What is the risk-neutral probability?
"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."
To get the probability of hitting a target price you need a little more math and an assumption about the expected return of your stock.  First let's examine the parts of this expression. IV is the implied volatility of the option.  That means it's the volatility of the underlying that is associated with the observed option price.  As a practical matter, volatility is the standard deviation of returns, expressed in annualized terms.  So if the monthly standard deviation is Y, then Y*SQRT(12) is the volatility. From the above you can see that IV*SQRT(DaysToExpire/356) de-annualizes the volatility to get back to a standard deviation.  So you get an estimate of the expected standard deviation of the return between now and expiration. If you multiply this by the stock price, then you get what you have called X, which is the standard deviation of the dollars gained or lost between now and expiration.  Denote the price change by A (so that the standard deviation of A is X). Note that we seek the expression for the probability of hitting a target level, Q, so mathematically we want 1 - Pr( A < Q - StockPrice)  We do 1 minus the probability of being below this threshold because cumulative distribution functions always find the probability of being BELOW a threshold, not above. If you are using excel and assuming a mean of zero for returns, the probability of hitting or exceeding Q at expiration, then, is That's your answer for the probability of exceeding Q. Accuracy is in the eye of the beholder.  You'd have to specify a criterion by which to judge it to know the answer.  I'm sure more sophisticated methods exist that are more unbiased and have less error, but I think it's a fine first approximation. This doesn't make sense in an economic context. For justice to provide sufficient deterrent, it needs to be rational to avoid a crime. If your probability of being caught is comparatively lower, the punishment needs to be higher in order to remain a deterrent.   Basically,   P(get caught) * Punitive Cost &gt; Profit Excellent, very sharp. No it will not be vega neutral exactly! If you think about it, what does a higher vol imply? That the delta of the option is higher than under BS model. Therefore, the vega should also be greater (simplistic explanation but generally accurate). So no, if you trade a 25-delta risky in equal size per leg, the vega will not be neutral. But, in reality, that is a very small portion of your risk. It plays a part, but in general the vanna position dominates by many many multiples.  What do you do that you asked such a question, if you don't mind? "Yes, and the math that tells you when is called the Kelly Criterion. The Kelly Criterion is on its face about how much you should bet on a positive-sum game. Imagine you have a game where you flip a coin, and if heads you are given 3 times your bet, and if tails you lose your bet.  Naively you'd think ""great, I should play, and bet every dollar I have!"" -- after all, it has a 50% average return on investment.  You get back on average 1.5$ for every dollar you bet, so every dollar you don't bet is a 0.5$ loss. But if you do this and you play every day for 10 years, you'll almost always end up bankrupt.  Funny that. On the other hand, if you bet nothing, you are losing out on a great investment.  So under certain assumptions, you neither want to bet everything, nor do you want to bet nothing (assuming you can repeat the bet almost indefinitely). The question then becomes, what percentage of your bankroll should you bet? Kelly Criterion answers this question.  The typical Kelly Criterion case is where we are making a bet with positive returns, not an insurance against loss; but with a bit of mathematical trickery, we can use it to determine how much you should spend on insuring against loss. An ""easy"" way to undertand the Kelly Criterion is that you want to maximize the logarithm of your worth in a given period.  Such a maximization results in the largest long-term value in some sense. Let us give it a try in an insurance case. Suppose you have a 1 million dollar asset.  It has a 1% chance per year of being destroyed by some random event (flood, fire, taxes, pitchforks). You can buy insurance against this for 2% of its value per year.  It even covers pitchforks. On its face this looks like a bad deal.  Your expected loss is only 1%, but the cost to hide the loss is 2%? If this is your only asset, then the loss makes your net worth 0.  The log of zero is negative infinity.  Under Kelly, any insurance (no matter how inefficient) is worth it.  This is a bit of an extreme case, and we'll cover why it doesn't apply even when it seems like it does elsewhere. Now suppose you have 1 million dollars in other assets.  In the insured case, we always end the year with 1.98 million dollars, regardless of if the disaster happens.  In the non-insured case, 99% of the time we have 2 million dollars, and 1% of the time we have 1 million dollars. We want to maximize the expected log value of our worth.  We have log(2 million - 20,000) (the insured case) vs 1% * log(1 million) + 99% * log(2 million). Or 13.7953 vs 14.49.  The Kelly Criterion says insurance is worth it; note that you could ""afford"" to replace your home, but because it makes up so much of your net worth, Kelly says the ""hit it too painful"" and you should just pay for insurance. Now suppose you are worth 1 billion.  We have log(1 billion - 20k) on the insured side, and 1%*log(999 million) + 99% * log(1 billion) on the uninsured side. The logs of each side are 21.42 vs 20.72.  (Note that the base of the logarithm doesn't matter; so long as you use the same base on each side). According to Kelly, we have found a case where insurance isn't worth it. The Kelly Criterion roughly tells you ""if I took this bet every (period of time), would I be on average richer after (many repeats of this bet) than if I didn't take this bet?""  When the answer is ""no"", it implies self-insurance is more efficient than using external insurance.  The answer is going to be sensitive to the profit margin of the insurance product you are buying, and the size of the asset relative to your total wealth. Now, the Kelly Criterion can easily be misapplied.  Being worth financially zero in current assets can easily ignore non-financial assets (like your ability to work, or friends, or whatever).  And it presumes repeat to infinity, and people tend not to live that long. But it is a good starting spot. Note that the option of bankruptcy can easily make insurance not ""worth it"" for people far poorer; this is one of the reasons why banks insist you have insurance on your proprety. You can use Kelly to calculate how much insurance you should purchase at a given profit margin for the insurance company given your net worth and the risk involved.  This can be used in Finance to work out how much you should hedge your bets in an investment as well; in effect, it quantifies how having money makes it easier to make money." Risk and return always go hand by hand.*  Risk is a measure of expected return volatility. The best investment at this stage is a good, easy to understand but thorough book on finance. *Applies to efficient markets only. 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. How about looking into the market price of risk?  Ive always wanted to know if risk is or should be priced the same across markets/asset classes/etc.  Eh? Let me know what you figure out.  Edit:  I just realized you people probably consider it to be the Sharpe ratio. That's not what I meant. I meant in the sense of option pricing. Someone entering a casino with $15 could employ a very simple strategy and have a better-than-90% chance of walking out with $16.  Unfortunately, the person would have a non-trivial chance (about one in 14) of walking out with $0. If after losing $15 the person withdrew $240 from the bank and tried to win $16, the person would have a better-than-90% chance of succeeding and ending up ahead (holding the original $15, plus the additional $240, plus $1) but would have at that point about a one in 14 chance from that point of losing the $240 along with the original $15.  Measured from the starting point, you'd have about a 199 out of 200 chance of gaining $1, and a one out of 200 chance of losing $240. Market-timing bets are like that.  You can arrange things so you have a significant chance of making a small profit, but at the risk of a large downside.  If you haven't firmly decided exactly how much downside you are willing to accept, it's very easy to simultaneously believe you don't have much money at risk, but that you'll be able to win back anything you lose.  The only way you can hope to win back anything you lose is by bringing a lot more money to the table, which will of course greatly increase your downside risk. The probability of making money for the person willing to accept $15 of downside risk to earn $1 is about 93%.  The probability of making money for the person willing to accept $255 worth of risk is about 99.5%.  It's easy to see that there are ways of playing which have a 99.5% chance of winning, and that there are ways of playing that only have a 15:1 downside risk.  Unfortunately, the ways of playing that have the smaller risk don't have anything near a 99.9% chance of winning, and those that have a better chance of winning have a much larger downside risk. "A subsidy is a benefit.  While you're right most of the time it is a crude ""transfer of wealth by the US government"" the formal intent of a subsidy is to assist so as to confer an advantage or mitigate a disadvantage.  If as you say ""discounting the risk premium"" actively by the US government so as to confer, in the words of Ueda and Di Mauro, a ""funding cost advantage [to] SIFIs [of] around 60bp in 2007 and 80bp in 2009"" is not a subsidy, then what is it?" I would say binomial tree, except your last sentence,  &gt; The probabilities of the various outcomes are unknown.  causes issues with that.   You could do a scenario analysis in which you compare values using different %chances of up vs. down. "This is the best tl;dr I could make, [original](https://www.richmondfed.org/-/media/richmondfedorg/publications/research/working_papers/2017/pdf/wp17-12.pdf) reduced by 98%. (I'm a bot) ***** &gt; 7 3 Local Dynamics The local dynamics of the simple search and matching model have been studied by Krause and Lubik.  &gt; In the previous literature, for example Mendes and Mendes and Bhattacharya and Bunzel, the backward dynamics are defined via the map g by rearranging to isolate &amp;theta;t :  1/&amp;xi;  &amp;theta;t = a&amp;theta;t+1  c&amp;theta;t+1 + d = g. 11 Under risk aversion, the dynamics depend on the time path of output yt.  &gt; 4.2 Stability Properties We now study the dynamics of the backward map zt = f. We first establish the properties of the function f. We then study the stability properties of the steady state, where we distinguish between two broad areas of dynamics in the backward map, namely stable and unstable.   ***** [**Extended Summary**](http://np.reddit.com/r/autotldr/comments/788alk/fed_global_dynamics_in_a_search_and_matching/) | [FAQ](http://np.reddit.com/r/autotldr/comments/31b9fm/faq_autotldr_bot/ ""Version 1.65, ~233564 tl;drs so far."") | [Feedback](http://np.reddit.com/message/compose?to=%23autotldr ""PM's and comments are monitored, constructive feedback is welcome."") | *Top* *keywords*: **dynamic**^#1 **model**^#2 **0.1**^#3 **1**^#4 **map**^#5" if I have a asset A with expected return of 100% and risk(measured by standard deviation) 1%, and asset B with expected return of 1% and risk 100%, would it be rational to put asset B into the portfolio ? No, because Modern Portfolio Theory would say that if there is another asset (B2) with the same (or higher) return but less risk (which you already have in asset A), you should invest in that. If those are the only two assets you can choose from, you would invest completely in Asset A. The point of diversification is that, so long as two assets aren't perfectly positively correlated (meaning that if one moves up the other always moves up), then losses in one asset will sometimes be offset by gains in another, reducing the overall risk. Can you give more detail on the problem? If you can model it with a one step binomial tree, then the price is favourable as long as the chance to multiply is P(S^1 = 10 S^0 ) &gt; 0.1.  If you don't know the probabilities, then the usual go-to is to determine what probability space is that would lead to an expected profit (plus an error, and a cushion for risk aversion if the bet is sufficiently large). Dummy example to explain this. Suppose your portfolio contained just two securities; a thirty year US government bond and a Tesla stock. Both of those position are currently valued at $1mm. The Tesla position however is very volatile with its daily volatility being about 5% (based on the standard deviation of its daily return) whole there bond's daily volatility is 1%. Then the Tesla position is 5/6 of your risk while being only 1/2 of the portfolio.   Now if in month the Tesla stock tanks to half is values then. Then it's risk is half as much as before and so it's total contribution to risk has gone down. Consider the black-scholes-merton result.  Notice that the expected value of the bond is its present value, discounted from the expiration date. The same is not applied to the price of the stock. The further in the future you go, the less value the bond carries because it's being discounted into oblivion. Now, looking at d1, as time tends towards infinity, so does d1. N(d1) is a probability.  The higher d1, the higher the probability and vice versa, so as time increases, the probability for S trends to 100% while K is discounted away. Note that the math doesn't yet fully model reality, as extremely long dated options such as the European puts Buffett wrote were traded at ~1/2 the value the model said he should've. He still had to take a GAAP loss: http://www.berkshirehathaway.com/letters/2008ltr.pdf