-performing assets. So called ""penny stocks"" have both low prices and low volumes and are susceptible to ""pump and dump"" schemes, where a manipulator buys a bunch of the stock, touts the stock to the world, pointing to the recent increase in price. They then sell out to all the new buyers, and the price collapses.  If you are going to invest in the stock market it's up to you to figure out which companies are which." (more like 10 years ago, but that's beside the point) Save, save, save! Both in the notion of squeezing as much value as you can out of every purchase and the notion of putting money away in a savings account.