Does the Black-Scholes Model apply to American Style options?
"The difference between an American and European option is that the American option can be exercised at any time, whereas the European option can be liquidated only on the settlement date. The American option is  ""continuous time"" instrument, while the European option is a ""point in time"" instrument. Black Scholes applies to the latter, European, option. Under ""certain"" (but by no means all) circumstances, the two are close enough to be regarded as substitutes. One of their disciples, Robert Merton, ""tweaked"" it to describe American options. There are debates about this, and other tweaks, years later." "Black-Scholes is ""close enough"" for American options since there aren't usually reasons to exercise early, so the ability to do so doesn't matter. Which is good since it's tough to model mathematically, I've read. Early exercise would usually be caused by a weird mispricing for some technical / market-action reason where the theoretical option valuations are messed up. If you sell a call that's far in the money and don't get any time value (after the spread), for example, you probably sold the call to an arbitrageur who's just going to exercise it. But unusual stuff like this doesn't change the big picture much." "A minor tangent. One can claim the S&P has a mean return of say 10%, and standard deviation of say 14% or so, but when you run with that, you find that the actual returns aren't such a great fit to the standard bell curve. Market anomalies producing the ""100-year flood"" far more often than predicted over even a 20 year period. This just means that the model doesn't reflect reality at the tails, even if the +/- 2 standard deviations look pretty. This goes for the Black-Sholes (I almost abbreviated it to initials, then thought better, I actually like the model) as well. The distinction between American and European is small enough that the precision of the model is wider than the difference of these two option styles. I believe if you look at the model and actual pricing, you can determine the volatility of a given stock by using prices around the strike price, but when you then model the well out of money options, you often find the market creating its own valuation." Just a few observations within the Black-Scholes framework: Next, you can now use the Black-Scholes framework (stock price is a Geometric Brownian Motion, no transaction costs, single interest rate, etc. etc.) and numerical methods (such as a PDE solver) to price American style options numerically, but not with a simple closed form formula (though there are closed-form approximations). as no advantage from exerting American call option early,we can use Black schole formula to evaluate the option.However, American put option is more likely to be exercised early which mean Black schole does not apply for this style of option
Here's another attempt at explanation: it's basically because parabolas are flat at the bottom. Let me explain.  As you might know, the variance of the log stock price in Black Scholes is vol^2 * T, in other words, variance of the log stock price is linear in time to expiry. Now, that means that the standard deviation of your log stock price is square root in time. This is consequential.  For normally distributed random variables, in 68% of cases we end up within one standard deviation. So, basically, we expect our log stock price to be within something something times square root of T. So, if your stock has a vol of 16%, it'll be plus/minus 32% in 4 years, plus/minus 16% for one year, plus/minus 8% for 3m, plus/minus 4% for 3-ish weeks, and plus/minus 1% for a business day. As you see, the decay is slow at first, but much more rapid as we get closer.  How does the square root function look? It's a sideways parabola. As we come closer to zero, the slope of the square root function goes to infinity. (That is related to the fact that Brownian motion is almost surely no-where differentiable - it just shoots off with infinite slope, returning immediately, of course :-) Another way of looking at it is the old traders rule of thumb that an at-the-money option is worth approximately S * 0.4 * vol * sqrt(T). (Just do a Taylor expansion of Black Scholes). Again, you have the square root of time to expiry in there, and as outlined above, as we get closer to zero, the square root drops slowly at first, and then precipitously. We have been become very popular, if you want a professional stylist wig at the Ewigsna.com. We have different hair wigs in  professional style such as remi hair, hair care or beauty products, we seek out the best black hair wig products for you. We believe our continued growth depends on our commitment to black full lace wigs, passionately working to surpass your expectations each and every time you shop with us.  If you require more information about the Ewigs, then you can visit our website. Note that this study isn't tracking graduates, just enrollment. Thomas Sowell pointed out that drop-out rates are higher for black Americans in many colleges, and believes that affirmative action is actually to blame - many of these upper tier institutions have lower academic requirements for black Americans applying, so these individuals are, on average, less prepared for that level to college, and drop out. He also felt that these students probably would have been prepared for an institution with lower requirements, and questioned whether it's better to drop out of Harvard or graduate from Brown. If you're saddled with a ton of debt and *also* don't have a degree to show for it, I'd say it's much more likely you're going to have trouble paying it off.  I wonder what the figures for black graduates looks like compared to white graduates? If black graduates are at a payoff rate comparable to that of white graduates, I think that would be a fairly telling piece of evidence that it's not, in fact, about race - but about whether or not you graduated at all. I think you need to be very careful here. Covered calls don't reduce risk or increase performance overall. If they did, every investment manager would be using them. In a typical portfolio, over the long term, the gains you give up when your stock goes beyond the strike of your calls will negate the premiums you receive over time. Psychologically, covered calls are appealing because your gains happen over a long period and this is why many people suggest it. But if you believe the Black-Scholes model (used for pricing options) this is what the model predicts over the long term - that you won't do any better than just holding stock (unless you have some edge other traders don't). Now you say you want to reduce diversification and raise your risk. Keeping in mind that there is no free lunch, there are several ways to reduce your risk but they all come at a price. For simplicity, there are three elements to consider - risk, potential gain and cash. These are tradeoffs and you can't simultaneously make them all favorable. You must trade one or more of them to gain in the others. Let's say you wanted to concentrate into a few stocks... how could you counteract the additional risk? 1) Covered calls: very popular strategy usually intended (erroneously) for increasing returns. You get the bonus of cash along with marginally less risk. But you give up a substantial amount of potential return. You won't have blowout returns if you do this. You still face substantial risk. 2) Collar your stock: You sell a covered call while using the cash from the sale to buy puts for protection. You give up potential gains, you're neutral on cash but gain significantly on reducing risk.  3) Use calls as proxy for stock: You don't hold stock but only calls in equivalent delta to the stock you would have held. Substantially lower risk while still having potential gain. Your tradeoff is the cash you have to pay for the calls. When using this, one must be very, very careful not to overleverage.  4) Puts as protection for stocks: This is basically the same as #3 in tradeoffs. You won't overleverage and you also get dividends. But for the most part it's the same. These are the main ways to reduce the risk you gain by concentrating. Options themselves are far broader. But keep in mind that there is no free money. All these techniques involve tradeoffs that you have to be aware of. "The CBOE states, in an investor's guide to Interest Rate Options: The Options’ Underlying Values   Underlying values for the option contracts are   10 times the underlying Treasury yields (rates)—   13-week T-bill yield (for IRX), 5-year T-note   yield (for FVX), 10-year T-note yield (for TNX)   and 30-year T-bond yield (for TYX). The Yahoo! rate listed is the actual Treasury yield; the Google Finance and CBOE rates reflect the 10 times value.  I don't think there's a specific advantage to ""being contrary"", more likely it's a mistake, or just different." If anyone is interested in looking into it, the company Pinnacle has actually been using theory from quantitative finance for a long time. I went to one of their talks during useR!2017 in Brussels, really interesting betting company. "Not all call options that have value at expiration, exercise by purchasing the security (or attempting to, with funds in your account).   On ETNs, they often (always?) settle in cash.  As an example of an option I'm currently looking at, AVSPY, it settles in cash (please confirm by reading the documentation on this set of options at http://www.nasdaqomxtrader.com/Micro.aspx?id=Alpha, but it is an example of this).  There's nothing it can settle into (as you can't purchase the AVSPY index, only options on it).  You may quickly look (wikipedia) at the difference between ""American Style"" options and ""European Style"" options, for more understanding here. Interestingly I just spoke to my broker about this subject for a trade execution.  Before I go into that, let me also quickly refer to Joe's answer:  what you buy, you can sell.  That's one of the jobs of a market maker, to provide liquidity in a market.  So, when you buy a stock, you can sell it.  When you buy an option, you can sell it.  That's at any time before expiration (although how close you do it before the closing bell on expiration Friday/Saturday is your discretion).  When a market maker lists an option price, they list a bid and an ask.  If you are willing to sell at the bid price, they need to purchase it (generally speaking).  That's why they put a spread between the bid and ask price, but that's another topic not related to your question -- just note the point of them buying at the bid price, and selling at the ask price -- that's what they're saying they'll do. Now, one major difference with options vs. stocks is that options are contracts. So, therefore, we can note just as easily that YOU can sell the option on something (particularly if you own either the underlying, or an option deeper in the money).  If you own the underlying instrument/stock, and you sell a CALL option on it, this is a strategy typically referred to as a covered call, considered a ""risk reduction"" strategy.  You forfeit (potential) gains on the upside, for money you receive in selling the option. The point of this discussion is, is simply: what one buys one can sell;  what one sells one can buy -- that's how a ""market"" is supposed to work.  And also, not to think that making money in options is buying first, then selling.  It may be selling, and either buying back or ideally that option expiring worthless. -- Now, a final example.  Let's say you buy a deep in the money call on a stock trading at $150, and you own the $100 calls.  At expiration, these have a value of $50.  But let's say, you don't have any money in your account, to take ownership of the underlying security (you have to come up with the additional $100 per share you are missing).  In that case, need to call  your broker and see how they handle it, and it will depend on the type of account you have (e.g. margin or not, IRA, etc).  Generally speaking though, the ""margin department"" makes these decisions, and they look through folks that have options on things that have value, and are expiring, and whether they have the funds in their account to absorb the security they are going to need to own.  Exchange-wise, options that have value at expiration, are exercised.  But what if the person who has the option, doesn't have the funds to own the whole stock? Well, ideally on Monday they'll buy all the shares with the options you have at the current price, and immediately liquidate the amount you can't afford to own, but they don't have to.  I'm mentioning this detail so that it helps you see what's going or needs to go on with exchanges and brokerages and individuals, so you have a broader picture." "I think what you really want to look into is put options. You can essentially replicate the same thing, without worrying about margin calls. Check out this site http://www.fundamentalfinance.com/options/options-basic-charts.php a quick glance seems to show it to be pretty good.  The way you would limit downside risk is to buy a put option, allowing you to sell anytime within the next n months for the current price (assuming american). This will allow you to limit downside risk, however, potential profits do go down due to fees as another answer suggests this could be cost prohibitive. This type of strategy is also known as a ""protective put"". http://www.optionseducation.org/strategies_advanced_concepts/strategies/protective_put.html If you wanted to be more refined you could use Ak's bands, although you have to be looking for that specific outcome. Also due to complexity, this can become a taxing (in terms of time invested) and risky (if you are wrong) investment. Either way I think you need to study payoff curves a little more."