How exactly is implied volatility assigned to an option's strike price?
"As I understand it, Implied Volatility represents the expected gyrations of an options contract over it's lifetime.  No, it represents that expected movement of the underlying stock, not the option itself.  Yes, the value of the option will move roughly in the same direction the value of the stock, but that's not what IV is measuring. I even tried staring at the math behind the Options pricing model to see if that could make more sense for me but that didn't help. That formula is correct for the Black-Scholes model - and it is not possible (or at least no one has done it yet) to solve for s to create a closed-form equation for implied volatility.  What most systems do to calculate implied volatility is plug in different values of s (standard deviation) until a value for the option is found that matches the quoted market value ($12.00 in this example).  That's why it's called ""implied"" volatility - the value is implied from market prices, not calculated directly. The thing that sticks out to me is that the ""last"" quoted price of $12 is outside of the bid-ask spread of $9.20 to $10.40, which tells me that the underlying stock has dropped significantly since the last actual trade.   If the Implied Vol is calculated based on the last executed trade, then whatever algorithm they used to solve for a volatility that match that price couldn't find a solution, which then choose to show as a 0% volatility.   In reality, the volatility is somewhere between the two neighbors of 56% and 97%, but with such a short time until expiry, there should be very little chance of the stock dropping below $27.50, and the value of the option should be somewhere around its intrinsic value (strike - stock price) of $9.18."
When you buy a call option, you transfer the risk to the owner of the asset. They are risking losing out on gains that may accumulate in addition to the strike price and paid premium. For example, if you buy a $25 call option on stock XYZ for $1 per contract, then any additional gain above $26 per share of XYZ is missed out on by the owner of the stock and solely benefits the option holder. In the first case, if you wish to own the stock, you just exercise the option, and buy it for the strike price. Else, you can sell the option just before expiration, it will be priced very close to its in-the-money value. "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." 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. I frequently do this on NADEX, selling out-of-the-money binary  calls. NADEX is highly illiquid, and the bid/ask is almost always from  the market maker. Out-of-the-money binary calls lose value quickly  (NADEX daily options exist for only ~21 hours). If I place an  above-ask order, it either gets filled quickly (within a few minutes)  due to a spike in the underlying, or not at all. I compensate by  changing my price hourly.  As Joe notes, one of Black-Scholes inputs is volatility, but price  determines (implied) volatility, so this is circular. In other words, you can  treat the bid/ask prices as bid/ask volatilities. This isn't as  far-fetched as it seems:  http://www.cmegroup.com/trading/fx/volatility-quoting-fx-options.html Use the Black-Scholes formula. If you know the current price, an options strike price, time until expiration, and risk-free interest rate, then knowing the market price of the option will tell you what the market's estimation of the volatility is. This does rely on a few assumptions, such as Gaussian random walk, but those are reasonable assumptions for most stocks. You can also get a list of past stock prices, put them in Excel, and ask Excel to calculate the standard deviation with stdev.s(), but that gives you the past volatility. The market's estimate of future volatility is more relevant. "How accurate is Implied Volatility in predicting future moves? How would you measure this?  If the implied volatility says that there's a 1% chance that a stock will double, and it doubles, was it ""right""?  You could also say that it says there's a 99% change that it doesn't double, so was it ""wrong""?   What you could measure is the variance of daily returns over a time period, and see how well that compares to implied volatility, but there's no way to compare IV with the absolute price movement.  If a stock goes up 0.01 each day, then the variance is 0 (the daily returns are the same each day), but over 250 the stock would go up $2.50." 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. "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" "I have traded options, but not professionally. I hadn't come across this terminology, but I expect it counts how far in-the-money, as an ordinal, an option is relative to the distinct strike prices offered for the option series — a series being the combination of underlying symbol, expiration date, and option type (call/put); e.g., all January 2015 XYZ calls, no matter the strike. For instance, if stock XYZ trades today at $11 and the available January 2015 XYZ calls have strike prices of $6, $8, $10, $12, $14, and $16, then I would expect the $10 call could be called one strike in the money, the $8 two strikes in the money, etc. Similarly, the $12 and $14 calls would be one and two strikes out of the money, respectively. However, if tomorrow XYZ moves to $13, then the $10 previously known as one strike in the money would now be two strikes in the money, and the $12 would be the new one strike in the money. Perhaps this terminology arose because many option strategies frequently involve using options that are at- or near-the-money, so the ""one strike in"" (or out) of the money contracts would tend to be those employed frequently? Perhaps it makes it easier for people to describe strategies in a more general sense, without citing specific examples. However, the software developer in me dislikes it, given that the measurement is relative to both the current underlying price (which changes quickly), and the strike prices available in the given option series. Hence, I wouldn't use this terminology myself and I suggest you eschew it, too, in favor of something concrete; e.g. specify your contract strikes in dollar terms — especially when it matters." No, something doesn't seem right here.  There would be virtually no time value to the option 10 minutes before market close on the expiration day. What option is it, and what is the expiration? EDIT: It appears you were looking only at the ASK price. It was $2.05. However, the BID price was only $1.35 and the last transaction was $1.40. So the true value is right about $1.35 to $1.40 at this second.  This is a pitfall that tends to occur when you trade options with almost no volume. For instance, the open interest in that option is only 1 contract (assuming that is yours). So the Bid and the Ask can often be very far apart as they are only being generated by computer traders or the result of outdated, irrelevant human orders. "Options are generally viewed as having two types of value: ""Intrinsic value"" and ""time value.""  The intrinsic value is based on the difference between the strike price on the option and the spot price of the underlying.  The time value is based on the volatility of the underlying and the amount of time left until expiration.  As the days pass toward expiration, the time value generally decreases, and the intrinsic value may move up or down depending on the spot price of the underlying. (In theory, time value could increase at some points if the volatility is also rising.) In your case, it looks like the time value is decreasing faster than the intrinsic value is increasing. This may happen because the volatility is also going down (as suggested in the answer by CQM) or may just happen because the time to expiration is getting shorter at equal volatility. As noted by DumbCoder in a comment to the original question, the Black-Scholes formula will give you more analytical insight into this if you're interested." In the scenario you describe, and really, in any scenario, by the nature of how option contracts work: a higher strike put will necessarily be more expensive than the lower strike put (everything else being equal). the lower strike call will necessarily be more expensive than the higher strike call (everything else being equal). In put options, the buyer has the right to sell stock for the strike price. So the higher the strike price, the more money the buyer of the put option can make by selling the shares of stock at a higher price. In call options, it's the exact opposite: buyers of the call option have the right to buy stock at the strike price. The lower the strike price, the better for the buyer: they have the right to buy stock for a lower amount of money. So it must be worth more. "You're mostly correct, although I think you're missing something essential about no-arbitrage versus an arbitrage argument.  Black-Scholes makes an arbitrage argument, which is that the value of an option should be the same as any portfolio that has identical cash flows, and this is generally a sound argument.  Notice, however, that BS is ultimately an equilibrium model: it tells you the ""correct"" price of an option if the assumptions of BS hold, and doesn't necessarily match observed market prices.  A no-arbitrage condition or model deliberately incorporates observed prices (or yields, or whatever) into the model, so that there cannot be an arbitrage opportunity implied by the model.  This comes up a lot in term structure of interest rate problems, where equilibrium models like Vasicek or Cox-Ingersoll-Ross won't perfectly reproduce the current, observed term structure, and so imply an arbitrage opportunity.  No-arbitrage models like Hull-White specifically match the model's term structure to observed yields/prices, so that there is no arbitrage opportunity between the observed term structure and your model of it.  It's important to note that this will still allow for arbitrage involving bonds that are *not* part of the observed term structure.  As for equivalent martingale measures, you might think about it more generally.  The process involves changing the probability distribution from the actual (which is hard to use for pricing) to a different one that's easier to use but will result in the same prices; this is nearly always a risk-neutral probability.  You can think of equivalent martingale  pricing as asking, ""how would this security behave, and be priced, in a world that is completely risk neutral"", and then making an argument that the prices are in fact equivalent.  EDIT: grammar" 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.