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."
"Following comments to your question here, you posted a separate question about why SPY, SPX, and the options contract don't move perfectly together.  That's here Why don't SPY, SPX, and the e-mini s&p 500 track perfectly with each other? I provided an answer to that question and will build on it to answer what I think you're asking on this question.  Specifically, I explained what it means that these are ""all based on the S&P.""  Each is a different entity, and different market forces keep them aligned.  I think talking about ""technicals"" on options contracts is going to be too confusing since they are really a very different beast based on forward pricing models, so, for this question, I'll focus on only SPY and SPX. As in my other answer, it's only through specific market forces (the creation / redemption mechanism that I described in my other answer), that they track at all.  There's nothing automatic about this and it has nothing to do with some issuer of SPY actually holding stock in the companies that comprise the SPX index. (That's not to say that the company does or doesn't hold, just that this doesn't drive the prices.)  What ever technical signals you're tracking, will reflect all of the market forces at play.  For SPX (the index), that means some aggregate behavior of the component companies, computed in a ""mathematically pure"" way.  For SPY (the ETF), that means (a) the behavior of SPX and (b) the behavior of the ETF as it trades on the market, and (c) the action of the authorized participants.  These are simply different things. Which one is ""right""?  That depends on what you want to do.  In theory you might be able to do some analysis of technical signals on SPY and SPX and, for example, use that to make money on the way that they fail to track each other.  If you figure out how to do that, though, don't post it here.  Send it to me directly. :)" "An option is a financial instrument instrument that gives you the right, but not the obligation, to do some transaction in the future at a given price. An employee stock option is a kind of ""call option"" -- it gives you the right, but not the obligation, to buy the stock at a certain price (the ""exercise price"", usually set as the price of the stock when the option was granted). The idea is that you would ""exercise"" the option (buy the stock at the given price as provided by the option), if the value of the stock is higher than the exercise price, and not if it is lower. The option is gifted to you. But that does not mean you get any stock. If and when you choose to exercise the option, you would buy the stock with your own money. At what time you can exercise the option (and how many shares you can exercise at a given time) will be specified in the agreement. Usually, you can only exercise a particular share after it has ""vested"" (according to some vesting schedule), and you lose the ability to exercise after you no longer work for the company (plus perhaps a grace period), or after the option expires." The price of a share has two components: Bid: The highest price that someone who wants to buy shares is willing to pay for them. Ask: The lowest price that someone who has a share is willing to sell it for. The ask is always higher than the bid, since if they were equal the buyer and seller would have a deal, make a transaction, and that repeats until they are not equal. For stock with high volume, there is usually a very small difference between the bid and ask, but a stock with lower volume could have a major difference. When you say that the share price is $100, that could mean different things. You could be talking about the price that the shares sold for in the most recent transaction (and that might not even be between the current bid and ask), or you could be talking about any of the bid, the ask, or some value in between them. If you have shares that you are interested in selling, then the bid is what you could immediately sell a share for. If you sell a share for $100, that means someone was willing to pay you $100 for it. If after buying it, they still want to buy more for $100 each, or someone else does, then the bid is still $100, and you haven't changed the price. If no one else is willing to pay more than $90 for a share, then the price would drop to $90 next time a transaction takes place and thats what you would be able to immediately sell the next share for. What explains the most of the future returns of a portfolio is the allocation between asset classes. In the long term, stock investments are almost certain to return more than any other kinds of investments. For 40+ years, I would choose a portfolio of 100% stocks. How to construct the portfolio, then? Diversification is the key. You should diversify in time (don't put a large sum of money into your stock portfolio immediately; if you have a large sum to invest, spread it around several years). You should diversify based on company size (invest in both large and small companies). You should also diversify internationally (don't invest in just US companies). If you prefer to pick individual stocks, 20 very carefully selected stocks may provide enough diversification if you keep diversification in mind during stock picking. However, careful stock picking cannot be expected to yield excess returns, and if you pick stocks manually, you need to rebalance your portfolio occasionally. Thus, if you're lazy, I would recommend a mutual fund, or many mutual funds if you have difficulty finding a low-cost one that is internationally diversified. The most important consideration is the cost. You cannot expect careful fund selection to yield excess returns before expenses. However, the expenses are certain costs, so prefer low-cost funds. Almost always this means picking index funds. Avoid funds that have a small number of stocks, because they typically invest only in the largest companies, which means you fail to get diversification in company size. So, instead of Euro STOXX 50, select STOXX 600 when investing to the European market. ETFs may have lower costs than traditional mutual funds, so keep ETFs in mind when selecting the mutual funds in which to invest. For international diversification, do not forget emerging markets. It is not excessive to invest e.g. 20% to emerging markets. Emerging markets have a higher risk but they also have a higher return. A portfolio that does not include emerging markets is not in my opinion well diversified. When getting close to retirement age, I would consider increasing the percentage of bonds in the portfolio. This should be done primarily by putting additional money to bonds instead of selling existing investments to avoid additional taxes (not sure if this applies to other taxation systems than the Finnish one). Bond investments are best made though low-cost mutual funds as well. Keep bond investments in your local currency and risk-free assets (i.e. select US government bonds). Whatever you do, remember that historical return is no guarantee of future return. Actually, the opposite may be true: there is a mean reversion law. If a particular investment has returned well in the past, it often means its price has gone up, making it more likely that the price goes down in the future. So don't select a fund based on its historical return; instead, select a fund based on low costs. However, I'm 99% certain that over a period of 40 years, stocks will return better than other investments. In addition to fund costs, taxes are the other certain thing that will be deducted from your returns. Research what options you have to reduce the taxes you need to pay. 401-K was explained in another answer; this may be a good option. Some things recommended in other answers that I would avoid: "I think ""Psychological Pricing"" is a similar phenomenon to what you are looking for.  This is where retailers use certain numbers in prices because those prices are more appealing to consumers.  Since stocks - and in your case bitcoin - have prices, they too will be more or less appealing at different prices based on psychology alone." "The correlation I heard most about in economics/finance was that stock prices and bond yields were negatively correlated; as the stock market does better, bond yields fall (company's doing well as evidenced by stocks, so it's a good credit risk, so YTM of its bonds on the market goes down). The correlation, if any, between the stock and futures market should be visible in the actual price histories. Index prices may be useful, but what's more likely is that various future prices have correlation with various companies' stocks. Where the future reflects the price of a raw material that is a significant cost of goods sold for a company, you'll see these two move inversely to each other in the short term. I think that if there is a causative relationship here, its that futures prices influence stock prices, not the other way around. The futures market generally represents the cost side of a consumer goods producer's bottom line. The stock market represents its profits. As futures go up, profit expectations go down, putting pressure on stock prices. Industries that deal in services, or in other types of goods, can still be affected because a rise in the cost of something consumers need will cause them to spend less on other things which affects margins in those other areas. So, in the short and medium term, when the futures market goes up the stock market sees a dip, and vice versa.  However, companies adapt; they can put upward pressure on prices for their goods to restore their desired margins, usually by slowly increasing them to prevent sticker shock (though elasticity of demand plays a part; the more we need something no matter what it costs, the faster prices can increase). To maintain costs, they can make things cheaper using less expensive materials (more plastic, less steel). They can restructure production processes (translated: move factories offshore, or at least to ""right-to-work"" states with less union strength) to save costs elsewhere. All of these reduce costs and thus increase profits, but take time to implement. Many of these things reduce direct costs, reducing demand for the commodity and causing the futures prices to go back down. So, over the long term, these differences even out, and it's down to the things that affect the entire market (inflation, consumer/investor confidence, monetary policy)." It's a way to help normalize the meaning of the earnings report. Some companies like Google have a small number of publicly traded shares (322 Million). Others like Microsoft have much larger numbers of shares (8.3 Billion). The meaning depends on the stock. If it's a utility company that doesn't really grow, you don't want to see lots of changes -- the earnings per share should be stable. If it's a growth company, earnings should be growing quickly, and flat growth means that the stock is probably going down, especially if slow growth wasn't expected. "You have actually asked several questions, so I think what I'll do is give you an intuition about risk-neutral pricing to get you started.  Then I think the answer to many of your questions will become clear. Physical Probability There is some probability of every event out there actually occurring, including the price of a stock going up.  That's what we call the physical probability.  It's very intuitive, but not directly useful for finding the price of something because price is not the weighted average of future outcomes.   For example, if you have a stock that is highly correlated with the market and has 50% chance of being worth $20 dollars tomorrow and a 50% chance of being worth $10, it's value today is not $15.  It will be worth less, because it's a risky stock and must earn a premium.  When you are dealing with physical probabilities, if you want to compute value you have to take the probability-weighted average of all the prices it could have tomorrow and then add in some kind of compensation for risk, which may be hard to compute. Risk-Neutral Probability Finance theory has shown that instead of computing values this way, we can embed risk-compensation into our probabilities.  That is, we can create a new set up ""probabilities"" by adjusting the probability of good market outcomes downward and increasing the probability of bad market outcomes.  This may sound crazy because these probabilities are no longer physical, but it has the desirable property that we then use this set of probabilities to price of every asset out there: all of them (equity, options, bonds, savings accounts, etc.).  We call these adjusted probabilities that risk-neutral probabilities.  When I say price I mean that you can multiply every outcome by its risk-neutral probability and discount at the risk-free rate to find its correct price.    To be clear, we have changed the probability of the market going up and down, not our probability of a particular stock moving independent of the market.  Because moves that are independent of the market do not affect prices, we don't have to adjust the probabilities of them happening in order to get risk-neutral probabilities. Anyway, the best way to think of risk-neutral probabilities is as a set of bogus probabilities that consistently give the correct price of every asset in the economy without having to add a risk premium. If we just take the risk-neutral probability-weighted average of all outcomes and discount at the risk-free rate, we get the price.  Very handy if you have them. Risk-Neutral Pricing We can't get risk-neutral probabilities from research about how likely a stock is to actually go up or down.  That would be the physical probability.  Instead, we can figure out the risk-neutral probabilities from prices. If a stock has only two possible prices tomorrow, U and D, and the risk-neutral probability of U is q, then Price = [ Uq + D(1-q) ] / e^(rt) The exponential there is just discounting by the risk-free rate.  This is the beginning of the equations you have mentioned.  The main thing to remember is that q is not the physical probability, it's the risk-neutral one.  I can't emphasize that enough.  If you have prespecified what U and D can be, then there is only one unknown in that equation: q. That means you can look at the stock price and solve for the risk neutral probability of the stock going up.   The reason this is useful is that you can same risk-neutral probability to price the associated option.  In the case of the option you don't know its price today (yet) but you do know how much money it will be worth if the stock moves up or down.  Use those values and the risk-neutral probability you computed from the stock to compute the option's price.  That's what's going on here.  To remember: the same risk-neutral probability measure prices everything out there.  That is, if you choose an asset, multiply each possibly outcome by its risk-neutral probability, and discount at the risk-free rate, you get its price.  In general we use prices of things we know to infer things about the risk-neutral probability measure in order to get prices we do not know."