What does an options premium really mean?
Intuitive? I doubt it. Derivatives are not the simplest thing to understand.  The price is either in the money or it isn't. (by the way, exactly 'at the money' is not 'in the money.') An option that's not in the money has time value only. As the price rises, and the option is more and more in the money, the time value drops. We have a $40 stock. It makes sense to me that a $40 strike price is all just a bet the stock will rise, there's no intrinsic value. The option prices at about $4.00 for one year out, with 25% volatility. But the strike of $30 is at $10.68, with $10 in the money and only .68 in time premium. There's a great calculator on line to tinker with.  Volatility is a key component of options trading. Think about it. If a stock rises 5%/yr but rarely goes up any more or less, just steady up, why would you even buy an option that was even 10% out of the money? The only way I can describe this is to look at a bell curve and how there's a 1/6 chance the event will be above one standard deviation. If that standard deviation is small, the chance of hitting the higher strikes is also small.  I wrote an article Betting on Apple at 9 to 2 in which I describe how a pair of option trades was set up so that a 35% rise in Apple stock would return 354% and Apple had two years to reach its target. I offer this as an example of options trading not being theory, but something that many are engaged in. What I found curious about the trade was that Apple's volatility was high enough that a 35% move didn't seem like the 4.5 to 1 risk the market said it was. As of today, Apple needs to rise 13% in the next 10 months for the trade to pay off. (Disclosure - the long time to expiration was both good and bad, two years to recover 35% seemed reasonable, but 2 years could bring anything in the macro sense. Another recession, some worldwide event that would impact Apple's market, etc. The average investor will not have the patience for these long term option trades.)
"First lets understand what convexity means: Convexity -  convexity refers to non-linearities in a financial model. In other   words, if the price of an underlying variable changes, the price of an   output does not change linearly, but depends on the second derivative   (or, loosely speaking, higher-order terms) of the modeling function.   Geometrically, the model is no longer flat but curved, and the degree   of curvature is called the convexity. Okay so for us idiots this means: if the price of ABC (we will call P) is determined by X and Y. Then if X decreases by 5 then the value of P might not necessarily decrease by 5 but instead is also dependent on Y (wtf$%#! is Y?, who cares, its not important for us to know, we can understand what convexity is without knowing the math behind it). So if we chart this the line would look like a curve. (clearly this is an over simplification of the math involved but it gives us an idea) So now in terms of options, convexity is also known as gamma, it will probably be easier to talk about gamma instead of using a confusing word like convexity(gamma is the convexity of options). So lets define Gamma: Gamma - The rate of change for delta with respect to the underlying asset's price. So the gamma of an option indicates how the delta of an option will change relative to a 1 point move in the underlying asset. In other words, the Gamma shows the option delta's sensitivity to market price changes. or Gamma shows how volatile an option is relative to movements in the underlying asset. So the answer is: If we are long gamma (convexity of an option) it simply means we are betting on higher volatility in the underlying asset(in your case the VIX). Really that simple? Well kinda, to fully understand how this works you really need to understand the math behind it. But yes being long gamma means being long volatility. An example of being ""long gamma"" is a ""long straddle"" Side Note: I personally do trade the VIX and it can be very volatile, you can make or lose lots of money very quickly trading VIX options. Some resources: What does it mean to be ""long gamma"" in options trading? Convexity(finance) Long Gamma – How to Make a Long Gamma Position Work for You Delta - Investopedia Straddles & Strangles - further reading if your interested. Carry(investment) - even more reading." The mathematics make it easier to understand why this is the case. Using very bad shorthand, d1 and d2 are inputs into N(), and N() can be expressed as the probability of the expected value or the most probable value which in this case is the discounted expected stock price at expiration. d1 has two σs which is volatility in the numerator and one in the denominator.  Cancelling leaves one on top.  Calculating when it's infinity gives an N() of 1 for S and 0 for K, so the call is worth S and the put PV(K).  At 0 for σ, it's the opposite. More concise is that any mathematical moment be it variance which mostly influences volatility, mean which determines drift, or kurtosis which mostly influences skew are all uncertanties thus costs, so the higher they are, the higher the price of an option. Economically speaking, uncertainties are costs.  Since costs raise prices, and volatility is an uncertainty, volatility raises prices. It should be noted that BS assumes that prices are lognormally distributed.  They are not.  The closest distribution, currently, is the logVariance Gamma distribution. Theta is a variable in options pricing. Theta aka time decay decreases over time. The reason of this is that you have to think of the option as insurance. It is a hedge against actual holdings in an asset. Would you pay more or less for insurance that covers you for a year's time, would you pay more or less for insurance that covers you for a week? The answer is that the market will pay less for insurance that covers them for a lower period of time. This is one of several ways of thinking about it. There is also the probability that the option will be profitable at all, the further out in the future, the more likely it will be profitable and people will pay a premium for it. There are other variables in the black-scholes formula and it is the most widely used options pricing formula. But keep in mind, the geniuses that made up the formula blew up their hedge fund thinking they could sell the options at an inflated premium from their own formula to everyone. Ironic really. "The tax comes when you close the position. If the option expires worthless it's as if you bought it back for $0. There's a short-term capital gain for the difference between your short-sale price and your buyback price on the option. I believe the capital gain is always short-term because short sales are treated as short-term even if you hold them open more than one year. If the option is exercised (calling away your stock) then you add the premium to your sale price on the stock and then compute the capital gain. So in this case you can end up treating the premium as a long-term capital gain. See IRS pub 550 http://www.irs.gov/publications/p550/ch04.html#en_US_2010_publink100010619 Search for ""Writers of puts and calls""" In addition to all these great answers, check out the Wikipedia entry on options. The most important thing to note from their definition is that an option is a derivative (and nothing about any derivative is simple).  Because it is a derivative, increases or decreases in the price of the underlying stock won't automatically result in the same amount of change in the value of the option. "First, your professor should learn proper grammar. Should be, ""Why **do**....""  Second, it looks like you're dealing with synthetic securities. You can create a synthetic T-Bill by doing a combination of long/short calls/puts.  But ignore all that. Just think about this without the technical jargon. We know that the risk free rate is typically what T-Bills are yielding right? And we know that since options are inherently more risky than US government debt, investors will demand a higher interest rate to compensate them for the risk. So, as the risk free rate increases, the value of a call will move the same direction, otherwise investors would stop dealing with call options and would instead buy safer, less-risky ""riskless"" investments.  It's not really an options question, just one of basic finance, risk, understanding of interest rates, etc." "I found the answer after some searching online. It turns out that when talking options, rarely is the current P/L line considered when talking about making adjustments/taking trades off. From Investopedia: http://www.investopedia.com/terms/b/breakevenpoint.asp ""... For options trading, the breakeven point is the market price that a stock must reach for an option buyer to avoid a loss if they exercise the option. For a call buyer, the breakeven point is the strike price plus the premium paid, while breakeven for a put position is the strike price minus the premium paid."" The first sentence sounds more like the current P/L line, but the bold section clearly states the rule I was looking for. In the example posted in my question above, the breakpoints labeled with ""1"" would be the break points I should consider." The entire premise of purchasing a call option is your expectation that the prices will rise. So even though there is a possibility of prices falling, you wouldn't mind paying higher premiums in a volatile market  for a call option because you're bullish and are expecting the volatility to eventually turn out in your favour i.e. prices to rise A derivative is a financial instrument of a special kind, the kind “whose price depends on, or is derived from, another asset”.  This definition is from John Hull, Options, Futures and Other Derivatives – a book definitely worth to own if you are curious about this, you can easily find old copies for a few dollars. The first point is that a derivative is a financial instrument, like credits, or insurances, the second point is that its price depends closely from the price of something else, the mentioned asset. In most cases derivatives can be understood as financial insurances against some risk bound to the asset. In the sequel I give a small list of derivatives and highlight the assets and the risk they can be bound to.  And first, let me point out that the definition is (marginally) wrong because some derivatives depend on things which are not assets, nor do they have a price, like temperature, sunlight, or even your own life in the case of mortgages.  But before going in this list, let me go through the remaining points of your question. What is the basic idea and concept behind a derivative? As already noted, in most cases, a derivative can be understood as a financial insurance compensating from a risk of some sort.  In a classical insurance contract, one party of the contract is an insurance company, but in the broader case of a derivative, that counterparty can be pretty anything: an insurance, a bank, a government, a large company, and most probably market makers. How is it really used, and how does this deviate from the first point? Briefly, how does is it affecting people, and how is it causing problems? An important point with derivatives is that it can be arbitrarily complicated to compute their prices.  Actually what is hidden in the attempt of giving a definition for derivatives, is that they are products whose price Y is a measurable function of one or several random variables X_1, X_2, … X_n on which we can use the theory of arbitrage pricing to get hints on the actual price Y of the asset – this is what the depends on means in technical terms.  In the most favorable case, we obtain an easy formula linking Y to the X_is which tells us what is the price of our financial instrument. But in practice, it can be very difficult, if at all possible, to determine a price for derivatives. This has two implications: Persons possessing sophisticated techniques to compute the price of derivatives have a strategic advantage on derivatives market, in comparison to less advanced actors on the market. Organisation owning assets they cannot price cannot compute their bilan anymore, so that they cannot know for sure their financial situation. They are somehow playing roulette. But wait, if derivatives are insurances they should help to mitigate some financial risk, which precisely means that they should help their owners to more accurately see their financial situation! How is this not a contradiction? Some persons with sophisticated techniques to compute the price of derivatives are actually selling complicated derivatives to less knowledgeable persons.  For instance, many communes in France and Germany have contracted credits whose reimbursements have a fixed interest part, like in a classical credit, and a variable interest part whose rate is computed against a complicated formula involving the value of the Swiss frank at each quarter starting from the inception of the credit.  (So, for a 25 years running credit of theis type, the price Y of the credit at its inception depends on 100 Xs, which are the uncertain prices for the Swiss frank each quarter of the 25 next years.)  Some of these communes can be quite small, with 5.000 inhabitants, and needless to say, do not have the required expertise to analyse the risks bound to such instruments, which in that special case led the court call the credit a swindling and to cancel the credit.  But what chain of events leads a 5.000 inhabitants city in France to own a credit whose reimbursements depends on the Swiss frank?  After the credit crunch in 2007 and the fall of Lehman Brothers in 2008, it has begun to be very hard to organise funding, which basically means to conclude credits running long in time on large amounts of money.  So, the municipality needs a 25 years credit of 10.000.000 EUROS and goes to its communal bank.  The communal bank has hundreds or thousands of municipalities looking for credits and needs itself a financing.  So the communal bank goes to one of the five largest financial institutions in the world, which insists on selling a huge credit whose reimbursements have a variable part depending on hundred of values the Swiss frank will have in the 25 next years. Since the the big bank has better computation techniques than the small bank it makes a big profit.  Since the small bank has no idea, how to compute the correct price of the credit it bought, it cuts this in pieces and sell it in the same form to the various communes it works with.  If we were to attribute this kind of intentions to the largest five banks, we could ask about the possibility that they designed the credit to take advantage of the primitive evaluation methods of the small bank. We could also ask if they organised a cartel to force communal banks to buy their bermudean snowballs. And we could also ask, if they are so influent that they eventually can manipulate the Swiss frank to secure an even higher profit.  But I will not go into this.  To the best of my understanding, the subprime crisis is a play along the same plot, with different actors, but I know this latter subject only by what I could read in French newspapers. So much for the “How is it causing problems?” part. What is some of the terminology in relation to derivatives (and   there meanings of course)? Answering this question is basically the purpose of the 7 first chapters of the book by Hull, along with deriving some important mathematical principles.  And I will not copy these seven chapters here! How would someone get started dealing in derivatives (I'm playing a     realistic stock market simulation, so it doesn't matter if your     answer to this costs me money)? If you ask the question, I understand that you are not a professional, so that your are actually trying to become the one that has money and zero knowledge in the play I outlined above. I would recommand not doing this.  That said, if you have a good mathematical background and can program well, once you are confindent with the books of Hull and Joshi, you can have fun implementing various market models and implementing trading strategies. Once you are confident with this, you can also read the articles on quantitative finance on arXiv.org. And once you are done with this, you can decide for yourself if you want to play the same market as the guys writing these articles. (And yes, even for the simplest options, they have better models than you have and will systematically outperform you in the long run, even if some random successes will give you the feeling that you do well and could do better.) (indeed, I've made it a personal goal to somehow lose every last cent of my money) You know your weapons! :) Two parties agree today on a price for one to deliver a commodity to the other at some future instant.  This is a classical future contract, it can be modified in every imaginable way, usually by embedding options.  For instance one party could have the option to choose between different delivery points or delivery days. Two parties write today a contract allowing the one party to buy at some future time a commodity to the the second party.  The price is written today, as part of the contract.  (There is the corresponding option entitling the owner to sell something.)  Unlike the future contract, only one party can be obliged to do something, the other jas a right but no obligation. If you buy and option, your are buying some sort of insurance against a change of price on some asset. This is the most familiar to anybody.  Credits can come in many different flavours, especially the formula to compute interests, or also embed options.  Common options are early settlement options or restructuration options. While this is not completely inutitive, the credit works like an insurance.  This is most easily understood from the side of the organisation lending the money, that speculates that the ratio of creanciers going bankrupt will be low enough for her to make profit, just like a fire insurance company speculates that the ratio of fire accidents will be low enough for her to make a profit. This is like a mortgage on a financial institution.  Two parties agree that one will recive an upfront today and give a compensation to the second one if some third party defaults. Here this is an explicit insurance against the unfortuante event, where a creancier goes bankrupt. One finds here more or less standard options on electricity. But electricity have delicious particularities as it can practically not be stored, and fallout is also (usually) avoided. As for classical options, these are insurances against price moves. A swap is like two complementary credits on the same amount of money, so that it ends up in the two parties not actually exchanging the credit nominal and only paying interest one to the other — which makes only sense if these interests are computed with different formulas. Typical example are fixed rate vs. EURIBOR on some given maturity, which we interpret as an insurance against fluctuations of the EURIBOR, or a fixed rate vs. the exchange ratio between two currencies, which we interpret as an insurance against the two currencies decorrelating. Swaps are the richest and the most generic category of financial derivatives.  The off-the-counter market features very imaginative, very customised insurance products. The most basic form is the insurance against drought, but you can image different dangers, and once you have it you can put it in options, in a swap, etc.  For instance, a restaurant with a terrasse could enter in a weather insurance, paying each year a fixed amount of money and becoming in return an amount of money based on the amount of rainy day in a year. Actually, this list is virtually without limits! The CBOE site, as well as some other sites and trading platforms, will show the bid/ask and statistics for that option at each individual options exchange, in addition to statistics and the best bid/offer across all exchanges. cboe.com: Delayed Quote Help lists what the single-letter codes mean. A is for the AMEX options exchange, B is for BOX, X is for PHLX, etc. Options reflect expectations about the underlying asset, and options are commonly priced using the Black-Scholes model:  N(d1) and N(d2) are probability functions, S is the spot (current) price of the asset, K is the strike price, r is the risk free rate, and T-t represents time to maturity.  Without getting into the mathematics, it suffices to say that higher volatility or expectation of volatility increases the perceived riskiness of the asset, so call options are priced lower  and put options are priced higher. Think about it intuitively. If the stock is more likely to go downwards, then there's an increased chance that the call option expires worthless, so call options must be priced lower to accommodate the relative change in expected value of the option.  Puts are priced similarly, but they move inversely with respect to call option prices due to Put-Call parity. So if call option prices are falling, then put option prices are rising (Note, however, that call prices falling does not cause put prices to rise. The inverse relationship exists because of changes in the underlying factors and how pricing works.) So the option action signifies that the market believes the stock is headed lower (in the given time frame). That does not mean it will go lower, and option traders assume risk whenever they take a particular position. Bottom line: gotta do your own homework! Best of luck. 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. I agree that high volatility just means the underlying stock price fluctuates more, and it does not imply if the stock is going up or down. But a high volatility in the price of an underlying also means that there is a higher chance that the underlying price could reach extreme prices (albeit in either direction). However, if you purchased a call option then if the underlying price reached an extremely high value, then you will be richly rewarded. But if the underlying price reached an extremely low value, you won't lose any more than the initial premium that you paid. There is no additional risk on your side, it's capped to the premium that you paid for the call option. It's this asymmetric outcome (Heads - I win, Tails - I don't lose) combined with high volatility that means that call options will increase in value when the underlying price becomes more volatile. If the optionality wasn't there then the price wouldn't be related to the volatility of the underlying. But that would be called a Future or a Forward :-) "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" "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."