# Does the Black-Scholes Model apply to American Style options?

After reading the Wikipedia article on the Black-Scholes model, it looks to me like it only applies to European options based on this quote:

The Black–Scholes model (pronounced /ˌblæk ˈʃoʊlz/1) is a mathematical model of a financial market containing certain derivative investment instruments. From the model, one can deduce the Black–Scholes formula, which gives the price of European-style options.

and

American options and options on stocks paying a known cash dividend (in the short term, more realistic than a proportional dividend) are more difficult to value, and a choice of solution techniques is available (for example lattices and grids).

Is this correct? If so, is there a similar model for American Style options? My previous understanding was that the options price was based on it's intrinsic value + the time value. I'm really not sure how these values are arrived at though.

I found this related question/answer, but it doesn't address this directly: Why are American-style options worth more than European-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.

• Nice use of the word arbitrageur! I hand't seen that word before; I had to go look that one up. Jun 10, 2011 at 17:33
• -1 "tough to model mathematically"?!? Sorry cannot understand that at all. You can easily model lattices and recursion eqs with spreadsheet after calcuating the rates and then last valuations applying the max(C-K, 0), where C is the forecoming exercising value and K is the purchase value for short option (inversersely for long option) and then just backward recursion with \frac{1}{1+r}(qC_{u} + (1-u)C_{d}) where C_{u} is the last upper value and C_{d} is the last down value and the q is the arbitrage-free rate (assuming non-arbitrage situation). Discreate model.
– user1770
Jun 10, 2011 at 19:29
• ...or did you mean by tough the partial derivatives and brownian z -function in Black-Scholes or something else? Mathematically the simplest model are not tough, just some stochastic processes, recursion and and partial derivatives.
– user1770
Jun 10, 2011 at 19:34
• The wikipedia on Black-Scholes says "American options... are more difficult to value, and a choice of solution techniques is available (for example lattices and grids)." en.wikipedia.org/wiki/Option_style says "There are no general formulae for American options, but a choice of models to approximate the price are available." Pretty sure I've read the same in stronger sources than Wikipedia. For most purposes you need to know the relationship among time, price, strike, interest rate, and volatility, that's why I say B-S is close enough, because that's the same for American options. Jun 10, 2011 at 22:57
• By "the relationship is the same" among those factors, I mean for practical purposes that I know of. I'm sure there are some scary computer trading systems and hedge funds that need to get more detailed, but for individual investors you just need to understand how time to expiration, strike price, underlying price, interest rates, and volatility factor into the option's value. Jun 11, 2011 at 2:50

Just a few observations within the Black-Scholes framework:

• American calls have the same price as European calls on non-dividend paying assets.
• The Black-Scholes formula is applicable only to European options (and, by the above, to American calls on non-dividend paying assets).
• By the call-put parity, if you have European call prices for some expiry dates and strikes, you also have the European put prices for those expiry dates and strikes.
• If you have European call prices for a given expiry date T for all strikes, you can easily compute the price of any "European" payoff for that expiry (for example, a digital call V = 1_{S>K}, or a parabola V = S^2, or whatever). Conceptually, you form butterfly spreads __/\_ for a series of increasing strikes, and they give you the "risk-neutral" probability that you end up there, and then you just integrate over your payoff.

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).

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.

• That makes perfect sense. If the prices were that predictable then the system wouldn't work. It turns out that the system actually works because prices are somewhat unpredictable. Jun 10, 2011 at 13:49
• I can also go on a bit about how for many strikes, the volume is so thin that the price can't be expected to reflect true value. If I had the skill and processing power, I'd scan for certain type of activity to find indications of unusual behavior. Than behavior may reflect illegal trading, so care is needed. If your trade follows and you have good records, you won't get nailed for the same insider trading the first guys did. Jun 10, 2011 at 14:08
• If I had the skill and processing power, I'd scan for certain type of activity to find indications of unusual behavior.` Aren't there online tools that will do this for you? Jun 10, 2011 at 15:11
• "10% mean return ... 14% standard deviation .. you find that the actual returns aren't a great fit to the standard bell curve". It seems that you think that the mean and standard deviation are exclusive to the bell (Gauss) curve. That's not true. There are an infinite number of distributions, even for a given mean and standard deviation. And for that reason alone you can't predict "100 year floods" from just mean and standard deviation; you need the actual distribution. Jul 28, 2014 at 9:36
• @MSalters - BS reflects the math of a bell curve. Not sure I get your point here. Jul 28, 2014 at 11:39

Yes, your understanding is correct. Strictly speaking, the Black-Scholes model is used to price European options. However, the payoff (price) of European and American options are close enough and can be used as an approximation if no dividends are paid on the underlying, and liquidity cost is close to zero (e.g. in a very low-interest rate scenario).

As of now, there are no closed-form methods to price American options. At least none that I know of. You should rely on lattices for multi-period binomial pricing, which is mostly recursive.

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