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
1.45 and 1.40 are the last trade prices.  The last trade (1.45) for the 27 strike call must have occurred earlier than the last trade (1.40) for the 26 strike call.  These options have low liquidity and don't trade very often.  You have to look at the bid and ask prices to see what people are currently bidding and asking for those options.  As you can see, the premium based on the bids and asks does decrease the further you go out of the money. "As already noted, options contain inherent leverage (a multiplier on the profit or loss). The amount of ""leverage"" is dictated primarily by both the options strike relative to the current share price and the time remaining to expiration. Options are a far more difficult investment than stocks because they require that you are right on both the direction and the timing of the future price movement. With a stock, you could choose to buy and hold forever (Buffett style), and even if you are wrong for 5 years, your unrealized losses can suddenly become realized profits if the shares finally start to rise 6 years later. But with options, the profits and losses become very final very quickly. As a professional options trader, the single best piece of advice I can give to investors dabbling in options for the first time is to only purchase significantly ITM (in-the-money) options, for both calls and puts. Do a web search on ""in-the-money options"" to see what calls or puts qualify.  With ITM options, the leverage is still noticeably better than buying/selling the shares outright, but you have a much less chance of losing all your premium. Also, by being fairly deep in-the-money, you reduce the constant bleed in value as you wait for the expected move to happen (the market moves sideways more than people usually expect). Fairly- to deeply-ITM options are the ones that options market-makers like least to trade in, because they offer neither large nor ""easy"" premiums. And options market-makers make their living by selling options to retail investors and other people that want them like you, so connect the dots. By trading only ITM options until you become quite experienced, you are minimizing your chances of being the average sucker (all else equal). Some amateur options investors believe that similar benefits could be obtained by purchasing long-expiration options (like LEAPS for 1+ years) that are not ITM (like ATM or OTM options). The problem here is that your significant time value is bleeding away slowly every day you wait.  With an ITM option, your intrinsic value is not bleeding out at all. Only the relatively smaller time value of the option is at risk. Thus my recommendation to initially deal only in fairly- to deeply-ITM options with expirations of 1-4 months out, depending on how daring you wish to be with your move timing." If the call is in the money and you believe the reason for the price jump was an overreaction with a pullback on the horizon or you anticipate downward movement for other reasons, I will roll (sometimes for a strike closer to at the money) as long as the trade results in a net credit!   You already have the statistical edge trading covered calls over everyone who purchased stock at the same point in time.  This is because covered calls reduce your cost basis and increase your probability of profit. For people reading this who are not interested in the math behind probability of profit(POP) for covered calls, you should be aware of why POP is higher for covered calls (CC).  With CCs you win when the stock price stays the same, you win when it goes down slightly, you win when the stock goes up.  You have two more ways to win than someone who just buys stock, therefore a higher probability of making a buck! Another option: If your stock is going to be called at a loss, or the strike you want to roll to results in a net debit, or  your cash funds are short of owning 100x shares and you are familiar with the stock, try writing a naked put for the price you want to buy at.  At experation, if the naked put is exercised, your basis is reduced by the premium of the put you sold, and you can write a covered call against the stock you now own.  If it expires worthless you keep the premium.  This is also another way to increase your POP. So, this isn't always the case, but in the example provided the option is most likely in the money or near the money since the delta is nearly 1 - indicating that a $1 move in the underlying results in a $0.92 move in the option - this will happen when the expiration is very far out or the option is in the money. As expiration gets closer, movements in the underlying become more pronounced in the options because the probability of the stock price moving from its current position is lower.    As the probability of the stock price moving goes down, the delta of in the money options approaches 100, eventually reaching 100 at expiration. Another way to word this is that the premium on in the money options shrinks as expiration approaches and the intrinsic value of the option increases as percentage of total value so that movements in the underlying stock price become a greater influence on the option price - hence a greater delta. Again, if the option is out of the money, this is not the case. "You'd need to know the delta and the theta of the option. You can either calculate them yourself using a model like Black-Scholes (assuming you have a market price and can imply a volatility, and know the other factors that go into the model) or, you can see if your broker quotes ""greeks"" as well (mine does). The delta is the sensitivity (rate of change in value) to the underlying stock price, and the theta is the sensitivity to time passing (usually expressed in $/day).  So if your option has a delta of .5 and a theta of -.04, when one day passes and the underlying stock goes up $3, the option will gain roughly $1.50 due to the underlying stock price and lose $0.04 due to time passing." Yes. It seems to me you got it right.  On my site, Stock Options Cafe, my last post was an illustration of a bullish call spread. In this case, I bought a 50 call, and sold the 60 call. This is a debit order as I was paying money, not collecting a new premium. 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. "Below I will try to explain two most common Binomial Option Pricing Models (BOPM) used. First of all, BOPM splits time to expiry into N equal sub-periods and assumes that in each period the underlying security price may rise or fall by a known proportion, so the value of an option in any sub-period is a function of its possible values in the following sub period. Therefore the current value of an option is found by working backwards from expiry date through sub-periods to current time. There is not enough information in the question from your textbook so we may assume that what you are asked to do is to find a value of a call option using just a Single Period BOPM. Here are two ways of doing this: First of all let's summarize your information: Current Share Price (Vs) = $70 Strike or exercise price (X) = $60 Risk-free rate (r) = 5.5% or 0.055 Time to maturity (t) = 12 months Downward movement in share price for the period (d) = $65 / $70 = 0.928571429 Upward movement in share price for the period (u) = 1/d = 1/0.928571429 = 1.076923077 ""u"" can be translated to $ multiplying by Vs => 1.076923077 * $70 = $75.38 which is the maximum probable share price in 12 months time. If you need more clarification here - the minimum and maximum future share prices are calculated from stocks past volatility which is a measure of risk. But because your textbook question does not seem to be asking this - you probably don't have to bother too much about it yet. Intrinsic Value: Just in case someone reading this is unclear - the Value of an option on maturity is the difference between the exercise (strike) price and the value of a share at the time of the option maturity. This is also called an intrinsic value.  Note that American Option can be exercised prior to it's maturity in this case the intrinsic value it simply the diference between strike price and the underlying share price at the time of an exercise. But the Value of an option at period 0 (also called option price) is a price you would normally pay in order to buy it. So, say, with a strike of $60 and Share Price of $70 the intrinsic value is $10, whereas if Share Price was $50 the intrinsic value would be $0. The option price or the value of a call option in both cases would be fixed.  So we also need to find intrinsic option values when price falls to the lowest probable and rises to the maximum probable (Vcd and Vcu respectively) (Vcd) = $65-$60 = $5 (remember if Strike was $70 then Vcd would be $0 because nobody would exercise an option that is out of the money) (Vcu) = $75.38-$60 = $15.38 1. Setting up a hedge ratio:  h = Vs*(u-d)/(Vcu-Vcd) h = 70*(1.076923077-0.928571429)/(15.38-5) = 1 That means we have to write (sell) 1 option for each share purchased in order to hedge the risks. You can make a simple calculation to check this, but I'm not going to go into too much detail here as the equestion is not about hedging. Because this position is risk-free in equilibrium it should pay a risk-free rate (5.5%). Then, the formula to price an option (Vc) using the hedging approach is: (Vs-hVc)(e^(rt))=(Vsu-hVcu) Where (Vc) is the value of the call option, (h) is the hedge ratio, (Vs) - Current Share Price, (Vsu) - highest probable share price, (r) - risk-free rate, (t) - time in years, (Vcu) - value of a call option on maturity at the highest probable share price. Therefore solving for (Vc): (70-1*Vc)(e^(0.055*(12/12))) = (75.38-1*15.38) => (70-Vc)*1.056540615 = 60 => 70-Vc = 60/1.056540615 => Vc = 70 - (60/1.056540615)  Which is similar to the formula given in your textbook, so I must assume that using 1+r would be simply a very close approximation of the formula above. Then it is easy to find that Vc = 13.2108911402 ~ $13.21 2. Risk-neutral valuation: Another way to calculate (Vc) is using a risk-neutral approach. We first introduce a variable (p) which is a risk-neutral probability of an increase in share price. p = (e^(r*t)-d)/(u-d) so in your case: p = (1.056540615-0.928571429)/(1.076923077-0.928571429) = 0.862607107 Therefore using (p) the (Vc) would be equal: Vc = [pVcu+(1-p)Vcd]/(e^(rt)) => Vc = [(0.862607107*15.38)+(0.137392893*5)]/1.056540615 => Vc = 13.2071229185 ~ $13.21 As you can see it is very close to the hedging approach. I hope this answers your questions. Also bear in mind that there is much more to the option pricing than this. The most important topics to cover are: Multi-period BOPM Accounting for Dividends Black-Scholes-Merton Option Pricing Model"