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."
For the type of market neutrality you desire, free from crash risk, it's best to hedge the shares with covered calls when implied volatility is expensive and puts when implied volatility is cheap with the nearest at the money expirations.  A put only strategy can be very expensive and should only be used with the longest term options available since they can cost many tens of % per year. Securities become almost perfectly correlated during a crash; therefore, market crash risk of one security is essentially equal to the market crash risk, so hedging the security itself makes a position market neutral for crash risk. This strategy will have intermittent opportunity cost risk in the form of slower returns during market expansion to pay for smaller losses during a crash; however, the expected long run return hedged this way should be greater than the underlying's expected long run return with less volatility. "What exactly is ""Call Protection""? Is ""NO"" normal for a CD?   A ""callable"" cd (or one without protection can be revoked before maturity by the issuer. For example, if interest rates drop they might recall the CD since they can borrow money at a lower rate. A callable CD should offer a better return in exchange for this. Is ""first coupon date"" the first time it actually pays interest?   Yes. Given that the first coupon date is in 6 months, and it's semiannual, is it fair to assume that this pays every six months?   Yes Given that the maturity date is 9/29/2025, this is really a 15-year CD, right?   It depends. CDs can be resold on the market. So it is AT LEAST a 15 year CD. It could also be a 20 year CD originally issued in 2005. What does ""Price (Ask)"" mean? It says 100 - does that mean you buy in chunks of $100?   Yes. More accurately, they are $100 per CD.  What is ""Yield to Worst (Ask)""? Is this the worst yield that this CD will return?   The lowest potential yield that can be received on a bond without the issuer actually defaulting. 3.25% is the best yield I've seen from something as stable as a CD. Would it be silly, though, to lock money up in an investment like this for 15 years?     There is no way to really say without a crystal ball. If you aren't willing to accept more risk and think that interest rates will remain at these historic lows for a long while, then it is probably a good deal. If you think interest rates are due to go up substantially, then it is probably smarter to ladder your investments in shorter term CDs. In investment circles this is known as ""Interest Rate Risk""" "Alternatively you could exercise 12000 shares for $36000 and immediately sell 7200 shares to recover your exercise price. Then you use the remaining 4800 share to pay the exercise price of the remaining 8000 options. Both scenarios are equivalent but may have different fees associated, so it's worth checking the fine print.  Tax wise: The above example is ""cash neutral before taxes"". The taxes associated with these transaction are substantial, so it's highly recommended to talk with a tax adviser. ""cash neutral after taxes"" depends highly on your specific tax situation." The answer is that the trader is hoping to profit from a potential rise in Implied volatility. He is isolating his exposure to IV only and mitigating his risk to the directional move of the underlying by hedging with the underlying. Basically, his delta is neutral. His gamma is positive and a potential source of profit, and his theta is negative which is a potential source of loss. He hopes that the profits from long gamma will overcome the loss from the short theta. he achieves this by actively gamma scalping to remain delta neutral over the life of the option. The relationship is not linear, and depends on a lot of factors. The term you're looking for is efficient frontier, the optimal rate of return for a given level of risk. The goal is to be on the efficient frontier, meaning that for the given level of risk, you're receiving the greatest possible rate of return (reward). http://www.investopedia.com/terms/e/efficientfrontier.asp Ditto Bill and I upvoted his answer. But let me add a bit. If everyone knew exactly what the risk was for every investment, then prices would be bid up or down until every stock (or bond or derivative or whatever) was valued at exactly risk times potential profit. (Or more precisely, integral of risk times potential profit.) If company A was 100% guaranteed to make $1 million profit this year, while company B had 50% change to make a $2 million profit and 50% to make $0, and every investor in the world knew that, then I'd expect the total price of all shares of the two stocks to stabilize at the same value. The catch to that, though, is that no one really knows the risk. The risk isn't like, we're going to roll a die and if it comes up even the company makes $1 million and if it comes up odd the company makes $0, so we could calculate the exact probability. The risk comes from lack of information. Will consumers want to buy this new product? How many? What are they willing to pay? How capable is the new CEO? Etc. It's very hard to calculate probabilities on these things. How can you precisely calculate the probability that unforeseen events will occur? So in real life prices are muddled. The risk/reward ratio should be roughly sort of approximately linear, but that's about the most one can say. &gt;Now introduce chance by randomly assigning each participant a “luck” score. That score, however, can play only a tiny role in the ultimate outcome, just 2 percent compared with 98 percent allotted to skill. This minor role for chance is enough to tilt the contest away from the top-skilled people. In a simulation with 1,000 participants, the person with the top skill score prevailed only 22 percent of the time. The more competition there is, the hardest it is for skill alone to win out. With 100,000 participants, the most skilled person wins just 6 percent of the time. http://www.bloomberg.om/news/articles/2016-09-01/why-luck-plays-a-big-role-in-making-you-rich    https://www.americanprogress.org/issues/economy/news/2006/04/26/1917/understanding-mobility-in-america/ &gt;Children from low-income families have only a 1 percent chance of reaching the top 5 percent of the income distribution, versus children of the rich who have about a 22 percent chance.  -- Children born to the middle quintile of parental family income ($42,000 to $54,300) had about the same chance of ending up in a lower quintile than their parents (39.5 percent) as they did of moving to a higher quintile (36.5 percent). Their chances of attaining the top five percentiles of the income distribution were just 1.8 percent.  Further, children born into the bottom quintile of income have a 42% chance of remaining there and only a 6% chance of making it to the top quintile. For those born  in the top quintile it's exactly the opposite: 42 percent remain in the top quintile as adults, and only 6 percent fall down to the lowest income bracket.   I'd imagine folks like you don't like to read so  https://www.mixcloud.com/rationallyspeaking/rationally-speaking-165-robert-frank-on-success-and-luck/ The other example I'd offer is the case for diversification. If one buys 10 well chosen stocks, i.e. stocks spread across different industries so their correlation to one another is low, they will have lower risk than each of the 10 folk who own one of those stocks per person. Same stocks, but lower risk when combined.