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
SPX options are cash settled European style.  You cannot exercise European style options before the expiration date.  Assuming it is the day of expiration and you own 2,000 strike puts and the index settlement value is 1,950 - you would exercise and receive cash for the in the money amount times the contract multiplier.  If instead you owned put options on the S&amp;P 500 SPDR ETF (symbol SPY) those are American style, physically settled options.  You can exercise a long American style option anytime between when your purchase it and when it expires.    If you exercised SPY puts without owning shares of SPY you would end up short stock at the strike price. &gt; “If the [Black-Scholes] formula is applied to extended time periods….it can produce absurd results.”  &gt; -Warren Buffett, 2008 Letter to Berkshire Hathaway Shareholders  I will give your question more thought, and come back with a quantitative solution. It may be most fruitful to apply a backward-induction options pricing model with detailed scenario-based discounted cash flow valuation models supported by pro forma financial statement and investment analysis.    Nonetheless, my initial reaction is inline with Warren Buffett's belief that in the long-run an assumption of the Black-Scholes options pricing model is invalid (see [here](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.613.1657&amp;rep=rep1&amp;type=pdf) and [here](http://www.elon.edu/docs/e-web/academics/business/economics/faculty/kurt/OptionValuationBuffetCritique_123113.pdf)). The perceived invalid assumption is that the distribution of expected future stock prices is *not* log-normal in the long-run. This non-log-normal view is especially true for a single company stock.   This invalid assumption results in over-valuation of options prices from Black-Scholes - which makes it much better to sell long-term options than to buy them if market participants are using Black-Scholes pricing models.  Again, without having done the math yet, my gut tells me if the option seller is using Black-Sholes pricing for long-dated options, you would be best to avoid buying them as the prices will be inflated compared to an economic reality fair price. /u/NotMyRealFaceBook  As a sniff test, I threw the following assumptions into a Black-Scholes options pricing calculator:  * Stock Price and Strike Price = $200  * Valuation date = 10/31/2017  * Expiration date = 10/29/2027 (European style - no early exercise)  * Volatility = 20% (on the low end for a tech. company - but again see the non-log-normal assumption point above)  * Interest rate = 4% (on the high end - which means the option seller is over-charging you)  * Dividend yield = 0%  The calculator's theoretical call option price results in $82.  Unless the company in question has a materially skewed upside given today's starting valuation, my opinion is the options you are offered are being priced via a Black-Scholes model and are over-priced. http://en.wikipedia.org/wiki/Black%E2%80%93Scholes  Start there, the black scholes formula is used to calculate the value of a call or put option and it incorporates the risk free rate in the formula.  I am currently a bit busy but I will write up a better answer later if I get time. "This chart concerns an option contract, not a stock. The method of analysis is to assume that the price of an option contract is normally distributed around some mean which is presumably the current price of the underlying asset. As the date of expiration of the contract gets closer the variation around the mean in the possible end price for the contract will decrease. Undoubtedly the publisher has measured typical deviations from the mean as a function of time until expiration from historical data. Based on this data, the program that computes the probability has the following inputs: (1) the mean (current asset price) (2) the time until expiration (3) the expected standard deviation based on (2) With this information the probability distribution that you see is generated (the green hump). This is a ""normal"" or Gaussian distribution. For a normal distribution the probability of a particular event is equal to the area under the curve to the right of the value line (in the example above the value chosen is 122.49). This area can be computed with the formula:  This formula is called the probability density for x, where x is the value (122.49 in the example above). Tau (T) is the reciprocal of the variance (which can be computed from the standard deviation). Mu (μ) is the mean. The main assumption such a calculation makes is that the price of the asset will not change between now and the time of expiration. Obviously that is not true in most cases because the prices of stocks and bonds constantly fluctuate. A secondary assumption is that the distribution of the option price around the mean will a normal (or Gaussian) distribution. This is obviously a crude assumption and common sense would suggest it is not the most accurate distribution. In fact, various studies have shown that the Burr Distribution is actually a more accurate model for the distribution of option contract prices." The value at expiration does not depend on the price path for a plain vanilla European or American option. At expiration, the value would simply be: max[K - S_T, 0], where: K is the strike price, and S_T is the underlying price at expiration. Thanks. Just to clarify I am looking for a more value-neutral answer in terms of things like Sharpe ratios. I think it's an oversimplification to say that on average you lose money because of put options - even if they expire uselessly 90% of the time, they still have some expected payoff that kicks in 10% of the time, and if the price is less than the expected payoff you will earn money in the long term by investing in put options (I am sure you know this as a PhD student I just wanted to get it out there.)I guess more formally my question would be are there studies on whether options prices correspond well to the diversification benefits they offer from an MPT point of view. Yes. If I own a call, an American call option can be exercised at my wish. A European call can only be exercised at expiration, by the way.  Your broker doesn't give you anything but a current quote for a given strike price. There are a number of good option related questions here. A bit of searching and reading will help you understand the process. 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 "I agree completely.  ""I don't always agree with John Cochrane, but when I do, I agree completely.""  I think heavy reliance on either approach to pricing is generally a bad idea.  Equilibrium models always include something that you're supposed to inherently know, but never do.  No-arbitrage models don't necessarily *say* anything that you don't (in some mathematical sense) already know.  So you're either stuck with unknown parameters, or you can't explain why you're something is worth what you say it is beyond, ""Herp derp, other people are doing it.""  So I think if buy-side people made some use of no-arbitrage models, they'd have a better understanding of the parameters they're making up, and if sell-side people sometimes used equilibrium models, they'd have a better grasp of what's going on economically.  Also, it would have the beneficial effect of reminding people that their models are always wrong, even if they're frequently useful." "Yes, long calls, and that's a good point. Let's see... if I bought one contract at the Bid price above...          $97.13 at expiry of $96.43 option = out of the money =- option price(x100)          = $113 loss.     $97.13 at expiry of $97.00 option = out of the money =- option price(x100)          = $77 loss.     $97.13 at expiry of $97.14 option = in the money by 1-cent=$1/contract profit - option price(x100) = $1-$58          = $57 loss   The higher strike prices have much lower losses if they expire with the  underlying stock at- or near-the-money. So, they carry ""gentler"" downside potential, and are priced much higher to reflect that ""controlled"" risk potential.  That makes sense. Thanks." It's a matter of risk and reward. And its origin goes back to the Black Scholles equation, which is sort of a bell curve of possible outcomes.  Do you see that from $36 to $34 strike, you are putting up over 35% more money to lower your break even by 30 cents?  If I were to bet* $2000, I could buy 3 of the $34 contracts but almost 5 of the $36 strike. If the stock went to $45, I'd be far better off.  *I say 'bet' because simply buying puts or calls, absent any underlying asset, is akin to gambling, not investing. I do it all the time, but with my Vegas money. Yes. It seems to me you got it right.  On my site, Stock Options Cafe, my last post was an illustration of a bullish call spread. In this case, I bought a 50 call, and sold the 60 call. This is a debit order as I was paying money, not collecting a new premium. An option is freely tradable, and all options (of the same kind) are equal. If your position is 0 and you sell 1 option, your new position in that option is -1. If the counterparty to your trade buys or sells more options to close, open, or even reopen their position afterwards, that doesn't matter to your position at all. Of course there's also the issue with American and European Options. European Options expire at their due date, but American Options expire at their due date or at any time before their due date if the holder decides they expire. With American Options, if a holder of an American Option decides to exercise the option, someone who is short the same option will be assigned as the counterparty (this is usually random). Expiry is after market close, so if one of your short American Options expires early, you will need to reopen the position the next day. Keep in mind dividends for slightly increased complexity. American and European Options do not in any way refer to the continents they are traded on, or to the location of the companies. These terms simply describe the expiry rules. You might have better luck using Quandl as a source.  They have free databases, you just need to register to access them.  They also have good api's, easier to use than the yahoo api's Their WIKI database of stock prices is curated and things like this are fixed (www.quandl.com/WIKI ), but I'm not sure that covers the London stock exchange.  They do, however, have other databases that cover the London stock exchange.