Option Theta: What conditions are needed for Theta > P/N, where P = option price, and N = days to expiration?
"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."
The time value decay is theoretically constant.  In reality, it is driven by supply and demand, just like everything else in the market.   For instance, if a big earnings announcement is coming out after the close for the day, you may see little or no time decay in the price of the options during the day before. Also, while in theory options have a set value as related to the trading price of the underlying security, that does not mean there will always be a buyer willing to pay a premium as they come close to expiration (in the last few minutes). You can't forget to account for the transaction fees associated with buying the options, or the risk factor involved.   It is rare, but there are times I've actually had to sell in the money calls at a penny or two LESS than they're actually worth at the time just to unload them in the last few minutes before the market closed on expiration day. To Chris' comment, find out if the assignment commission is the same as the commission for an executed trade. If that does affect the profit, just let it expire.  I've had spreads (buy a call, sell a higher strike call, same dates) so deep in the money, I just made sense to let both exercise at expiration. Don't panic if all legs ofthe trade don't show until Sunday or even Monday morning. Volume @ 0 doesn't mean that there are no buyers and sellers, it just means that there hasn't been any trades done yet. What you need to look for are the bids and offers (for selling and buying, respectively). For further expiration and NTM or IT options there will almost always be a bid and an offer (but it may be very wide). Now, in case where there is 0 bid (no one is willing to buy), you may still have a chance if the option has some value in it. For that - you need your broker to try to shop it to market making firms. Now, depending on who your broker is, this may or may not be possible. Alternatively, if you have DMA (direct market access) to the options exchanges, you can try to put in an offer of your own and wait for someone to execute against you, however do not expect to be traded with unless your price is out of line with the cost. However, in wide markets, you can try Lampost options (they may give you price improvement) or try to offer very close to the bid. You may save yourself a penny or two and perhaps get a rebate if you are using BATSO or NASDAQO markets (if you have DMA and pass-through exchange fees). As other uses have pointed out, your example is unusual in that is does not include any time value or volatility value in the quoted premiums, the premiums you quote are only intrinsic values.  For well in-the-money options, the intrinsic value will certainly be the vast majority of the premium, but not the sole component. Having said that, the answer would clearly be that the buyer should buy the $40 call at a premium of $10.  The reason is that the buyer will pay less for the option and therefore risk less money, or buy more options for the same amount of money.  Since the buyer is assuming that the price will rise, the return that will be realised will be the same in gross terms, but higher in relative terms for the buyer of the $40 call.   For example, if the underlying price goes to $60, then the buyer of the $40 call would (potentially) double their money when the premium goes from $10 to $20, while the buyer of the $30 call would realise a (potential) 50% profit when the premium goes from $20 to $30. Considering the situation beyond your scenario, things are more difficult if the bet goes wrong.  If the underlying prices expires at under $40, then the buyer of the $40 call will be better off in gross terms but may be worse off in relative terms (if it expires above $30).  If the underlying price expires between $40 and $50, then the buy of the $30 will be better off in relative term, having lost a smaller percentage of their money. yes, you are double counting. Your profit is between ($7.25 and $8) OR ($7.75 and $8.50). in other words, you bought the stock at $7.75 and sold at $8.00 and made $0.50 on top. Profit =  $8.00-$7.75+$0.50 (of course all this assumes that the stock is at or above $8.00 when the option expires. If it's below, then your profit = market price - $7.75 + $0.50 by the way the statement  won't call me away until the stock reaches $8.50 is wrong. They already paid $0.50 for the right to buy the stock at $8.00. If the stock is $8.01 on the day of expiration your options will be executed(automatically i believe). I understand that ITM have little time value, so they will have small time decay(theta), but why OTM has a lesser theta than ATM? The Time value represents uncertainty.  That uncertainty decreases the farther away from ATM you get (in either direction).  At-the-money, there is roughly a 50% chance that the option expires worthless. As you get deeper in-the-money, the change that is expires worthless decreases, so there is less uncertainty (there is more certainty that the option will pay off).  As you go deeper OTM, the probability that the option expires worthless increases, so there is also less uncertainty. At the TTM decreases, the uncertainty (theta) decreases as well, since there is less time for the option to cross the strike from either direction.  Similarly, as volatility decreases, theta decreases, since low-volatility stocks have a less change of crossing the strike. But if underlying goes to 103 at expiration, both the call and the put   expire worthless If the stock closes at 103 on expiration, the 105 put is worth $2, not worthless. "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" In the money puts and calls are subject to automatic execution at expiration. Each broker has its own rules and process for this.  For example, I am long a put. The strike is $100. The stock trades at the close, that final friday for $90. I am out to lunch that day. Figuratively, of course. I wake up Saturday and am short 100 shares. I can only be short in a margin account. And similarly, if I own calls, I either need the full value of the stock (i.e. 100*strike price) or a margin account.  I am going to repeat the key point. Each broker has its own process for auto execution. But, yes, you really don't want a deep in the money option to expire with no transaction.  On the flip side, you don't want to wake up Monday to find they were bought out by Apple for $150. "Consider the futures market. Traders buy and sell gold futures, but very few contracts, relatively speaking, result in delivery. The contracts are sold, and ""Open interest"" dwindles to near zero most months as the final date approaches. The seller buys back his short position, the buyer sells off his longs.  When I own a call, and am 'winning,' say the option that cost me $1 is now worth $2, I'd rather sell that option for even $1.95 than to buy 100 shares of a $148 stock. The punchline is that very few option buyers actually hope to own the stock in the end. Just like the futures, open interest falls as expiration approaches." For futures, you are obligated to puchase the security at $x when the contract expires.  For an option, you have the right or option to do so if it's favorable to you. Also, this would be a sick way to predict the weather. What if there were options on rain that had week long maturities? One could back out from the price and different maturities the probability of it raining on a certain day. There is no reason to roll an option if the current market value is lower than the strike sold. Out-of-the-money strikes (as is the $12 strike) are all time value which is decaying constantly and that is to our advantage. If share price remains below the strike, the option will expire worthless, you will still have your shares and free to sell another option the Monday after expiration Friday. If share price is > $12 on expiration Friday and you want to keep those shares, you can roll out or out-and-up depending on your outlook for the stock. Good luck, Alan "The other two answers seem basically correct, but I wanted to add on thing:  While you can exercise an ""American style"" option at any time, it's almost never smart to do so before expiration.  In your example, when the underlying stock reaches $110, you can theoretically make $2/share by exercising your option (buying 100 shares @ $108/share) and immediately selling those 100 shares back to the market at $110/share.  This is all before commission. In more detail, you'll have these practical issues: You are going to have to pay commissions, which means you'll need a bigger spread to make this worthwhile.  You and those who have already answered have you finger on this part, but I include it for completeness. (Even at expiration, if the difference between the last close price and the strike price is pretty close, some ""in-the-money"" options will be allowed to expire unexercised when the holders can't cover the closing commission costs.) The market value of the option contract itself should also go up as the price of the underlying stock goes up.  Unless it's very close to expiration, the option contract should have some ""time value"" in its market price, so, if you want to close your position at this point, earlier then expiration, it will probably be better for you to sell the contract back to the market (for more money and only one commission) than to exercise and then close the stock position (for less money and two commissions). If you want to exercise and then flip the stock back as your exit strategy, you need to be aware of the settlement times.  You probably are not going to instantly have those 100 shares of stock credited to your account, so you may not be able to sell them right away, which could leave you subject to some risk of the price changing.  Alternatively, you could sell the stock short to lock in the price, but you'll have to be sure that your brokerage account is set up to allow that and understand how to do this." It's not actually fighting the paradox of choice, it's lowering the costs of supply chain management. A regular supermarket, for example, has to source 50 different varieties of hot sauce from different manufacturers. Trader Joe's has a couple of varieties of their own sauce, and it's delivered in the same truck that delivers 99% of their food.   People overlook this fact a lot - Trader Joes is a business that is vertically integrated, unlike any other supermarket.