How is a probability cone read?
A number of ways exist to calculate the chances of a particular outcome. Options, for example, use current price, cost of money, and volatility among other factors to price the chance of an underlying asset reaching a certain price in a certain timeframe.  A graphical forecast simply puts these calculations into a visual format. That said, it appears the image you offer shows the prediction as it existed in the past along with how the stock has done since.  A disclaimer - The odds of a fair die being rolled to a given number are 1 in 6. It's a fact. With stocks, on the other hand, models try to simulate real life and many factors can't be accounted for.
I'm pretty sure my dog is better at catching balls than I am, also dogs don't think that the alignment of the stars during their birth had any affect on how happy they are.   I read a book called *The Drunkard's Walk* and in it they talk about a experiment where humans and rats were shown a deck of cards that was 2/3 red and 1/3 black. When humans were asked to guess the cards, they guessed black 1/3 of the time, and red 2/3 of the time, while rats picked red every time. Humans were right about 60% of the time, rats were right 2/3 (66.7%) of the time. Rat brains are more correct than human brains. Quick and dirty paper but pretty interesting.. I'm not in Portfolio Management but I probably would have ended up at the modal number as well.   I don't know the subject deeply enough to answer my own question, but is the bias always toward underestimation of variance? Or is that a complex of the way the problem was set up?   Another question I have for those in investment management; Would this impact asset allocation? Why are you reading them? Most undergrad books are good enough for a general understanding of a subject. Graduate books, on the other hand, can sometimes get a bit difficult to read and start to throw in measure theory and other abstract concepts, and they're more rigorous with definitions compared to the intuitive explanations of undergrad books (this can be a good or bad thing). I probably wouldn't bother unless you're looking to take graduate level classes in the subjects.  With that said, Hull's book isn't too advanced and you would probably not have too much difficulty with it, so recommended. Not sure if I've read the corp fin book. Start with real analysis before you jump into stochastic calculus. The two books I have (Oksendal: Stochastic Differential Equations and Steele: Stochastic Calculus and Financial Applications) both begin with formal algebraic set-theory definitions of probability spaces, random variables, functions, and stochastic processes. If you're not comfortable with that sort of language it's hard to gain traction in any further reading. "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." Stocks, Bonds, Bills, and Lottery Tickets notes the work of Fama and French who researched the idea of a small-cap premium along with a value premium that may be useful to note in terms of what has outperformed if one looks from 1926 to present. Slice and dice would also be another article about an approach that over weights the small-cap and value sides of things if you want another resource here. "First, to mention one thing - better analysis calls for analyzing a range of outcomes, not just one; assigning a probability on each, and comparing the expected values. Then moderating the choice based on risk tolerance. But now, just look at the outcome or scenario of 3% and time frame of 2 days. Let's assume your investable capital is exactly $1000 (multiply everything by 5 for $5,000, etc.). A. Buy stock: the value goes to 103; your investment goes to $1030; net return is $30, minus let's say $20 commission (you should compare these between brokers; I use one that charges 9.99 plus a trivial government fee). B. Buy an call option at 100 for $0.40 per share, with an expiration 30 days away (December 23). This is a more complicated. To evaluate this, you need to estimate the movement of the value of a 100 call, $0 in and out of the money, 30 days remaining, to the value of a 100 call, $3 in the money, 28 days remaining. That movement will vary based on the volatility of the underlying stock, an advanced topic; but there are techniques to estimate that, which become simple to use after you get the hang of it. At any rate, let's say that the expected movement of the option price in this scenario is from $0.40 to $3.20. Since you bought 2500 share options for $1000, the gain would be 2500 times 2.8 = 7000. C. Buy an call option at 102 for $0.125 per share, with an expiration 30 days away (December 23). To evaluate this, you need to estimate the movement of the value of a 102 call, $2 out of the money, 30 days remaining, to the value of a 102 call, $1 in the money, 28 days remaining. That movement will vary based on the volatility of the underlying stock, an advanced topic; but there are techniques to estimate that, which become simple to use after you get the hang of it. At any rate, let's say that the expected movement of the option price in this scenario is from $0.125 to $ 1.50. Since you bought 8000 share options for $1000, the gain would be 8000 times 1.375 = 11000. D. Same thing but starting with a 98 call. E. Same thing but starting with a 101 call expiring 60 days out. F., ... Etc. - other option choices. Again, getting the numbers right for the above is an advanced topic, one reason why brokerages warn you that options are risky (if you do your math wrong, you can lose. Even doing that math right, with a bad outcome, loses). Anyway you need to ""score"" as many options as needed to find the optimal point. But back to the first paragraph, you should then run the whole analysis on a 2% gain. Or 5%. Or 5% in 4 days instead of 2 days. Do as many as are fruitful. Assess likelihoods. Then pull the trigger and buy it. Try these techniques in simulation before diving in! Please! One last point, you don't HAVE to understand how to evaluate projected option price movements if you have software that  does that for you. I'll punt on that process, except to mention it. Get the general idea? Edit P.S. I forgot to mention that brokers need love for handling Options too. Check those commission rates in your analysis as well." Crystal Ball is really easy to use. I go to a top 3 b-school and we did some mid-level modeling work with it. The professor taught it to us over like 2 weeks (in addition to some other statistical probability stuff) and once you start working with it, it becomes pretty second nature. "The other answers here do an excellent job of laying out the mathematics of the expected value. Here is a different take on the question of whether lottery tickets are a sensible investment. I used to have the snobbish attitude that many mathematically literate people have towards lotteries: that they are ""a tax on the mathematically illiterate"", and so on. As I've gotten older I've realized that though, yes, it is certainly true that humans are staggeringly bad at estimating risks, that people actually are surprisingly rational when they spend their money. What then is the rational basis for buying lottery tickets, beyond the standard explanation of ""it's cheap entertainment""? Suppose you are a deeply poor person in America. Your substandard education prepared you for a job in manufacturing which no longer exists, you're working several minimum wage jobs just to keep food on the table, and you're one fall off a ladder from medical-expense-induced total financial disaster.  Now suppose you have things that you would like to spend truly enormous amounts of money on, like, say, sending your children to schools with ever-increasing tuitions, or a home in a safe neighbourhood.  Buying lottery tickets is a bad investment, sure. Name another legal investment strategy that has a million-dollar payout that is accessible to the poor in America. Even if you could invest 10% of your minimum-wage salary without missing the electricity bill, that's still not going to add up to a million bucks in your lifetime. Probably not even $100K.  When given a choice between no chance whatsoever at achieving your goals and a cheap chance that is literally a one-in-a-million chance at achieving your goals the rational choice is to take the bad investment option over no investment at all." "Their algorithm may be different (and proprietary), but how I would to it is to assume that daily changes in the stock are distributed normally (meaning the probability distribution is a ""bell curve"" - the green area in your chart).  I would then calculate the average and standard deviation (volatility) of historical returns to determine the center and width of the bell curve (calibrating it to expected returns and implied volaility based on option prices), then use standard formulas for lognormal distributions to calculate the probability of the price exceeding the strike price. So there are many assumptions involved, and in the end it's just a probability, so there's no way to know if it's right or wrong - either the stock will cross the strike or it won't." I don't know too much about the kelly criterion, but going by the other answers it sounds like it could be quite risky depending how you use it. I have been taught the first thing you do in trading is protect your existing capital and any profits you have made, and for this reason I prefer and use Position Sizing (PS). The concept with PS is that you only risk a small % of your capital on every trade, usually not more than 1%, however if you want to be very aggressive then not more than 2%. I use 1% of my capital for every trade. So if you are trading with an account of $40,000 and your risk R on every trade is 1%, then R = $400. As an example, say you decide to buy a stock at $10 and you work out your initial stop to be at $9.50, then our maximum risk R of $400 is divided by the stop distance of $0.50 to get your PS = $400/$0.50 = 800 shares. If the price then drops after your purchase, your maximum loss (subject to no slippage) would be $400.  If the price moves up you would raise your stop until your potential loss becomes smaller and smaller and then becomes a gain once your stop moves above your initial purchase price. The aim is to make your gains be larger than your losses. So if your average loss is kept to 1R or less then you should aim to get your average gains to 2R, 3R or more. This would be considered a good trading system where you will make regular profits even with a win ratio of 50%. "You make several good points.  I'll start with Black-Scholes; the arbitrage argument in Black-Scholes is between the option and a hypothetical and unobserved portfolio with identical cash flows (the replicating portfolio).  We then value the replicating portfolio via an equilibrium model.  When you use BS, there isn't necessarily *any* observed security whose is guaranteed not imply some arbitrage opportunity, because BS makes no reference to observed prices.  A no-arbitrage modification might look like this: ""I observe the prices (and implied volatilities) of some options, and use the implied volatilities to price another option (maybe with a different strike).  Doing so ensures that there is no arbitrage between my price and the market prices implied by my model.""  Realistically, the problem arbitrage-free models are addressing is that our models and assumptions are wrong, even though they're reasonable approximations a lot of the time.  A no-arbitrage model removes some set of obvious deficiencies, but at the cost of not being able to explain why things are priced as they are.  So, for instance, Vasicek won't reproduce the observed term structure, and Hull-White fixes this, but Hull-White doesn't explain where the term structure comes from (i.e., what the term structure *should* be)." Just for clarification, delta and probability of expiring in the money are not the same thing. What the guy meant was that delta is usually a close enough approximation to the probability. One way to think about it is to look at the probabilities and deltas of In the Money, Out of the Money, and At the Money options. In these cases, the delta and probabilities are about the same. In fact if you look at an options chain with delta and probabilities, you can see that they are all about the same. In other words, there is a linear relationship between delta and probability. Here are a couple links to other answers around the web: Hope this answer helps! Something that is missing from the discussion is the actual market for the lottery ticket -- if a market existed for the tickets themselves, that would make this far more obvious, but since there isn't one; buying a single ticket gives different Expected Values, but since the ticket has a defined 'game' instance, a single ticket is a gamble. Playing the lottery in the long run could be part of a high risk investment portfolio. [edited for clarity] "@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)."