The Black-Scholes-Merton model provides a mathematical formula for estimating the price of call and put options based on certain variables. It assumes stock prices follow a log-normal distribution and uses variables like the current stock price, strike price, risk-free interest rate, time to expiration, and implied volatility to estimate an option's price. While widely used, it relies on assumptions that are not always accurate to real market conditions, such as constant volatility and a log-normal stock price distribution.

1. Black-Scholes-Merton
Christopher Williams, Patrick Corn, and Jimin Khim contributed
The Black-Scholes-Merton model, sometimes just called the Black-Scholes model,
is a mathematical model of financial derivative markets from which the Black-Scholes
formula can be derived. This formula estimates the prices of call and put options.
Originally it priced European options, and was the first widely adopted mathematical
formula for pricing options. Some credit this model for the significant increase in options
trading, and name it a significant influence in modern financial pricing. Prior to invention
of this formula and model, options traders didn't all use a consistent mathematical way
to value options, and empirical analysis has shown that price estimates produced by this
formula are close to observed prices.
In their initial formulation of the model, Fischer Black and Myron Scholes (the
economists who originally formulated the model) came up with a partial differential
equation known as the Black-Scholes equation[1]
, and later Robert Merton published a
mathematical understanding of their model, using stochastic calculus[2]
that helped
formulate what became known as the Black-Scholes-Merton formula. Both Myron
Scholes and Robert Merton split the 1997 Nobel Prize in Economists, listing Fischer
Black as a contributor, though he was ineligible for the prize as he had passed away
before it was awarded.
Roughly, their model determines the price of an option by calculating the return an
investor gets less the amount that investor has to pay, using log-normal
distributionprobabilities to account for volatility in the underlying asset. The log-normal
distribution of returns used in the model are based on theories of brownian motion, with
asset prices exhibiting similar behavior to organic movement in Brownian motion.
The formula helped legitimize options trading, making it seem less like gambling and
more like science. Today the Black-Scholes-Merton formula is widely used, though in
individually modified ways, by traders and investors. As is its fundamental strategy of
hedging to best control, or "eliminate", risks associated with volatility in the assets that
underlie the option.
Contents
 The Black-Scholes-Merton Formula
 On Volatility
 High-level Explanation of the Black-Scholes-Merton Formula
 Example + Problem
 Hedging to "eliminate" risk
 Criticisms of the Black-Scholes-Merton Model

2.  References
The Black-Scholes-Merton Formula
Again, the Black-Scholes-Merton formula is an estimate of the prices of European call
and put options, with the core difference between American and European options being
that European options can only be exercised on their one exercise date versus American
call options that can be exercised any time up to that expiration date. It's also used only
to determine prices of non-dividend paying assets.
The Black–Scholes-Merton formula of value for a European call option is:
(note: the formula for a European put option is similar)
where
 is the stock price.
 is the price of the call option as a formulation of the stock price and time.
 is the exercise price.
 is the time to maturity, i.e. the exercise date, , less the amount of time between
now, , and then. Generally, this is represented in years with one month
equaling or .
 and are cumulative distribution functions for a standard normal distribution with
the following formulation:where
 represents the underlying volatility (a standard deviation of log returns),
and
 is the risk-free interest rate, i.e. the rate of return an investor could get on
an investment assumed to be risk free (like a T-bill).
On Volatility

3. Price of an Oil and a Cow ETF over three years.
Volatility, in the case of financial assets, is the measure of how much and how quickly
the asset's price changes. It's a measure of uncertainty. If traders were certain that an
asset was worth a certain amount, then they'd buy at that price and sell below it. They'd
sit on it. But highly uncertain assets get traded at a wider range of prices. Implied
volatility, what options use, is the value of the volatility of the underlying asset.
Prior to the Black-Scholes-Merton formula, investors had their own ways of estimating
the price of options. These methods varied, but generally they incorporated some
measure of implied volatility. Stocks with more volatility had a higher chance of having
a very high value in the future, or a very low value. In the above graph, an OIL ETF and
an ETF for Live Cattle (COW) are graphed, with the Oil ETF being a much more volatile
asset (spiking up and down more). The price not only decreases more drastically (larger
slope in the trend line), but on a daily basis the stock fluctuates more significantly (the
price fluctuates from the trend line more drastically).
Because of the way options work, the buyer of a call only makes money if an asset is
above the strike price. If it's below that price, they don't care how much below the strike
it is, they have spent the same amount. But they do care how much above the strike
price it is. As such, highly volatile assets (options with higher implied volatility) are more
likely to make investors more money, and are more valuable.
Technically, and in the case of the Black-Scholes-Merton model, implied volatility is the
annualized standard deviation of the return on the asset, and is expressed as a decimal
percentage. This will be explained more below. But in the B-S-M formula, is both a
measure of implied volatility and the standard deviation. This is because, in a measure
of possible returns for an asset, highly volatile assets will have a wider standard
deviation than less volatile ones. The graph below shows the periodic daily returns for
the previous Cow and Oil ETF, essentially a count of how many days the price changed ,
or , or , etc. Because the Cow ETF is a less volatile stock, the graph of its normal
distribution is narrower, and the standard deviation is lower at ~; expressed as a
percentage that's from the mean, meaning that the price of the ETF on any given day

4. (because this is a graph of periodic daily returns) is highly unlikely to more than than
the mean. One standard deviation in a normal distribution is , and the mean was , so the
expect prices for of the days was . And indeed Compared to this, the Oil ETF's graph is
much wider; in fact it goes beyond the current -axis of this graph (there is a wider graph
of this same ETF over this same time period in the section below). It has a standard
deviation of , or . The Oil ETF had a mean of , so for of the days prices were: , a much
wider range of prices (both absolutely and relatively).
High-level Explanation of the Black-Scholes-Merton Formula
Overall: Intuitively, and roughly, the Black-Scholes-Merton formula subtracts , the
exercise price discounted back to present value times the probability that the option is
above the strike price at maturity, from , the stock price today times a probability that
is if the stock is below the strike price but is some probability representing the stock's
value if it's above the strike price. Roughly, it's an investor's return, minus the cost of the
option.
Discounting to Present Value: The portion of the formulation is simply a calculation
of the present value of that strike price. It compounds the risk-free interest rate over the
period between now (when the calculation is done) and the future expiration date. This
is done because the price of this option should reflect that alternative risk-free choice an
investor has. If an investor could put some money in a risk-free T-Bill and get a return
over one year, then an option needs to generate additional return above and beyond
that to justify the increased risk associated with it.

5. The periodic daily returns for an Oil ETF for every day of the three years between Nov 1, 2013 and Nov 1, 2016.
Roughly, this distribution shows the amount of each day's gain or loss and the number of times that gain/loss
happened. For instance, there were 98 days with 0% gain or loss in this 755 trading day period.
Probability: Those probability weightings, and , come from a normal probability
distribution curve. If an investor graphed the periodic daily returns (the returns for this
option each day) the resulting graph would be a normal distribution, a bell shaped curve,
like the one for the Oil ETF to the right. Just as the historical prices were normally
distributed, the B-S-M model assumes that future prices will be normally distributed.
Therefore is, roughly, looking for , the area under the bell curve up to some -score, or
the probability that the future price will be above the strike price on the expiration date.
The standard notation for -score is

6. The and functions are simply calculations of area on the curve. For instance a
calculation of for this Oil ETF is represented in the image to the right. If the curve is the
normal distribution of all probabilities for the option, then is the percentage of
probabilities that the option will expire in the money.
Example + Problem
Given the complexity of the model it's always good to see it in action:
Smart investors calculate the price of an option for themselves, before they buy. If you
have the chance to buy a European call option with the following parameters, what cost
should you pay less than to make it worth it?
 stock price: $50
 strike price: $45
 time to expiration: 80 days
 risk-free interest rate: 2%
 implied volatility: 30%
In other words, using the B-S-M formula, what should the cost of this call option be?
While the formula is involved, this is essentially a matter of plugging in the given
variables:
and can be found by looking at a -score table:
Note: In the case of this stock, there is a probability of that represents the expected
value at expiration, which is multiplied by to yield a return. The strike price is and its
present value is which is multiplied by probability of the call option expiring in the
money, to yield .

7. 10.0%25.2%1.0%50.6%16.5%15.7%25.4%
Suppose you're an investor and are curious what the market thinks the implied volatility
of the S&P 500 is today. You know a few things:
 You can assume that every other
investor is using black-scholes-merton
formula for pricing.
 You look a common ETF of the S&P
500, the SPY spider.
 Today it's priced at: $216
 A European Call Option has a strike
price of: $210
 To expire: 30 days from today.
 The risk-free interest rate is: 1.8%
 The market is pricing this European call
option at: $7.93
What is the implied volatility?
(all answers are truncated)
Note: Using Excel's "Goal Seek" may be helpful, as would guessing and adjusting in a
manner consistent with the newton raphson method.
Hedging to "eliminate" risk
Once an asset is priced, the key idea is to hedge the option by buying and selling the
underlying asset in just the right way so as to "eliminate risk". This is referred to as delta
hedging or dynamic hedging. The idea is to maintain a zero option greeks - delta, where
delta is the sensitivity of an option to changes in the price of the underlying assets. This
is a fairly complex form of hedging, and is principally performed by large investment
institutions (investment banks, hedge funds, private equity funds, etc.). The formal
calculation for delta is:That is the first derivative of the value of the option over the first
derivative of the value of the underlying asset. As such, the basic strategy of delta
hedging is to buy or sell some of the underlying asset (the denominator) in responses to
changes in the value of that asset, it is to keep static, even though the value of that
asset change regularly.
One important note, is that the risk eliminated here is not the risk that the underlying
asset will go down in value, this is not to prevent normal negative returns from assets
simply not performing, but to eliminate the more sharp shifts in price not correlated to
changed in underlying value - the tail ends of that periodic daily return graph above
where in a single day the price of an asset can change significantly, and then correct

8. back to the original price in following days. Also, risk is never really "eliminated". That is
the goal, but there are always risks, like default risk, that are harder to control for.
Criticisms of the Black-Scholes-Merton Model
Nassim Nicholas Taleb, famous for his 2007 bestselling book "Black Swan" which
discussed unpredictable events in financial markets, along with Espen Gaarder Haug
have criticized the Black-Scholes-Merton model, saying that it is "fragile to jumps and
tail events" and can only handle "mild randomness."[3]
. This is one of a few known
challenges to the model:
 Fragility to "tail-risk" or other extreme randomness: In general, returns do
not absolutely follow a normal distribution. The p-value on the Anderson-Darling
Normality Test is 0.000 when applied to S&P returns, showing that market returns
are leptokurtic (having greater kurtosis, or more concentrated about the mean
with fat tails.)[4]
 The structure of B-S-M doesn't reflect present realities: The B-S-M model
assumes a market using European Call options, when most options traded today
are American call options that can be sold at any point. It also does not allow for
dividends, something that is commonly found in options.
 Assumption of a risk free interest rate: A theoretical calculation of risk-free
rates is hard to come up with and in practice, investors use proxies like the long-
term yield on the US Treasury coupon bonds (generally 10 year bonds). However
this assumes that US Treasury bonds are "risk-free", when a more accurate
statement would be that they're what the market assumes are the least risky
investment vehicles.
 Assumption of cost-less trading: Trading generally comes with exchange fees,
the costs to buy or sell stocks and options, and the cost of time, the time it takes
for the order to go through may result in changes to the price on the market.
These costs can be managed, but are not included in the model.
 Gap-Risk: Also the model assumes that trading occurs continuously, unlike
reality, where markets shut down for the night and then can reopen at significantly
different prices to reflect new information.
Empirically, significant pricing discrepancies between B-S-M and reality have occurred
more often than if returns were log-normal. But the B-S-M model continues to be used.
It is simple, easy to determine and can be adjusted for various inadequacies.

9. A Volatility Smile[5]
One of the more common criticisms of the B-S-M model is the existence of a volatility
smile. The Black-Scholes-Merton pricing model suggests a constant volatility and log-
normal distributions of returns, where, in reality, implied volatility varies widely. Options
whose strike price are said to be "deep-in-the-money" or "out-of-the-money", i.e. whose
strike price is further away from the assumed underlying asset price, command higher
prices than a flat volatility would suggest - their implied volatility is higher.
References
1. Black, F., & Scholes, M. The Pricing of Options and Corporate Liabilities.
Retrieved November 1, 2016, from http://www.jstor.org/stable/1831029
2. Merton, R. Theory of Rational Option Pricing. Retrieved November 1, 2016,
from http://www.jstor.org/stable/3003143
3. Haug, E., & Taleb, N. Why We Have Never Used the Black–Scholes–Merton
Option Pricing Formula. Retrieved November 9th 2016,
from http://polymer.bu.edu/hes/rp-haug08.pdf
4. Hurvich, C. SOME DRAWBACKS OF BLACK-SCHOLES. Retrieved November
9th 2016,
from http://people.stern.nyu.edu/churvich/Forecasting/Handouts/Scholes.pdf
5. Brianegge, . Volatility Smile. Retrieved November 9th 2016,
from https://en.wikipedia.org/wiki/File:Volatility_smile.svg
Cite as: Black-Scholes-Merton. Brilliant.org. Retrieved 10:5