Why Gamma is highest for an option that is at the money
Yes, you've got it right.  The change in price is less meaningful as the instrument is further from the price of the underlying.  As the delta moves less, the gamma is much less. Gamma is to delta as acceleration is to speed.  Speed is movement relative to X, and acceleration is rate of change in speed.  Delta is movement relative to S, and gamma is the rate of change in delta. Delta changes quickly when it is around the money, which is another way of saying gamma is higher.  Delta is the change of the option price relative to the change in stock price.  If the strike price is near the market price, then the odds of being in or out of the money could appear to be changing very quickly - even going back and forth repeatedly. Gamma is the rate of change of the delta, so these sudden lurches in pricing are by definition the gamma.  This is to some extent a little mundane and even obvious.  But it's a useful heuristic for analyzing prices and movement, as well as for focusing analyst attention on different pricing aspects. You've got it right.  If delta is constant (zero 'speed' for the change in price) then gamma is zero (zero 'acceleration').
"Some thoughts on your questions in order, Duration: You might want to look at the longest-dated option (often a ""LEAP""), for a couple reasons. One is that transaction costs (spread plus commission, especially spread) are killer on options, so a longer option means fewer transactions, since you don't have to keep rolling the option. Two is that any fundamentals-based views on stocks might tend to require 3-5 years to (relatively) reliably work out, so if you're a fundamental investor, a 3-6 month option isn't great. Over 3-6 months, momentum, short-term news, short squeezes, etc. can often dominate fundamentals in determining the price. One exception is if you just want to hedge a short-term event, such as a pending announcement on drug approval or something, and then you would buy the shortest option that still expires after the event; but options are usually super-expensive when they span an event like this. Strike: Strike price on a long option can be thought of as a tradeoff between the max loss and minimizing ""insurance costs."" That is, if you buy a deeply in-the-money put or call, the time value will be minimal and thus you aren't paying so much for ""insurance,"" but you may have 1/3 or 1/2 of the value of the underlying tied up in the option and subject to loss. If you buy a put or call ""at the money,"" then you might have only say 10% of the value of the underlying tied up in the option and subject to loss, but almost the whole 10% may be time value (insurance cost), so you are losing 10% if the underlying stock price stays flat. I think of the deep in-the-money options as similar to buying stocks on margin (but the ""implied"" interest costs may be less than consumer margin borrowing rates, and for long options you can't get a margin call). The at-the-money options are more like buying insurance, and it's expensive. The commissions and spreads add significant cost, on top of the natural time value cost of the option. The annual costs would generally exceed the long-run average return on a diversified stock fund, which is daunting. Undervalued/overvalued options, pt. 1: First thing is to be sure the options prices on a given underlying make sense at all; there are things that ""should"" hold, for example a synthetic long or short should match up to an actual long or short. These kinds of rules can break, for example on LinkedIn (LNKD) after its IPO, when shorting was not permitted, the synthetic long was quite a bit cheaper than a real long. Usually though this happens because the arbitrage is not practical. For example on LNKD, the shares to short weren't really available, so people doing synthetic shorts with options were driving up the price of the synthetic short and down the price of the synthetic long. If you did actually want to be long the stock, then the synthetic long was a great deal. However, a riskless arbitrage (buy synthetic long, short the stock) was not possible, and that's why the prices were messed up. Another basic relationship that should hold is put-call parity: http://en.wikipedia.org/wiki/Put%E2%80%93call_parity Undervalued/overvalued options, pt. 2: Assuming the relationship to the underlying is sane (synthetic positions equivalent to actual positions) then the valuation of the option could focus on volatility. That is, the time value of the option implies the stock will move a certain amount. If the time value is high and you think the stock won't move much, you might short the option, while if the time value is low and you think the stock will move a lot, you might buy the option. You can get implied volatility from your broker perhaps, or Morningstar.com for example has a bunch of data on option prices and the implied components of the price model. I don't know how useful this really is though. The spreads on options are so wide that making money on predicting volatility better than the market is pretty darn hard. That is, the spread probably exceeds the amount of the mispricing. The price of the underlying is more important to the value of an option than the assumed volatility. How many contracts: Each contract is 100 shares, so you just match that up. If you want to hedge 100 shares, buy one contract. To get the notional value of the underlying multiply by 100. So say you buy a call for $30, and the stock is trading at $100, then you have a call on 100 shares which are currently priced at $10,000 and the option will cost $30*100=3,000. You are leveraged about 3 to 1. (This points to an issue with options for individual investors, which is that one contract is a pretty large notional value relative to most portfolios.)" Convexity refers to vega. Gamma refers to delta. Negative carry refers to time decay. Understanding the BS equation is not needed. What is needed is an understanding of the bell curve.   You seem to understand volatility. 68% of the time an event will fall inside one standard deviation. 16% of the time it will be higher, 16%, lower.  Now, if my $100 stock has a STD of $10, there's a 16% chance it will trade above $110. But if the STD is $5, the chance is 2.3% per the chart below. The higher volatility makes the option more valuable as there's a highr chance of it being 'in the money.' My answer is an over simplification, per your request. It really centers on the probability of your position falling to $0 and your level of comfort if that were to happen.  There are a plethora of situations that could cause an option contract to become worthless.  The application of leverage to a position also increases the risk. Zero risk would be an FDIC insured savings account, high risk would be buying options on margin, and there's a very wide grey area in between.  I agree that the whole process of assigning a risk level is dubious at best.  As you say, it seems using past data could help assign a risk level, look to beta values if you believe in that.  The problem here is the main disclaimer in use is that past performance cannot be relied upon for future gains. As an aside, if the US government files bankruptcy you'll have a whole host of more immediate problems than the value of your t-bills.  At that point dollars would have been a risky investment. Don't know the name but it means you're long with conviction :P  Unlimited gains, maximum loss of 95$  + (8-6) = 97$.  Basically You are long @ 107  - -2 from 105 to 95.  You would have to be ULTRA bullish to initiate this strategy. "So, if an out-of-the-money option (all time value) has a price P (say $3.00), and there are N days... The extrinsic value isn't solely determined by time value as your quote suggests. It's also based on volatility and demand. Here is a quote from http://www.tradingmarkets.com/options/trading-lessons/the-mystery-of-option-extrinsic-value-767484.html distinguishing between extrinsic time value and extrinsic non-time value: The time value of an option is entirely predictable. Time value premium declines at an accelerating rate, with most time decay occurring in the last one to two months before expiration. This occurs on a predictable curve. Intrinsic value is also predictable and easily followed. It is worth one point for every point the option is in the money. For example, a call with a strike of 30 has three points of intrinsic value when the current value of the underlying stock is $33 per share; and a 40 put has two points of intrinsic value when the underlying stock is worth $38. The third type of premium, extrinsic value, increases or decreases when the underlying stock changes and when the distance between current value of stock and strike of the option get closer together. As a symptom of volatility, extrinsic value may be greater for highly volatile underlying stock, and lower for less volatile stocks. Extrinsic value is the only classification of option premium that is unpredictable. The SPYs you point out probably had a volatility component affecting value. This portion is a factor of expectations or uncertainty. So an event expected to conclude prior to expiration, but of unknown outcome can cause theta to be higher than p/n. For example, a drug company is being sued and the outcome of a trial will determine whether that company pays out millions or not. The extrinsic will be higher than p/n prior to the outcome of the trial then drops after.  Of course, the most common situation where this happens is earnings. After the announcement, it's not unusual to see a dramatic drop in the extrinsic portion of options. This is why sometimes a new option trader gets angry when buying calls prior to earnings. When 'surprise' good earnings are announced as hoped, the rise is stock price is largely offset by a fall in extrinsic value giving call holders little or no gain! As for the reverse situation where theta is lower than p/n would expect? Well you can actually have negative theta meaning the extrinsic portion rises over time. (this statement is a little confusing because theta is usually described as negative, but since you describe it as a positive number, negative here means the opposite of what you'd expect).  This is a quote from ""Option Volatility & Pricing"". Keep in mind that they use 'positive' theta to mean the time value increases up over time: Is it ever possible for an option to have a positive theta such that if nothing changes the option will be worth more tomorrow than it is today? When futures options are subject to stock-type settlement, as they currently are in the United States, the carrying cost on a deeply in-the-money option, either a call or a put, can, under some circumstances, be greater than the volatility component. If this happens, and the option is European (no early exercise permitted), it will have a theoretical value less than parity (less than intrinsic value). As expiration approaches, the value of the option will slowly rise to parity. Hence, the option will have a positive theta. Sheldon Natenberg. Option Volatility & Pricing: Advanced Trading Strategies and Techniques (Kindle Locations 1521-1525). Kindle Edition." 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? "Just as a matter of research, apparently there is a way to find high option volumes such as a site here: https://www.barchart.com/options/volume-leaders/stocks However, that information is going to be heavily skewed by ""underlying security that moved a lot more than expected and probably got a lot of positions filled incidentally today"", but I think it is a good place to start building up a list of securities with a lot of option interest. There is also a tab there for ETFs. This will not tell you exactly that a particular stock always has high option volume, but most of the ones that show up there repeatedly and across multiple strike prices will meet your criteria." This is a Short Diagonal Calender Put Spread Generally, you're writing that long dated one at the money, and buying the short dated one out of the money.  The maximum amount that can be made is if the stock breaks out strongly to the upside, and you keep the upfront credit minus whatever small amount it took to buy the April puts back.  You can also make money if it breaks strongly to the downside, but only if the credit when you opened your positions was more than $10. Example:  Now say the stock falls to $500 by the time of that march expiration. You'd make $90/share on the march put, and lose $100/share on the April put (or a little more; but that deep in the money, there won't be much premium on it). That's a loss of $10/share, or -$1000. So: I make a point of pointing this out because in that article I linked to the fact that your upfront credit needs to be greater than the strike spread in order to profit to the downside is not clearly mentioned. Those are the expected yields; they are not guaranteed. This was actually the bread and butter of Graham Newman, mispriced bonds.  Graham's writings in the Buffett recommended edition of Securities Analysis are invaluable to bond valuation. The highest yielder now is a private subsidiary of Société Générale.  A lack of financial statements availability and the fact that this is the US derivatives markets subsidiary are probably the cause of the higher rates. The cost is about a million USD to buy them. The rest will be similar cases, but Graham's approach could find a diamond; however, bonds are big ticket items, so one should expect to pay many hundreds of thousands of USD per trade. One reason might be the 100% margin requirement on long options.  Suppose I want to go long AAPL.   I could get a deep ITM call or buy shares. $12,700 for 100 shares, with it's 25% margin requirement is like around $3200 locked up cash.  Combine with a deep OTM Jan 2017 $70 strike put for $188, would give a $3400 margin requirement to enter the trade. or I could be in the JAN 2017 $70 strike for nearer $5800, but with a 100% margin requirement due to being a long call. So (3400/5800) = 59% increase in margin requirement for Deep ITM calls. Plus long term the shares will pay dividends, while a LEAP CALL does not. You don't see Buying and Selling. You see Bid and Ask. Best Bid--Highest Price someone is willing to pay to buy a stock. Best Ask - Lowest price someone is willing to accept to sell a stock. As for your second question, if you can look up Accumulation/Distribution Algorithm and Iceberg Order, you will get basic idea. Theoretically, it's a question of rate of return. If a desired or acceptable rate of return for market makers' capital is X, and X is determined by the product of margin & turnover then higher turnover means lower margin for a constant X.  Margin, in the case of trading, is the bid/ask spread, and turnover, in the case of trading, is volume. Empirically, it has been noted in the last markets still offering such wide-varying evidence, equity options: http://faculty.baruch.cuny.edu/lwu/890/mayhew_jf2002.pdf On July 20, when you posted this question, AAPL was trading almost at 115.  The market charges an extra premium for buying an option that is in the money (or on the money like this case) over one that is out of the money.  In order for the 130 Call to be worth something the market has to go up 15 points. Otherwise you lose 100% of your premium. On the other hand with the 115 every point that the market goes up means that you recover some of that premium. It is much more likely that you recover part of your premium with the 115 than with the 130.  With the higher probability of losing part of the premium, the sellers are going to be reluctant to write the option unless they receive larger compensation. One should also point out that you make a major assumption in that the high of the day doesn't occur on a gap up in morning trading. It's unlikely that you'd fill at a reasonable price, thereby throwing your strategy into disarray.