How is the time-premium on PUT options calculated
I asked a friend and he gave me a good explanation, so I'm just gonna paste it here for others: There is a simple and a complex answer depending on how much you want to understand the pricing dynamic of options.  LEAPs don't react 1:1 with a stock move because the probability of your option being in the money at expiry is still very much up in the air so you basically don't get full credit for a move in the stock this far out from expiry. The more complex answer involves a discussion of option 'greeks'.  Delta, Gamma, Theta, Vega, and Rho are variables that affect the pricing of all options.  The key greek in this case is Delta because it describes mathematically the expected move of an option as a ratio vs changes in stock price.  For put options the ratio is -1 to 0 where -1 is direct correlation between stock price and option price and 0 is no correlation.  The Delta increases as an option gets deeper in the money and also as it gets closer to expiry and reflects the probability of the option expiring in the money.  For your option contract the current Delta is -0.5673  so -3.38 * -0.5673 = 1.9 which is close.  Also keep in mind that that strike price had a last trade at 12:03 when the stock was at 13.3 and the current ask price is 22.30 so the last price isn't a true reflection of the market value.   As for the other greeks, Gamma is a reflection of volatility in the sense that it affects the rate of change of Delta as price and time changes.  Theta is the value of the time component of the option and is expressed as the expected time decay per day.  The problem is that the time premium is really some arbitrary number that the market maker seems to be able to change at will without justification and it can fluctuate wildly over short periods of time and I think this may explain some of the discrepancy.   If you bought the options when AAPL was $118.68 a couple weeks ago (option price of $18.85) and now AAPL is at $112.34 and the Delta over that time averaged at -0.55 then your expected option price would be $22.34 (($118.68 - $112.34) * 0.55 + 18.85 = $22.34) so you lost around $0.24 in time premium or 'Theta burn' over the last 2 weeks assuming it opens trading around 22.1 on Monday. Your broker should have information about the option contract greeks somewhere.  For my platform I have to put the cursor over top of the option contract for it to show me the greeks.  If your broker doesn't have this then you can get it from nasdaq.com. This is another reason that I only invest in deep in the money LEAPs because the time premium is much much lower than near the money and also because delta is much higher so if I want to trade out of it early I don't feel like I'm getting ripped off not getting paid for a stock price move.  For example look at the Jan 17 175 put.  The Delta is -0.9 and the time premium is only $0-1 depending if you are looking at the bid or ask.  The only downside is expected returns are lower for deep in the money contracts and they are expensive to buy. "According to Yahoo, AAPL was trading at $113.26 at 1:10 PM on 11/13/15, which is the approximate time of your option quote.  You provided a quote for AAPL at 4:15, and the stock happened to keep going down most of the that afternoon.  To make a sensible comparison, you need to take contemporary prices on both the stock and the option.  The quote on the option also shows the ""price"" being outside of the bid-ask range, which suggests that the option was trading thinly and that the last price occurred sometime earlier in the day.  If you use a price in the bid-ask range ($21.90-$22.30) and use the price of AAPL at the time of the put quote, you'll come up with a price that's much closer to your expectation."
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. 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. "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." In addition to the other answers, here's a proper strategy that implements your idea: If the options are priced properly they should account for future dividend payments, so all other things aside, a put option that is currently at the money should be in the money after the dividend, and hence more expensive than a put option that is out of the money today but at the money after the dividend has been paid. The unprotected futures (if priced correctly) should account for dividend payments based on the dividend history and, since maturing after the payment, should earn you (you sell them) less money because you deliver the physical after the dividend has been paid. The protected ones should reflect the expected total return value of the stock at the time of maturity (i.e. the dividend is mentally calculated into the price), and any dividend payments that happen on the way will be debited from your cash (and credited to the counterparty). Now that's the strategy that leaves you with nearly no risk (the only risk you bear is that the dividend isn't as high as you expected).  But for that comfort you have to pay premiums.  So to see if you're smarter than the market, subtract all the costs for the hedging instruments from your envisaged dividend yield and see if your still better than the lending rate.  If so, do the trade. For a deep in the money, it almost makes no difference because the intrinsic value, the price of the option, is seldom far above the liquidation value, the price of the underlying less the strike price. For an at the money, ceteris paribus, an early exercise would immediately cut the value of the option to 0; however, life is not so simple as JB King has shown. Purely theoretically, for an at or near the money option, an early exercise will be an instantaneous cost because the value after exercise is less than the previously trading or implied option price. "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." You own the stock at $29.42 At $40, the stocks is called at $26. You can't add the call premium, as it's already accounted for. The trade is biased towards being bearish on the stock. (I edited and added the graph the evening I answered)  Not the pretiest graph, but you get the idea. With that $29.42 cost, you are in the money till about $30, then go negative until the most you lose is $3.42. "This is because volatility is cumulative and with less time there is less cumulative volatility. The time value and option value are tied to the value of the underlying.  The value of the underlying (stock) is quite influenced by volatility, the possible price movement in a given span of time.  Thirty days of volatility has a much broader spread of values than two days, since each day benefits from the possible price change of the prior days.  So if a stock could move up to +/- 1% in a day, then compounded after 5 days it could be +5%, +0%, or -5%.  In other words, this is compounded volatility. Less time means far less volatility, which is geometric and not linear.  Less volatility lowers the value of the underlying.  See Black-Scholes for more technical discussion of this concept. A shorter timeframe until option expiration means there are fewer days of compounded volatility.  So the expected change in the underlying will decrease geometrically.  The odds are good that the price at T-5 days will be close to the price at T-0, much more so than the prices at T-30 or T-90. Additionally, the time value of an American option is the implicit put value (or implicit call).  While an ""American"" option lets you exercise prior to expiry (unlike a ""European"" option, exercised only at expiry), there's an implicit put option in a call (or an implicit call in a put option).  If you have an American call option of 60 days and it goes into the money at 30 days, you could exercise early.  By contract, that stock is yours if you pay for it (or, in a put, you can sell whenever you decide). In some cases, this may make sense (if you want an immediate payoff or you expect this is the best price situation), but you may prefer to watch the price.  If the price moves further, your gain when you use the call may be even better.  If the price goes back out of the money, then you benefited from an implicit put.  It's as though you exercised the option when it went in the money, then sold the stock and got back your cash when the stock went out of the money, even though no actual transaction took place and this is all just implicit. So the time value of an American option includes the implicit option to not use it early.  The value of the implicit option also decreases in a nonlinear fashion, since the value of the implicit option is subject to the same valuation principles. But the larger principle for both is the compounded volatility, which drops geometrically." The value of an option has 2 components, the extrinsic or time value element and the intrinsic value from the difference in the strike price and the underlying asset price. With either an American or European option the intrinsic value of a call option can be 'locked in' any time by selling the same amount of the underlying asset (whether that be a stock, a future etc).  Further, the time value of any option can be monitised by delta hedging the option, i.e. buying or selling an amount of the underlying asset weighted by the measure of certainty (delta) of the option being in the money at expiry. Instead, the extra value of the American option comes from the financial benefit of being able to realise the value of the underlying asset early. For a dividend paying stock this will predominantly be the dividend. But for non-dividend paying stocks or futures, the buyer of an in-the-money option can realise their intrinsic gains on the option early and earn interest on the profits today. But what they sacrifice is the timevalue of the option.  However when an option becomes very in the money and the delta approaches 1 or -1, the discounting of the intrinsic value (i.e. the extra amount a future cash flow is worth each day as we draw closer to payment) becomes larger than the 'theta' or time value decay of the option. Then it becomes optimal to early exercise, abandon the optionality and realise the monetary gains upfront. For a non-dividend paying stock, the value of the American call option is actually the same as the European. The spot price of the stock will be lower than the forward price at expiry discounted by the risk free rate (or your cost of funding). This will exactly offset the monetary gain by exercising early and banking the proceeds. However for an option on a future, the value today of the underlying asset (the future) is the same as at expiry and its possible to fully realise the interest earned on the money received today. Hence the American call option is worth more. For both examples the American put option is worth more, slightly more so for the stock. As the stock's spot price is lower than the forward price, the owner of the put option realises a higher (undiscounted) intrinsic profit from selling the stock at the higher strike price today than waiting till expiry, as well as realising the interest earned. Liquidity may influence the perceived value of being able to exercise early but its not a tangible factor that is added to the commonly used maths of the option valuation, and isn't really a consideration for most of the assets that have tradeable option markets. It's also important to remember at any point in the life of the option, you don't know the future price path. You're only modelling the distribution of probable outcomes. What subsequently happens after you early exercise an American option no longer has any bearing on its value; this is now zero! Whether the stock subsequently crashes in price is irrelevent. What is relevant is that when you early exercise a call you 'give up' all potential upside protected by the limit to your downside from the strike price. "Some thoughts on your questions in order, Duration: You might want to look at the longest-dated option (often a ""LEAP""), for a couple reasons. One is that transaction costs (spread plus commission, especially spread) are killer on options, so a longer option means fewer transactions, since you don't have to keep rolling the option. Two is that any fundamentals-based views on stocks might tend to require 3-5 years to (relatively) reliably work out, so if you're a fundamental investor, a 3-6 month option isn't great. Over 3-6 months, momentum, short-term news, short squeezes, etc. can often dominate fundamentals in determining the price. One exception is if you just want to hedge a short-term event, such as a pending announcement on drug approval or something, and then you would buy the shortest option that still expires after the event; but options are usually super-expensive when they span an event like this. Strike: Strike price on a long option can be thought of as a tradeoff between the max loss and minimizing ""insurance costs."" That is, if you buy a deeply in-the-money put or call, the time value will be minimal and thus you aren't paying so much for ""insurance,"" but you may have 1/3 or 1/2 of the value of the underlying tied up in the option and subject to loss. If you buy a put or call ""at the money,"" then you might have only say 10% of the value of the underlying tied up in the option and subject to loss, but almost the whole 10% may be time value (insurance cost), so you are losing 10% if the underlying stock price stays flat. I think of the deep in-the-money options as similar to buying stocks on margin (but the ""implied"" interest costs may be less than consumer margin borrowing rates, and for long options you can't get a margin call). The at-the-money options are more like buying insurance, and it's expensive. The commissions and spreads add significant cost, on top of the natural time value cost of the option. The annual costs would generally exceed the long-run average return on a diversified stock fund, which is daunting. Undervalued/overvalued options, pt. 1: First thing is to be sure the options prices on a given underlying make sense at all; there are things that ""should"" hold, for example a synthetic long or short should match up to an actual long or short. These kinds of rules can break, for example on LinkedIn (LNKD) after its IPO, when shorting was not permitted, the synthetic long was quite a bit cheaper than a real long. Usually though this happens because the arbitrage is not practical. For example on LNKD, the shares to short weren't really available, so people doing synthetic shorts with options were driving up the price of the synthetic short and down the price of the synthetic long. If you did actually want to be long the stock, then the synthetic long was a great deal. However, a riskless arbitrage (buy synthetic long, short the stock) was not possible, and that's why the prices were messed up. Another basic relationship that should hold is put-call parity: http://en.wikipedia.org/wiki/Put%E2%80%93call_parity Undervalued/overvalued options, pt. 2: Assuming the relationship to the underlying is sane (synthetic positions equivalent to actual positions) then the valuation of the option could focus on volatility. That is, the time value of the option implies the stock will move a certain amount. If the time value is high and you think the stock won't move much, you might short the option, while if the time value is low and you think the stock will move a lot, you might buy the option. You can get implied volatility from your broker perhaps, or Morningstar.com for example has a bunch of data on option prices and the implied components of the price model. I don't know how useful this really is though. The spreads on options are so wide that making money on predicting volatility better than the market is pretty darn hard. That is, the spread probably exceeds the amount of the mispricing. The price of the underlying is more important to the value of an option than the assumed volatility. How many contracts: Each contract is 100 shares, so you just match that up. If you want to hedge 100 shares, buy one contract. To get the notional value of the underlying multiply by 100. So say you buy a call for $30, and the stock is trading at $100, then you have a call on 100 shares which are currently priced at $10,000 and the option will cost $30*100=3,000. You are leveraged about 3 to 1. (This points to an issue with options for individual investors, which is that one contract is a pretty large notional value relative to most portfolios.)" 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. 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. "RoR for options you bought is fairly easy: (Current Value-Initial Cost)/Initial Cost gives you the actual return. If you want the rate of return, you need to annualize that number: You divide the return you got above by the number of days the investment was in place, and then multiply that number by the number of days in a year. (365 if you're using calendar days, about 255 if you're using trading days.) RoR for options you sold is much more complex: The problem is that RoR is basically calculating the size of your return relative to the capital it tied up to earn it.   That's simple when you bought something; the capital tied up is the money you put up.   It's more complex on a position like a short option, where the specific transaction in question generates cash when it's put on.    The correct way to deal with this is to A) Bundle your strategy (options, stock and collateral) into one RoR where appropriate, and B) include any needed collateral to support the short option in the calculation. So, if you sell a ""cash-secured"" put, where you have to post the money that you'd need to take delivery of the shares if they were put to you, the initial cost is the total amount you'd need to put the trade on:   in this case, it's the cash amount, less the premium you collected for selling the put.  That's just one example.    But the approach holds more broadly:  if you're using covered calls, your original cost is the cost of the stock less the premium generated by the sale of the call." Consider the black-scholes-merton result.  Notice that the expected value of the bond is its present value, discounted from the expiration date. The same is not applied to the price of the stock. The further in the future you go, the less value the bond carries because it's being discounted into oblivion. Now, looking at d1, as time tends towards infinity, so does d1. N(d1) is a probability.  The higher d1, the higher the probability and vice versa, so as time increases, the probability for S trends to 100% while K is discounted away. Note that the math doesn't yet fully model reality, as extremely long dated options such as the European puts Buffett wrote were traded at ~1/2 the value the model said he should've. He still had to take a GAAP loss: http://www.berkshirehathaway.com/letters/2008ltr.pdf As the option approaches expiry, the delta will approach zero or one, depending on whether you're in or out of the money. This might be easiest to conceptualise if you look at the option value as a function of the stock price, and then realise that the delta is the slope of that curve. Now, as we get closer to expiry, time value fades away, and we get closer and closer to the intrinsic value, which looks like this hockey stick: __/ As you see, close to expiry, if you're out of the money, you have nothing (with delta zero), while if you're in the money, you have a forward (with delta one).