Does the premium of an option of a certain strike price increase at a slower rate from OTM to ITM as gamma affects delta?
If we assume constant volatility, gamma increases as the stock gets closer to the strike price. Thus, delta is increasing at a faster rate as the stock reaches closer to ITM because gamma is the derivative of delta. As the stock gets deeper ITM, the gamma will slow down as delta reaches 1 or -1 (depends if a call or a put). Thus, the value of the option will change depending upon the level of the delta. I am ignoring volatility and time for this description.  See this diagram from Investopedia: Gamma
"The CBOE states, in an investor's guide to Interest Rate Options: The Options’ Underlying Values   Underlying values for the option contracts are   10 times the underlying Treasury yields (rates)—   13-week T-bill yield (for IRX), 5-year T-note   yield (for FVX), 10-year T-note yield (for TNX)   and 30-year T-bond yield (for TYX). The Yahoo! rate listed is the actual Treasury yield; the Google Finance and CBOE rates reflect the 10 times value.  I don't think there's a specific advantage to ""being contrary"", more likely it's a mistake, or just different." "One thing I would like to clear up here is that Black Scholes is just a model that makes some assumptions about the dynamics of the underlying + a few other things and with some rather complicated math, out pops the Black Scholes formula. Black Scholes gives you the ""real"" price under the assumptions of the model. Your definition of what a ""real"" price entails will depend on what assumptions you make. With that being said, Black Scholes is popular for pricing European options because of the simplicity and speed of using an analytic formula as opposed to having a more complex model that can only be evaluated using a numerical method, as DumbCoder mentioned (should note that, for many other types of derivative contracts, e.g. American or Bermudan style exercise, the Black Scholes analytic formula is not appropriate). The other important thing to note here is that the market does not necessarily need to agree with the assumptions made in the Black Scholes model (and they most certainly do not) to use it. If you look at implied vols for a set of options which have the same expiration but differing strike prices, you may find that the implied vols for each contract differ and this information is telling you to what degree the traders in the market for those contracts disagree with the lognormal distribution assumption made by Black Scholes. Implied vol is generally the thing to look at when determining cheapness/expensiveness of an option contract. With all that being said, what I'm assuming you are interested in is either called a ""delta-gamma approximation"" or more generally ""Greek/sensitivities based profit and loss attribution"" (in case you wanted to Google some more about it). Here is an example that is relevant to your question. Let's say we had the following European call contract: Popping this in to BS formula gives you a premium of $4.01, delta of 0.3891 and gamma of 0.0217. Let's say you bought it, and the price of the stock immediately moves to 55 and nothing else changes, re-evaluating with the BS formula gives ~6.23. Whereas using a delta-gamma approximation gives: The actual math doesn't work out exactly and that is due to the fact that there are higher order Greeks than gamma but as you can see here clearly they do not have much of an impact considering a 10% move in the underlying is almost entirely explained by delta and gamma." "people are willing to pay higher premiums for options when stocks go down. Obviously the time value and intrinsic value and interests rates of the option doesn't change because of this so the miscalculation remainder is priced into the implied volatility part of the formula.  Basically, anything that suggests the stock price will get volatile (sharp moves in either direction) will increase the implied volatility of the option. For instance, around earnings reports, the IV in both calls and puts in the nearest expiration dates are very high. When stocks go down sharply, the volatility is high because some people are buying puts for protection and others are buying calls because they think there will be a rebound move in the other direction. People (the ""sleep-at-night"" investors, not the derivatives traders ;) )  tend to be calm when stocks are going up, and fearful when they are going down. The psychology is important to understand and observe and profit from, not to quantitatively prove. The first paragraph should be your qualitative answer" "Is there ever any ambiguity on what that that exact strike is in delta space, or does everyone back it out from the pricing model the same way?  I ask, because in my product nearly everyone runs a heavier delta to the put (the severity of that varies). So on trades that are ""tied up"", everyone participating on it can have slightly different deltas that they are modeling" "Number 2 cannot occur. You can buy the call back and sell the stock, but the broker won't force that #2 choice. To trade options, you must have a margin account. No matter how high the stock goes, once ""in the money"" the option isn't going to rise faster, so your margin % is not an issue.  And your example is a bit troublesome to me. Why would a $120 strike call spike to $22 with only a month left? You've made the full $20 on the stock rise and given up any gain after that. That's all. The call owner may exercise at any time. Edit: @jaydles is right, there are circumstances where an option price can increase faster than the stock price. Options pricing generally follows the Black-Scholes model. Since the OP gave us the current stock price, option strike price, and time to expiration, and we know the risk free rate is <1%, you can use the calculator to change volatility. The number two scenario won't occur, however, because a covered call has no risk to the broker, they won't force you to buy the option back, and the option buyer has no motive to exercise it as the entire option value is time premium." I don't think user4358's explanation is correct.  A trailing LIT Sell Order adjusts downwards, i.e. if you place the order with an Aux price (in TWS it's trigger price) of 105.00 and a trailing amount of 6.00 then, assuming the ask is 100.00, TWS will add the trailing amount to the ask price and if it's less than the trigger price it will adjust. So in my example, if the market (ask) goes straight up to 105.00, nothing will be adjusted, the trigger is touched and the limit order will be placed (see below). If on the the other hand the market goes down to 99.00 then trlng amt + ask is 105.00, if it goes further down to 98.00 then the trigger price will be adjusted to 104.00 (because it's less than the current trigger), and so on. For the LIT part you have either an absolute limit price you can enter, or you have an offset limit which will be subtracted from the trigger price, in which case it is adjusted as well. So back to my example, the trigger is now 104.00 and the limit offset is say 1.00, so my limit order would be placed at 103.00 if the ask ever touches 104.00, and that in turn is only visible if the bid touches 103.00 (because it's limit-if-touched). For a buy just use the same explanation with some swapped roles, the trigger price adjust upwards when the trailing amount plus bid is larger than the current trigger, and the limit offset will be added to the trigger price. Edit Also quite succinct and worth having a look at: http://www.interactivebrokers.com/en/trading/orders/trailingLimitTouched.php Guesswork, highly subjective As for why this might be good, well, you have to believe in momentum strategies, i.e. a market that goes down, will continue to go down, if you believe that and you believe in mean reversion as well, then a trailing limit order can assist you in not buying/selling impulsively, but closer to the mean.  I've never used it that way though. What I have done, even just now to get the explanation right, is to place trailing buy and sell orders simultaneously.  You will find that you can just go in with coarse estimates and because the adjustments will go towards each other, you will end up with a narrowing band of trigger prices (as opposed to trailing stop orders which will give you a widening band of trigger prices).  If you believe in overshooting and equilibria then this can be one easy way to profit from it.  I've just sold EURUSD for 1.26420 and bought it back at 1.26380 with a trailing amount of 5pips and a limit offset of 2pips within the time of writing this. At the higher level - yes. The value of an OTM (out of the money) option is pure time value. It's certainly possible that when the stock price gets close to that strike, the value of that option may very well offer you a chance to sell at a profit. Look at any OTM strike bid/ask and see if you can find the contract low for that option. Most will show that there was an opportunity to buy it lower at some point in the past.  Your trade. Ask is meaningless when you own an option. A thinly traded one can be bid $0 /ask $0.50. What is the bid on yours? The CBOE had a great article on this. I will search for it and edit. The normal dividends are not adjusted. Which is why you see early exercise of just out of the money options sometimes. To get that dividend. A special dividend, say a $50 stock with $1/yr dividend but now has a $3 one time dividend would likely result in an option strike adjustment. 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. NL7 is right and his B-S reference, a good one.   Time decay happens to occur in a way that 2X the time gives an option 1.414X (the square root of 2) times the value, so half the time means about .707 of the value.  This valuation model should help the trader decide on exactly how far out to go for a given trade. No, it means what it says. Prices change, hence price of the derivative can go down even if the price of the underlying doesn't change (e.g. theta decay in options). Well, yes -- you've implicitly made many assumptions (such as that the embedded option has longer maturity). The important thing to consider is when this option pays out; the premium will obviously be adjusted.  For a concrete example, consider an equity option-on-an-option. The outer option has strike 110, the inner option has strike 100 (spot = forward = 100). Then the inner strike pays out when spot_T &gt; 100, but the outer option has zero value there; the overall option only pays out if spot_T &gt; 110, reducing the structure to a call option with strike 110. "Yes, long calls, and that's a good point. Let's see... if I bought one contract at the Bid price above...          $97.13 at expiry of $96.43 option = out of the money =- option price(x100)          = $113 loss.     $97.13 at expiry of $97.00 option = out of the money =- option price(x100)          = $77 loss.     $97.13 at expiry of $97.14 option = in the money by 1-cent=$1/contract profit - option price(x100) = $1-$58          = $57 loss   The higher strike prices have much lower losses if they expire with the  underlying stock at- or near-the-money. So, they carry ""gentler"" downside potential, and are priced much higher to reflect that ""controlled"" risk potential.  That makes sense. Thanks." "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." "There is a white paper on ""The weekend effect of equity options"" it is a good paper and shows that (for the most part) option values do lose money from Friday to Monday. Which makes sense because it is getting closer to expiration. Of course this not something that can be counted on 100%.  If there is some bad news and the stock opens down on a Monday the puts would have increased and the calls decreased in value. Article Summary (from the authors): ""We find that returns on options on individual equities display markedly lower returns over weekends (Friday close to Monday close) relative to any other day of the week. These patterns are observed both in unhedged and delta-hedged positions, indicating that the effect is not the result of a weekend effect in the underlying securities. We find even stronger weekend effects in implied volatilities, but only after an adjustment to quote implied volatilities in terms of trading days rather than calendar days.""  ""Our results hold for puts and calls over a wide range of maturities and strike prices, for both equally weighted portfolios and for portfolios weighted by the market value of open interest, and also for samples that include only the most liquid options in the market. We find no evidence of a weekly seasonal in bid-ask spreads, trading volume, or open interest that could drive the effect. We also find little evidence that weekend returns are driven by higher levels of risk over the weekend.  ""The effect is particularly strong over expiration weekends, and it is also present to a lesser degree over mid-week holidays. Finally, the effect is stronger when the TED spread and market volatility are high, which we interpret as providing support for a limits to arbitrage explanation for the persistence of the effect."" - Christopher S. Jones & Joshua Shemes You can read more about this at this link for Memphis.edu"