The Black–Scholes /ˌblæk
Based on works previously developed by market researchers and practitioners, such as Louis Bachelier, Sheen Kassouf and Ed Thorp among others, Fischer Black and Myron Scholes demonstrated in the late 1960s that a dynamic revision of a portfolio removes the expected return of the security, thus inventing the risk neutral argument.^{[4]}^{[5]} In 1970, after they attempted to apply the formula to the markets and incurred financial losses due to lack of risk management in their trades, they decided to focus in their domain area, the academic environment.^{[6]} After three years of efforts, the formula named in honor of them for making it public, was finally published in 1973 in an article entitled "The Pricing of Options and Corporate Liabilities", in the Journal of Political Economy.^{[7]}^{[8]}^{[9]} Robert C. Merton was the first to publish a paper expanding the mathematical understanding of the options pricing model, and coined the term "Black–Scholes options pricing model". Merton and Scholes received the 1997 Nobel Memorial Prize in Economic Sciences for their work, the committee citing their discovery of the risk neutral dynamic revision as a breakthrough that separates the option from the risk of the underlying security.^{[10]} Although ineligible for the prize because of his death in 1995, Black was mentioned as a contributor by the Swedish Academy.^{[11]}
The key idea behind the model is to hedge the option by buying and selling the underlying asset in just the right way and, as a consequence, to eliminate risk. This type of hedging is called "continuously revised delta hedging" and is the basis of more complicated hedging strategies such as those engaged in by investment banks and hedge funds.
The model's assumptions have been relaxed and generalized in many directions, leading to a plethora of models that are currently used in derivative pricing and risk management. It is the insights of the model, as exemplified in the Black–Scholes formula, that are frequently used by market participants, as distinguished from the actual prices. These insights include noarbitrage bounds and riskneutral pricing (thanks to continuous revision). Further, the Black–Scholes equation, a partial differential equation that governs the price of the option, enables pricing using numerical methods when an explicit formula is not possible.
The Black–Scholes formula has only one parameter that cannot be directly observed in the market: the average future volatility of the underlying asset, though it can be found from the price of other options. Since the option value (whether put or call) is increasing in this parameter, it can be inverted to produce a "volatility surface" that is then used to calibrate other models, e.g. for OTC derivatives.
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Transcription
Voiceover: We're now gonna talk about probably the most famous formula in all of finance, and that's the BlackScholes Formula, sometimes called the BlackScholesMerton Formula, and it's named after these gentlemen. This right over here is Fischer Black. This is Myron Scholes. They really laid the foundation for what led to the BlackScholes Model and the BlackScholes Formula and that's why it has their name. This is Bob Merton, who really took what BlackScholes did and took it to another level to really get to our modern interpretations of the BlackScholes Model and the BlackScholes Formula. All three of these gentlemen would have won the Nobel Prize in Economics, except for the unfortunate fact that Fischer Black passed away before the award was given, but Myron Scholes and Bob Merton did get the Nobel Prize for their work. The reason why this is such a big deal, why it is Nobel Prize worthy, and, actually, there's many reasons. I could do a whole series of videos on that, is that people have been trading stock options, or they've been trading options for a very, very, very long time. They had been trading them, they had been buying them, they had been selling them. It was a major part of financial markets already, but there was no really good way of putting our mathematical minds around how to value an option. People had a sense of the things that they cared about, and I would assume especially options traders had a sense of the things that they cared about when they were trading options, but we really didn't have an analytical framework for it, and that's what the BlackScholes Formula gave us. Let's just, before we dive into this seemingly hairy formula, but the more we talk about it, hopefully it'll start to seem a lot friendlier than it looks right now. Let's start to get an intuition for the things that we would care about if we were thinking about the price of a stock option. You would care about the stock price. You would care about the exercise price. You would especially care about how much higher or lower the stock price is relative to the exercise price. You would care about the riskfree interest rate. The riskfree interest rate keeps showing up when we think about taking a present value of something, If we want to discount the value of something back to today. You would, of course, think about how long do I have to actually exercise this option? Finally, this might look a little bit bizarre at first, but we'll talk about it in a second. You would care about how volatile that stock is, and we measure volatility as a standard deviation of log returns for that security. That seems very fancy, and we'll talk about that in more depth in future videos, but at just an intuitive level, just think about 2 stocks. So let's say that this is stock 1 right over here, and it jumps around, and I'll make them go flat, just so we make no judgment about whether it's a good investment. You have one stock that kind of does that, and then you have another stock. Actually, I'll draw them on the same, so let's say that is stock 1, and then you have a stock 2 that does this, it jumps around all over the place. So this green one right over here is stock 2. You could imagine stock 2 just in the way we use the word 'volatile' is more volatile. It's a wilder ride. Also, if you were looking at how dispersed the returns are away from their mean, you see it has, the returns have more dispersion. It'll have a higher standard deviation. So, stock 2 will have a higher volatility, or a higher standard deviation of logarithmic returns, and in a future video, we'll talk about why we care about log returns, Stock 1 would have a lower volatility, so you can imagine, options are more valuable when you're dealing with, or if you're dealing with a stock that has higher volatility, that has higher sigma like this, this feels like it would drive the value of an option up. You would rather have an option when you have something like this, because, look, if you're owning the stock, man, you have to go after, this is a wild ride, but if you have the option, you could ignore the wildness, and then it might actually make, and then you could exercise the option if it seems like the right time to do it. So it feels like, if you were just trading it, that the more volatile something is, the more valuable an option would be on that. Now that we've talked about this, let's actually look at the BlackScholes Formula. The variety that I have right over here, this is for a European call option. We could do something very similar for a European put option, so this is right over here is a European call option, and remember, European call option, it's mathematically simpler than an American call option in that there's only one time at which you can exercise it on the exercise date. On an American call option, you can exercise it an any point. With that said, let's try to at least intuitively dissect the BlackScholes Formula a little bit. So the first thing you have here, you have this term that involved the current stock price, and then you're multiplying it times this function that's taking this as an input, and this as how we define that input, and then you have minus the exercise price discounted back, this discounts back the exercise price, times that function again, and now that input is slightly different into that function. Just so that we have a little bit of background about what this function N is, N is the cumulative distribution function for a standard, normal distribution. I know that seems, might seem a little bit daunting, but you can look at the statistics playlist, and it shouldn't be that bad. This is essentially saying for a standard, normal distribution, the probability that your random variable is less than or equal to x, and another way of thinking about that, if that sounds a little, and it's all explained in our statistics play list if that was confusing, but if you want to think about it a little bit mathematically, you also know that this is going to be, it's a probability. It's always going to be greater than zero, and it is going to be less than one. With that out of the way, let's think about what these pieces are telling us. This, right over here, is dealing with, it's the current stock price, and it's being weighted by some type of a probability, and so this is, essentially, one way of thinking about it, in very rough terms, is this is what you're gonna get. You're gonna get the stock, and it's kind of being weighted by the probability that you're actually going to do this thing, and I'm speaking in very rough terms, and then this term right over here is what you pay. This is what you pay. This is your exercise price discounted back, somewhat being weighted, and I'm speaking, once again, I'm handweaving a lot of the mathematics, by like are we actually going to do this thing? Are we actually going to exercise our option? That makes sense right over there, and it makes sense if the stock price is worth a lot more than the exercise price, and if we're definitely going to do this, let's say that D1 and D2 are very, very large numbers, we're definitely going to do this at some point in time, that it makes sense that the value of the call option would be the value of the stock minus the exercise price discounted back to today. This right over here, this is the discounting, kind of giving us the present value of the exercise price. We have videos on discounting and present value, if you find that a little bit daunting. It also makes sense that the more, the higher the stock price is, so we see that right over here, relative to the exercise price, the more that the option would be worth, it also makes sense that the higher the stock price relative to the exercise price, the more likely that we will actually exercise the option. You see that in both of these terms right over here. You have the ratio of the stock price to the exercise price. A ratio of the stock price to the exercise price. We're taking a natural log of it, but the higher this ratio is, the larger D1 or D2 is, so that means the larger the input into the cumulative distribution function is, which means the higher probabilities we're gonna get, and so it's a higher chance we're gonna exercise this price, and it makes sense that then this is actually going to have some value. So that makes sense, the relationship between the stock price and the exercise price. The other thing I will focus on, because this tends to be a deep focus of people who operate with options, is the volatility. We already had an intuition, that the higher the volatility, the higher the option price, so let's see where this factors into this equation, here. We don't see it at this first level, but it definitely factors into D1 and D2. In D1, the higher your standard deviation of your log returns, so the higher sigma, we have a sigma in the numerator and the denominator, but in the numerator, we're squaring it. So a higher sigma will make D1 go up, so sigma goes up, D1 will go up. Let's think about what's happening here. Well, here we have a sigma. It's still squared. It's in the numerator, but we're subtracting it. This is going to grow faster than this, but we're subtracting it now, so for D2, a higher sigma is going to make D2 go down because we are subtracting it. This will actually make, can we actually say this is going to make, a higher sigma's going to make the value of our call option higher. Well, let's look at it. If the value of our sigma goes up, then D1 will go up, then this input, this input goes up. If that input goes up, our cumulative distribution function of that input is going to go up, and so this term, this whole term is gonna drive this whole term up. Now, what's going to happen here. Well, if D2 goes down, then our cumulative distribution function evaluated there is going to go down, and so this whole thing is going to be lower and so we're going to have to pay less. If we get more and pay less, and I'm speaking in very handwavy terms, but this is just to understand that this is as intuitively daunting as you might think, but it looks definitively, that if the standard deviation, if the standard deviation of our log returns or if our volatility goes up, the value of our call option, the value of our European call option goes up. Likewise, using the same logic, if our volatility were to be lower, then the value of our call option would go down. I'll leave you there. In future videos, we'll think about this in a little bit more depth.
Contents
 1 The Black–Scholes world
 2 Notation
 3 Black–Scholes equation
 4 Black–Scholes formula
 5 The Greeks
 6 Extensions of the model
 7 Black–Scholes in practice
 8 Criticism and comments
 9 See also
 10 Notes
 11 References
 12 External links
The Black–Scholes world
The Black–Scholes model assumes that the market consists of at least one risky asset, usually called the stock, and one riskless asset, usually called the money market, cash, or bond.
Now we make assumptions on the assets (which explain their names):
 (riskless rate) The rate of return on the riskless asset is constant and thus called the riskfree interest rate.
 (random walk) The instantaneous log return of stock price is an infinitesimal random walk with drift; more precisely, it is a geometric Brownian motion, and we will assume its drift and volatility are constant (if they are timevarying, we can deduce a suitably modified Black–Scholes formula quite simply, as long as the volatility is not random).
 The stock does not pay a dividend.^{[Notes 1]}
Assumptions on the market:
 There is no arbitrage opportunity (i.e., there is no way to make a riskless profit).
 It is possible to borrow and lend any amount, even fractional, of cash at the riskless rate.
 It is possible to buy and sell any amount, even fractional, of the stock (this includes short selling).
 The above transactions do not incur any fees or costs (i.e., frictionless market).
With these assumptions holding, suppose there is a derivative security also trading in this market. We specify that this security will have a certain payoff at a specified date in the future, depending on the value(s) taken by the stock up to that date. It is a surprising fact that the derivative's price is completely determined at the current time, even though we do not know what path the stock price will take in the future. For the special case of a European call or put option, Black and Scholes showed that "it is possible to create a hedged position, consisting of a long position in the stock and a short position in the option, whose value will not depend on the price of the stock".^{[12]} Their dynamic hedging strategy led to a partial differential equation which governed the price of the option. Its solution is given by the Black–Scholes formula.
Several of these assumptions of the original model have been removed in subsequent extensions of the model. Modern versions account for dynamic interest rates (Merton, 1976),^{[citation needed]} transaction costs and taxes (Ingersoll, 1976),^{[citation needed]} and dividend payout.^{[13]}
Notation
The notation used throughout this page will be defined as follows:
 , the price of the underlying asset at time t.;
 , the price of the option as a function of the underlying asset, S at time, t;
 , the price of a European call option and the price of a European put option;
 , the strike price of the option, also known as the exercise price;
 , the annualized riskfree interest rate, continuously compounded Also known as the force of interest;
 , the drift rate of , annualized;
 , the standard deviation of the stock's returns; this is the square root of the quadratic variation of the stock's log price process;
 , a time in years; we generally use: now , expiry ;
 , the value of the portfolio.
We will use to denote the standard normal cumulative distribution function,
will denote the standard normal probability density function,
Black–Scholes equation
As above, the Black–Scholes equation is a partial differential equation, which describes the price of the option over time. The equation is:
The key financial insight behind the equation is that one can perfectly hedge the option by buying and selling the underlying asset in just the right way and consequently "eliminate risk".^{[citation needed]} This hedge, in turn, implies that there is only one right price for the option, as returned by the Black–Scholes formula (see the next section).
Black–Scholes formula
The Black–Scholes formula calculates the price of European put and call options. This price is consistent with the Black–Scholes equation as above; this follows since the formula can be obtained by solving the equation for the corresponding terminal and boundary conditions.
The value of a call option for a nondividendpaying underlying stock in terms of the Black–Scholes parameters is:
The price of a corresponding put option based on put–call parity is:
For both, as above:
 is the cumulative distribution function of the standard normal distribution
 is the time to maturity (expressed in years)
 is the spot price of the underlying asset
 is the strike price
 is the risk free rate (annual rate, expressed in terms of continuous compounding)
 is the volatility of returns of the underlying asset
Alternative formulation
Introducing some auxiliary variables allows the formula to be simplified and reformulated in a form that is often more convenient (this is a special case of the Black '76 formula):
The auxiliary variables are:
 is the time to expiry (remaining time, backwards time)
 is the discount factor
 is the forward price of the underlying asset, and
with d_{+} = d_{1} and d_{−} = d_{2} to clarify notation.
Given put–call parity, which is expressed in these terms as:
the price of a put option is:
Interpretation
The Black–Scholes formula can be interpreted fairly handily, with the main subtlety the interpretation of the (and a fortiori ) terms, particularly and why there are two different terms.^{[14]}
The formula can be interpreted by first decomposing a call option into the difference of two binary options: an assetornothing call minus a cashornothing call (long an assetornothing call, short a cashornothing call). A call option exchanges cash for an asset at expiry, while an assetornothing call just yields the asset (with no cash in exchange) and a cashornothing call just yields cash (with no asset in exchange). The Black–Scholes formula is a difference of two terms, and these two terms equal the value of the binary call options. These binary options are much less frequently traded than vanilla call options, but are easier to analyze.
Thus the formula:
breaks up as:
where is the present value of an assetornothing call and is the present value of a cashornothing call. The D factor is for discounting, because the expiration date is in future, and removing it changes present value to future value (value at expiry). Thus is the future value of an assetornothing call and is the future value of a cashornothing call. In riskneutral terms, these are the expected value of the asset and the expected value of the cash in the riskneutral measure.
The naive, and not quite correct, interpretation of these terms is that is the probability of the option expiring in the money , times the value of the underlying at expiry F, while is the probability of the option expiring in the money times the value of the cash at expiry K. This is obviously incorrect, as either both binaries expire in the money or both expire out of the money (either cash is exchanged for asset or it is not), but the probabilities and are not equal. In fact, can be interpreted as measures of moneyness (in standard deviations) and as probabilities of expiring ITM (percent moneyness), in the respective numéraire, as discussed below. Simply put, the interpretation of the cash option, , is correct, as the value of the cash is independent of movements of the underlying, and thus can be interpreted as a simple product of "probability times value", while the is more complicated, as the probability of expiring in the money and the value of the asset at expiry are not independent.^{[14]} More precisely, the value of the asset at expiry is variable in terms of cash, but is constant in terms of the asset itself (a fixed quantity of the asset), and thus these quantities are independent if one changes numéraire to the asset rather than cash.
If one uses spot S instead of forward F, in instead of the term there is which can be interpreted as a drift factor (in the riskneutral measure for appropriate numéraire). The use of d_{−} for moneyness rather than the standardized moneyness – in other words, the reason for the factor – is due to the difference between the median and mean of the lognormal distribution; it is the same factor as in Itō's lemma applied to geometric Brownian motion. In addition, another way to see that the naive interpretation is incorrect is that replacing N(d_{+}) by N(d_{−}) in the formula yields a negative value for outofthemoney call options.^{[14]}^{:6}
In detail, the terms are the probabilities of the option expiring inthemoney under the equivalent exponential martingale probability measure (numéraire=stock) and the equivalent martingale probability measure (numéraire=risk free asset), respectively.^{[14]} The risk neutral probability density for the stock price is
where is defined as above.
Specifically, is the probability that the call will be exercised provided one assumes that the asset drift is the riskfree rate. , however, does not lend itself to a simple probability interpretation. is correctly interpreted as the present value, using the riskfree interest rate, of the expected asset price at expiration, given that the asset price at expiration is above the exercise price.^{[15]} For related discussion – and graphical representation – see section "Interpretation" under Datar–Mathews method for real option valuation.
The equivalent martingale probability measure is also called the riskneutral probability measure. Note that both of these are probabilities in a measure theoretic sense, and neither of these is the true probability of expiring inthemoney under the real probability measure. To calculate the probability under the real ("physical") probability measure, additional information is required—the drift term in the physical measure, or equivalently, the market price of risk.
Derivations
A standard derivation for solving the Black–Scholes PDE is given in the article Black–Scholes equation.
The Feynman–Kac formula says that the solution to this type of PDE, when discounted appropriately, is actually a martingale. Thus the option price is the expected value of the discounted payoff of the option. Computing the option price via this expectation is the risk neutrality approach and can be done without knowledge of PDEs.^{[14]} Note the expectation of the option payoff is not done under the real world probability measure, but an artificial riskneutral measure, which differs from the real world measure. For the underlying logic see section "risk neutral valuation" under Rational pricing as well as section "Derivatives pricing: the Q world" under Mathematical finance; for detail, once again, see Hull.^{[16]}^{:307–309}
The Greeks
"The Greeks" measure the sensitivity of the value of a derivative or a portfolio to changes in parameter value(s) while holding the other parameters fixed. They are partial derivatives of the price with respect to the parameter values. One Greek, "gamma" (as well as others not listed here) is a partial derivative of another Greek, "delta" in this case.
The Greeks are important not only in the mathematical theory of finance, but also for those actively trading. Financial institutions will typically set (risk) limit values for each of the Greeks that their traders must not exceed. Delta is the most important Greek since this usually confers the largest risk. Many traders will zero their delta at the end of the day if they are speculating and following a deltaneutral hedging approach as defined by Black–Scholes.
The Greeks for Black–Scholes are given in closed form below. They can be obtained by differentiation of the Black–Scholes formula.^{[17]}
Calls  Puts  

Delta  
Gamma  
Vega  
Theta  
Rho 
Note that from the formulae, it is clear that the gamma is the same value for calls and puts and so too is the vega the same value for calls and put options. This can be seen directly from put–call parity, since the difference of a put and a call is a forward, which is linear in S and independent of σ (so a forward has zero gamma and zero vega). N' is the standard normal probability density function.
In practice, some sensitivities are usually quoted in scaleddown terms, to match the scale of likely changes in the parameters. For example, rho is often reported divided by 10,000 (1 basis point rate change), vega by 100 (1 vol point change), and theta by 365 or 252 (1 day decay based on either calendar days or trading days per year).
(Vega is not a letter in the Greek alphabet; the name arises from reading the Greek letter ν (nu) as a V.)
Extensions of the model
The above model can be extended for variable (but deterministic) rates and volatilities. The model may also be used to value European options on instruments paying dividends. In this case, closedform solutions are available if the dividend is a known proportion of the stock price. 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).
Instruments paying continuous yield dividends
For options on indices, it is reasonable to make the simplifying assumption that dividends are paid continuously, and that the dividend amount is proportional to the level of the index.
The dividend payment paid over the time period is then modelled as
for some constant (the dividend yield).
Under this formulation the arbitragefree price implied by the Black–Scholes model can be shown to be
and
where now
is the modified forward price that occurs in the terms :
and
 .^{[18]}
Instruments paying discrete proportional dividends
It is also possible to extend the Black–Scholes framework to options on instruments paying discrete proportional dividends. This is useful when the option is struck on a single stock.
A typical model is to assume that a proportion of the stock price is paid out at predetermined times . The price of the stock is then modelled as
where is the number of dividends that have been paid by time .
The price of a call option on such a stock is again
where now
is the forward price for the dividend paying stock.
American options
The problem of finding the price of an American option is related to the optimal stopping problem of finding the time to execute the option. Since the American option can be exercised at any time before the expiration date, the Black–Scholes equation becomes an inequality of the form
 ^{[19]}
with the terminal and (free) boundary conditions: and where denotes the payoff at stock price .
In general this inequality does not have a closed form solution, though an American call with no dividends is equal to a European call and the RollGeskeWhaley method provides a solution for an American call with one dividend;^{[20]}^{[21]} see also Black's approximation.
BaroneAdesi and Whaley^{[22]} is a further approximation formula. Here, the stochastic differential equation (which is valid for the value of any derivative) is split into two components: the European option value and the early exercise premium. With some assumptions, a quadratic equation that approximates the solution for the latter is then obtained. This solution involves finding the critical value, , such that one is indifferent between early exercise and holding to maturity.^{[23]}^{[24]}
Bjerksund and Stensland^{[25]} provide an approximation based on an exercise strategy corresponding to a trigger price. Here, if the underlying asset price is greater than or equal to the trigger price it is optimal to exercise, and the value must equal , otherwise the option "boils down to: (i) a European upandout call option… and (ii) a rebate that is received at the knockout date if the option is knocked out prior to the maturity date". The formula is readily modified for the valuation of a put option, using put–call parity. This approximation is computationally inexpensive and the method is fast, with evidence indicating that the approximation may be more accurate in pricing long dated options than BaroneAdesi and Whaley.^{[26]}
Binary options
By solving the Black–Scholes differential equation, with for boundary condition the Heaviside function, we end up with the pricing of options that pay one unit above some predefined strike price and nothing below.^{[27]}
In fact, the Black–Scholes formula for the price of a vanilla call option (or put option) can be interpreted by decomposing a call option into an assetornothing call option minus a cashornothing call option, and similarly for a put – the binary options are easier to analyze, and correspond to the two terms in the Black–Scholes formula.
Cashornothing call
This pays out one unit of cash if the spot is above the strike at maturity. Its value is given by
Cashornothing put
This pays out one unit of cash if the spot is below the strike at maturity. Its value is given by
Assetornothing call
This pays out one unit of asset if the spot is above the strike at maturity. Its value is given by
Assetornothing put
This pays out one unit of asset if the spot is below the strike at maturity. Its value is given by
Foreign exchange
If we denote by S the FOR/DOM exchange rate (i.e., 1 unit of foreign currency is worth S units of domestic currency) we can observe that paying out 1 unit of the domestic currency if the spot at maturity is above or below the strike is exactly like a cashor nothing call and put respectively. Similarly, paying out 1 unit of the foreign currency if the spot at maturity is above or below the strike is exactly like an assetor nothing call and put respectively. Hence if we now take , the foreign interest rate, , the domestic interest rate, and the rest as above, we get the following results.
In case of a digital call (this is a call FOR/put DOM) paying out one unit of the domestic currency we get as present value,
In case of a digital put (this is a put FOR/call DOM) paying out one unit of the domestic currency we get as present value,
While in case of a digital call (this is a call FOR/put DOM) paying out one unit of the foreign currency we get as present value,
and in case of a digital put (this is a put FOR/call DOM) paying out one unit of the foreign currency we get as present value,
Skew
In the standard Black–Scholes model, one can interpret the premium of the binary option in the riskneutral world as the expected value = probability of being inthemoney * unit, discounted to the present value. The Black–Scholes model relies on symmetry of distribution and ignores the skewness of the distribution of the asset. Market makers adjust for such skewness by, instead of using a single standard deviation for the underlying asset across all strikes, incorporating a variable one where volatility depends on strike price, thus incorporating the volatility skew into account. The skew matters because it affects the binary considerably more than the regular options.
A binary call option is, at long expirations, similar to a tight call spread using two vanilla options. One can model the value of a binary cashornothing option, C, at strike K, as an infinitessimally tight spread, where is a vanilla European call:^{[28]}^{[29]}
Thus, the value of a binary call is the negative of the derivative of the price of a vanilla call with respect to strike price:
When one takes volatility skew into account, is a function of :
The first term is equal to the premium of the binary option ignoring skew:
is the Vega of the vanilla call; is sometimes called the "skew slope" or just "skew". If the skew is typically negative, the value of a binary call will be higher when taking skew into account.
Relationship to vanilla options' Greeks
Since a binary call is a mathematical derivative of a vanilla call with respect to strike, the price of a binary call has the same shape as the delta of a vanilla call, and the delta of a binary call has the same shape as the gamma of a vanilla call.
Black–Scholes in practice
The assumptions of the Black–Scholes model are not all empirically valid. The model is widely employed as a useful approximation to reality, but proper application requires understanding its limitations – blindly following the model exposes the user to unexpected risk.^{[30]} Among the most significant limitations are:
 the underestimation of extreme moves, yielding tail risk, which can be hedged with outofthemoney options;
 the assumption of instant, costless trading, yielding liquidity risk, which is difficult to hedge;
 the assumption of a stationary process, yielding volatility risk, which can be hedged with volatility hedging;
 the assumption of continuous time and continuous trading, yielding gap risk, which can be hedged with Gamma hedging.
In short, while in the Black–Scholes model one can perfectly hedge options by simply Delta hedging, in practice there are many other sources of risk.
Results using the Black–Scholes model differ from real world prices because of simplifying assumptions of the model. One significant limitation is that in reality security prices do not follow a strict stationary lognormal process, nor is the riskfree interest actually known (and is not constant over time). The variance has been observed to be nonconstant leading to models such as GARCH to model volatility changes. Pricing discrepancies between empirical and the Black–Scholes model have long been observed in options that are far outofthemoney, corresponding to extreme price changes; such events would be very rare if returns were lognormally distributed, but are observed much more often in practice.
Nevertheless, Black–Scholes pricing is widely used in practice,^{[3]}^{:751}^{[31]} because it is:
 easy to calculate
 a useful approximation, particularly when analyzing the direction in which prices move when crossing critical points
 a robust basis for more refined models
 reversible, as the model's original output, price, can be used as an input and one of the other variables solved for; the implied volatility calculated in this way is often used to quote option prices (that is, as a quoting convention).
The first point is selfevidently useful. The others can be further discussed:
Useful approximation: although volatility is not constant, results from the model are often helpful in setting up hedges in the correct proportions to minimize risk. Even when the results are not completely accurate, they serve as a first approximation to which adjustments can be made.
Basis for more refined models: The Black–Scholes model is robust in that it can be adjusted to deal with some of its failures. Rather than considering some parameters (such as volatility or interest rates) as constant, one considers them as variables, and thus added sources of risk. This is reflected in the Greeks (the change in option value for a change in these parameters, or equivalently the partial derivatives with respect to these variables), and hedging these Greeks mitigates the risk caused by the nonconstant nature of these parameters. Other defects cannot be mitigated by modifying the model, however, notably tail risk and liquidity risk, and these are instead managed outside the model, chiefly by minimizing these risks and by stress testing.
Explicit modeling: this feature means that, rather than assuming a volatility a priori and computing prices from it, one can use the model to solve for volatility, which gives the implied volatility of an option at given prices, durations and exercise prices. Solving for volatility over a given set of durations and strike prices, one can construct an implied volatility surface. In this application of the Black–Scholes model, a coordinate transformation from the price domain to the volatility domain is obtained. Rather than quoting option prices in terms of dollars per unit (which are hard to compare across strikes, durations and coupon frequencies), option prices can thus be quoted in terms of implied volatility, which leads to trading of volatility in option markets.
The volatility smile
One of the attractive features of the Black–Scholes model is that the parameters in the model other than the volatility (the time to maturity, the strike, the riskfree interest rate, and the current underlying price) are unequivocally observable. All other things being equal, an option's theoretical value is a monotonic increasing function of implied volatility.
By computing the implied volatility for traded options with different strikes and maturities, the Black–Scholes model can be tested. If the Black–Scholes model held, then the implied volatility for a particular stock would be the same for all strikes and maturities. In practice, the volatility surface (the 3D graph of implied volatility against strike and maturity) is not flat.
The typical shape of the implied volatility curve for a given maturity depends on the underlying instrument. Equities tend to have skewed curves: compared to atthemoney, implied volatility is substantially higher for low strikes, and slightly lower for high strikes. Currencies tend to have more symmetrical curves, with implied volatility lowest atthemoney, and higher volatilities in both wings. Commodities often have the reverse behavior to equities, with higher implied volatility for higher strikes.
Despite the existence of the volatility smile (and the violation of all the other assumptions of the Black–Scholes model), the Black–Scholes PDE and Black–Scholes formula are still used extensively in practice. A typical approach is to regard the volatility surface as a fact about the market, and use an implied volatility from it in a Black–Scholes valuation model. This has been described as using "the wrong number in the wrong formula to get the right price".^{[32]} This approach also gives usable values for the hedge ratios (the Greeks). Even when more advanced models are used, traders prefer to think in terms of Black–Scholes implied volatility as it allows them to evaluate and compare options of different maturities, strikes, and so on. For a discussion as to the various alternative approaches developed here, see Financial economics § Challenges and criticism.
Valuing bond options
Black–Scholes cannot be applied directly to bond securities because of pulltopar. As the bond reaches its maturity date, all of the prices involved with the bond become known, thereby decreasing its volatility, and the simple Black–Scholes model does not reflect this process. A large number of extensions to Black–Scholes, beginning with the Black model, have been used to deal with this phenomenon.^{[33]} See Bond option: Valuation.
Interestrate curve
In practice, interest rates are not constant – they vary by tenor (coupon frequency), giving an interest rate curve which may be interpolated to pick an appropriate rate to use in the Black–Scholes formula. Another consideration is that interest rates vary over time. This volatility may make a significant contribution to the price, especially of longdated options.This is simply like the interest rate and bond price relationship which is inversely related.
Short stock rate
It is not free to take a short stock position. Similarly, it may be possible to lend out a long stock position for a small fee. In either case, this can be treated as a continuous dividend for the purposes of a Black–Scholes valuation, provided that there is no glaring asymmetry between the short stock borrowing cost and the long stock lending income.^{[citation needed]}
Criticism and comments
Espen Gaarder Haug and Nassim Nicholas Taleb argue that the Black–Scholes model merely recasts existing widely used models in terms of practically impossible "dynamic hedging" rather than "risk", to make them more compatible with mainstream neoclassical economic theory.^{[34]} They also assert that Boness in 1964 had already published a formula that is "actually identical" to the Black–Scholes call option pricing equation.^{[35]} Edward Thorp also claims to have guessed the Black–Scholes formula in 1967 but kept it to himself to make money for his investors.^{[36]} Emanuel Derman and Nassim Taleb have also criticized dynamic hedging and state that a number of researchers had put forth similar models prior to Black and Scholes.^{[37]} In response, Paul Wilmott has defended the model.^{[31]}^{[38]}
British mathematician Ian Stewart published a criticism in which he suggested that "the equation itself wasn't the real problem" and he stated a possible role as "one ingredient in a rich stew of financial irresponsibility, political ineptitude, perverse incentives and lax regulation" due to its abuse in the financial industry.^{[39]}
In his 2008 letter to the shareholders of Berkshire Hathaway, Warren Buffett wrote: "I believe the Black–Scholes formula, even though it is the standard for establishing the dollar liability for options, produces strange results when the longterm variety are being valued... The Black–Scholes formula has approached the status of holy writ in finance ... If the formula is applied to extended time periods, however, it can produce absurd results. In fairness, Black and Scholes almost certainly understood this point well. But their devoted followers may be ignoring whatever caveats the two men attached when they first unveiled the formula."^{[40]}
See also
 Binomial options model, a discrete numerical method for calculating option prices
 Black model, a variant of the Black–Scholes option pricing model
 Black Shoals, a financial art piece
 Brownian model of financial markets
 Financial mathematics (contains a list of related articles)
 Fuzzy payoff method for real option valuation
 Heat equation, to which the Black–Scholes PDE can be transformed
 Jump diffusion
 Monte Carlo option model, using simulation in the valuation of options with complicated features
 Multiplicative calculus
 Real options analysis
 Stochastic volatility
Notes
 ^ Although the original model assumed no dividends, trivial extensions to the model can accommodate a continuous dividend yield factor.
References
 ^ "Scholes on merriamwebster.com". Retrieved March 26, 2012.
 ^ MacKenzie, Donald (2006). An Engine, Not a Camera: How Financial Models Shape Markets. Cambridge, MA: MIT Press. ISBN 0262134608.
 ^ ^{a} ^{b} Bodie, Zvi; Alex Kane; Alan J. Marcus (2008). Investments (7th ed.). New York: McGrawHill/Irwin. ISBN 9780073269672.
 ^ Taleb, 1997. pp. 91 and 110–111.
 ^ Mandelbrot & Hudson, 2006. pp. 9–10.
 ^ Mandelbrot & Hudson, 2006. p. 74
 ^ Mandelbrot & Hudson, 2006. pp. 72–75.
 ^ Derman, 2004. pp. 143–147.
 ^ Thorp, 2017. pp. 183–189.
 ^ https://www.nobelprize.org/nobel_prizes/economicsciences/laureates/1997/press.html
 ^ "Nobel Prize Foundation, 1997" (Press release). October 14, 1997. Retrieved March 26, 2012.
 ^ Black, Fischer; Scholes, Myron. "The Pricing of Options and Corporate Liabilities". Journal of Political Economy. 81 (3): 637–654. doi:10.1086/260062.
 ^ Merton, Robert. "Theory of Rational Option Pricing". Bell Journal of Economics and Management Science. 4 (1): 141–183. doi:10.2307/3003143.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} Nielsen, Lars Tyge (1993). "Understanding N(d_{1}) and N(d_{2}): RiskAdjusted Probabilities in the Black–Scholes Model" (PDF). Revue Finance (Journal of the French Finance Association). 14 (1): 95–106. Retrieved Dec 8, 2012, earlier circulated as INSEAD Working Paper 92/71/FIN (1992); abstract and link to article, published article.
 ^ Don Chance (June 3, 2011). "Derivation and Interpretation of the Black–Scholes Model" (PDF). Retrieved March 27, 2012.
 ^ Hull, John C. (2008). Options, Futures and Other Derivatives (7th ed.). Prentice Hall. ISBN 0135052831.
 ^ Although with significant algebra; see, for example, HongYi Chen, ChengFew Lee and Weikang Shih (2010). Derivations and Applications of Greek Letters: Review and Integration, Handbook of Quantitative Finance and Risk Management, III:491–503.
 ^ "Extending the Black Scholes formula". finance.bi.no. October 22, 2003. Retrieved July 21, 2017.
 ^ André Jaun. "The Black–Scholes equation for American options". Retrieved May 5, 2012.
 ^ Bernt Ødegaard (2003). "Extending the Black Scholes formula". Retrieved May 5, 2012.
 ^ Don Chance (2008). "ClosedForm American Call Option Pricing: RollGeskeWhaley" (PDF). Retrieved May 16, 2012.
 ^ Giovanni BaroneAdesi & Robert E Whaley (June 1987). "Efficient analytic approximation of American option values". Journal of Finance. 42 (2): 301–20. doi:10.2307/2328254.
 ^ Bernt Ødegaard (2003). "A quadratic approximation to American prices due to BaroneAdesi and Whaley". Retrieved June 25, 2012.
 ^ Don Chance (2008). "Approximation Of American Option Values: BaroneAdesiWhaley" (PDF). Retrieved June 25, 2012.
 ^ Petter Bjerksund and Gunnar Stensland, 2002. Closed Form Valuation of American Options
 ^ American options
 ^ Hull, John C. (2005). Options, Futures and Other Derivatives. Prentice Hall. ISBN 0131499084.
 ^ Breeden, D. T., & Litzenberger, R. H. (1978). Prices of statecontingent claims implicit in option prices. Journal of business, 621651.
 ^ Gatheral, J. (2006). The volatility surface: a practitioner's guide (Vol. 357). John Wiley & Sons.
 ^ Yalincak, Hakan, "Criticism of the Black–Scholes Model: But Why Is It Still Used? (The Answer is Simpler than the Formula)" <<http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2115141>>
 ^ ^{a} ^{b} Paul Wilmott (2008): In defence of Black Scholes and Merton Archived 20080724 at the Wayback Machine, Dynamic hedging and further defence of Black–Scholes^{[permanent dead link]}
 ^ Riccardo Rebonato (1999). Volatility and correlation in the pricing of equity, FX and interestrate options. Wiley. ISBN 0471899984.
 ^ Kalotay, Andrew (November 1995). "The Problem with Black, Scholes et al" (PDF). Derivatives Strategy.
 ^ Espen Gaarder Haug and Nassim Nicholas Taleb (2011). Option Traders Use (very) Sophisticated Heuristics, Never the Black–Scholes–Merton Formula. Journal of Economic Behavior and Organization, Vol. 77, No. 2, 2011
 ^ Boness, A James, 1964, Elements of a theory of stockoption value, Journal of Political Economy, 72, 163–175.
 ^ A Perspective on Quantitative Finance: Models for Beating the Market, Quantitative Finance Review, 2003. Also see Option Theory Part 1 by Edward Thorpe
 ^ Emanuel Derman and Nassim Taleb (2005). The illusions of dynamic replication, Quantitative Finance, Vol. 5, No. 4, August 2005, 323–326
 ^ See also: Doriana Ruffinno and Jonathan Treussard (2006). Derman and Taleb's The Illusions of Dynamic Replication: A Comment, WP2006019, Boston University  Department of Economics.
 ^ Ian Stewart (2012) The mathematical equation that caused the banks to crash, The Observer, February 12.
 ^ [1]
Primary references
 Black, Fischer; Myron Scholes (1973). "The Pricing of Options and Corporate Liabilities". Journal of Political Economy. 81 (3): 637–654. doi:10.1086/260062. [2] (Black and Scholes' original paper.)
 Merton, Robert C. (1973). "Theory of Rational Option Pricing". Bell Journal of Economics and Management Science. The RAND Corporation. 4 (1): 141–183. doi:10.2307/3003143. JSTOR 3003143. [3]
 Hull, John C. (1997). Options, Futures, and Other Derivatives. Prentice Hall. ISBN 0136015891.
Historical and sociological aspects
 Bernstein, Peter (1992). Capital Ideas: The Improbable Origins of Modern Wall Street. The Free Press. ISBN 0029030129.
 Derman, Emanuel. "My Life as a Quant" John Wiley & Sons, Inc. 2004. ISBN 0471394203
 MacKenzie, Donald (2003). "An Equation and its Worlds: Bricolage, Exemplars, Disunity and Performativity in Financial Economics". Social Studies of Science. 33 (6): 831–868. doi:10.1177/0306312703336002. [4]
 MacKenzie, Donald; Yuval Millo (2003). "Constructing a Market, Performing Theory: The Historical Sociology of a Financial Derivatives Exchange". American Journal of Sociology. 109 (1): 107–145. CiteSeerX 10.1.1.461.4099. doi:10.1086/374404. [5]
 MacKenzie, Donald (2006). An Engine, not a Camera: How Financial Models Shape Markets. MIT Press. ISBN 0262134608.
 Mandelbrot & Hudson, "The (Mis)Behavior of Markets" Basic Books, 2006. ISBN 9780465043552
 Szpiro, George G., Pricing the Future: Finance, Physics, and the 300Year Journey to the Black–Scholes Equation; A Story of Genius and Discovery (New York: Basic, 2011) 298 pp.
 Taleb, Nassim. "Dynamic Hedging" John Wiley & Sons, Inc. 1997. ISBN 0471152803
 Thorp, Ed. "A Man for all Markets" Random House, 2017. ISBN 9781400067961
Further reading
 Haug, E. G (2007). "Option Pricing and Hedging from Theory to Practice". Derivatives: Models on Models. Wiley. ISBN 9780470013229. The book gives a series of historical references supporting the theory that option traders use much more robust hedging and pricing principles than the Black, Scholes and Merton model.
 Triana, Pablo (2009). Lecturing Birds on Flying: Can Mathematical Theories Destroy the Financial Markets?. Wiley. ISBN 9780470406755. The book takes a critical look at the Black, Scholes and Merton model.
External links
Discussion of the model
 Ajay Shah. Black, Merton and Scholes: Their work and its consequences. Economic and Political Weekly, XXXII(52):3337–3342, December 1997
 The mathematical equation that caused the banks to crash by Ian Stewart in The Observer, February 12, 2012
 When You Cannot Hedge Continuously: The Corrections to Black–Scholes, Emanuel Derman
 The Skinny On Options TastyTrade Show (archives)
Derivation and solution
 Derivation of the Black–Scholes Equation for Option Value, Prof. Thayer Watkins
 Solution of the Black–Scholes Equation Using the Green's Function, Prof. Dennis Silverman
 Solution via risk neutral pricing or via the PDE approach using Fourier transforms (includes discussion of other option types), Simon Leger
 Stepbystep solution of the Black–Scholes PDE, planetmath.org.
 The Black–Scholes Equation Expository article by mathematician Terence Tao.
Computer implementations
 Black–Scholes in Multiple Languages
 Black–Scholes in Java moving to link below
 Black–Scholes in Java
 Chicago Option Pricing Model (Graphing Version)
 Black–Scholes–Merton Implied Volatility Surface Model (Java)
 Online Black–Scholes Calculator
Historical
 Trillion Dollar Bet—Companion Web site to a Nova episode originally broadcast on February 8, 2000. "The film tells the fascinating story of the invention of the Black–Scholes Formula, a mathematical Holy Grail that forever altered the world of finance and earned its creators the 1997 Nobel Prize in Economics."
 BBC Horizon A TVprogramme on the socalled Midas formula and the bankruptcy of LongTerm Capital Management (LTCM)
 BBC News Magazine Black–Scholes: The maths formula linked to the financial crash (April 27, 2012 article)