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Language: en
this single equation spawned four
multi-trillion dollar Industries and
transformed everyone's approach to risk
do you think that most people are aware
of the size scale utility of derivatives
no no idea but at its core this equation
comes from physics from discovering
atoms understanding how heat is
transferred and how to beat the casino
at Blackjack so maybe it shouldn't be
surprising that some of the best to beat
the stock market were not veter Traders
but physicists scientists and
mathematicians in 1988 a mathematics
professor named Jim Simons set up The
Medallion investment fund and every year
for the next 30 Years The Medallion fund
delivered higher returns than the market
average and not just by a little bit it
returned 66% per year at that rate of
growth $100 invested in 1988 would be
worth $8.4 billion today this made Jim
Simons easily the richest mathematician
of all time but being good at math
doesn't guarantee success in financial
markets just ask Isaac
Newton in 1720 Newton was 77 years old
and he was rich he had made a lot of
money working as a professor at
Cambridge for decades and he had a side
hustle as the master of the Royal Mint
his net worth was
£30,000 the equivalent of $6 million
today
now to grow his fortune Newton invested
in stocks one of his big bets was on the
south sea company their business was
shipping enslaved Africans across the
Atlantic business was booming and the
share price grew rapidly by April of
1720 the value of Newton's shares had
doubled so he sold his stock but the
stock price kept going up and by June
Newton bought back in and he kept buying
shares even as the price peaked
when the price started to fall Newton
didn't sell he bought more shares
thinking he was buying the dip but there
was no rebound and ultimately he lost
around a third of his wealth when asked
why he didn't see it coming Newton
responded I can calculate the Motions of
the heavenly bodies but not the madness
of
people so what did Simons get right that
Newton got
wrong well for one thing Simons was able
to stand on the the shoulders of
giants the pioneer of using math to
model financial markets was Lou bashier
born in 1870 both of his parents died
when he was 18 and he had to take over
his father's wine business he sold the
business a few years later and moved to
Paris to study Physics but he needed a
job to support himself and his family
and he found one at the borse the Paris
Stock Exchange and inside was Newton's
Madness of people in its rawest form
hundreds of Trad ERS screaming prices
making hand signals and doing deals the
thing that captured Bell's interest were
contracts known as
options the earliest known options were
bought around 600 BC by the Greek
philosopher thies of militus he believed
that the coming summer would yield a
bumper crop of Olives to make money off
this idea he could have purchased Olive
presses which if he were right would be
in great demand but he didn't have
enough money to buy the machines so
instead he went to all the existing
Olive press owners and paid them a
little bit of money to secure the option
to rent their presses in the summer for
a specified price when the Harvest came
th was right there were so many olives
that the price of renting a press
skyrocketed thies paid the Press owners
their pre-agreed price and then he
rented out the machines at a higher rate
and pocketed the difference th had
executed the first known call
option A call option gives you the right
but not the obligation to buy something
at a later date for a set price known as
the strike price you can also buy a put
option which gives you the right but not
the obligation to sell something at a
later date for the strike price put
options are useful if you expect the
price to go down call options are useful
if you expect the price to go up for
example let's say the current price of
Apple stock is $100 but you expect it to
go up you could buy a call option for
$10 that gives you the right but not the
obligation to buy Apple stock in one
year for $100 that is the strike price
just a little side note American options
can be exercised on any date up to the
expiry whereas European options must be
exercised on the expiry date to keep
things simple we'll stick to European
options so if in a year the price of
Apple stock has gone up to
$130 you can use the option to buy
shares for $100 and then immediately
sell them for $130 after you take take
into account the $10 you paid for the
option you've made a $20 profit
alternatively if in a year the stock
price has dropped to $70 you just
wouldn't use the option and you've lost
the $10 you paid for it so the profit
and loss diagram looks like this if the
stock price ends up below the strike
price you lose what you paid for the
option but if the stock price is higher
than the strike price then you earn that
difference minus the cost of the
option there are at least least three
advantages of options one is that it
limits your downside if you had bought
the stock instead of the option and it
went down to $70 you would have lost $30
and in theory you could have lost 100 if
the stock went to zero the second
benefit is options provide leverage if
you had bought the stock and it went up
to $130 then your investment grew by 30%
but if you had bought the option you
only had to put up $10 so your profit of
$20 is actually a 200% return on
investment on the downside if you had
owned the stock your investment would
have only dropped by 30% whereas with
the option you lose all
100% so with options trading there's a
chance to make much larger profits but
also much bigger losses the third
benefit is you can use options as a
hedge I think the original motivation
for options was to figure out a way to
reduce risk and then of course once
people decided they wanted to buy
insurance that meant that there are
other people out there that wanted to
sell it for a profit and that's how
markets get
created so options can be an incredibly
useful investing tool but what bellier
saw on the trading floor was chaos
especially when it came to the price of
stock options even though they had been
around for hundreds of years no one had
found a good way to price them Traders
would just bargain to come to an
agreement about what the price should be
given the option to buy or sell
something in the future seems like a
very amorphous kind of a trade and so
coming up with prices for these rather
strange objects has been a challenge
that's plagued a number of economists
and business people for
centuries now Basher already interested
in probability thought there had to be a
mathematical solution to this problem
and he proposed this as his PhD topic to
his adviser HRI panker looking into the
math of Finance wasn't really something
people did back then but to bisher's
surprise panker agreed to accurately
price an option first you need to know
what happens to stock prices over time
the price of a stock is basically set by
a tug-of-war between buyers and sellers
when more people want to buy a stock the
price goes up when more people want to
sell a stock the price goes down but the
number of buyers and sellers can be
influenced by almost anything like the
weather politics new competitors
innovation and so on so bellier realized
that it's virtually impossible to
predict all these factors accurately so
the best you can do is assume that at
any point in time the stock price is
just as likely to go up as down and
therefore over the long term stock
prices follow a random walk moving up
and down as if their next move is
determined by the flip of a coin
Randomness is a Hallmark of an efficient
market by efficient Economist typically
mean that you can't make money by
trading the idea that you shouldn't be
able to buy an asset and sell it
immediately for a profit is known as the
efficient market hypothesis the more
people try to make money by predicting
the stock market and then trading on
those predictions the less predictable
those prices are if you and I could
predict the stock market tomorrow then
we would do it we would start trading
today on stocks that we thought were
going to go up tomorrow well if we did
that then instead of going up tomorrow
they would go up now as we bought more
and more of the stock so the very Act of
predicting actually affects the quality
of the future outcomes and so in a
totally efficient market the the prices
tomorrow can't possibly have any
predictive power if they did we would
have taken advantage of it
today this is a gon board it's got rows
of pegs arranged in a triangle and
around 6,000 tiny ball bearings that I
can pour through the pegs now each time
a ball hits a peg there's a 50/50 chance
it goes to the left or the right so each
ball follows a random walk as it passes
through these pegs which makes it
basically impossible to predict the path
of any individual Ball but if I flip
this over what you can see is that all
the balls together always create a
predictable pattern that is a collection
of random walks creates a normal
distribution
it's centered around the middle because
the number of passive ball could take to
get here is the greatest and the further
out you go the fewer the passive ball
could take to get there like if you want
to end up here well the ball would have
to go left left left left all the way
down so there's only one way to get here
but to get into the middle there are
thousands of paths that a ball could
take now bellier believed a stock price
is just like a ball going through a gton
board each additional layer of pegs
represents a time step so after a short
time the stock price could only move up
or down a little but after more time a
wider range of prices is possible
according to Baler the expected future
price of a stock is described by a
normal distribution centered on the
current price which spreads out over
time Balia realized he had rediscovered
the exact equation which describes how
heat radiates from regions of high
temperature to regions of low
temperature this was first discovered by
Joseph Fier back in
1822 so bellier called his Discovery the
radiation of
probabilities since he was writing about
Finance the physics Community didn't
take any notice but the mathematics of
the random walk would go on to solve an
almost Century old mystery in
physics in 1827 Scottish botanist Robert
Brown was looking at pollen grains under
the microscope and he noticed that the
particles suspended in water on the
microscope slide were moving around
randomly because he didn't know whether
it was something to do with the pollen
being living material he tested
non-organic particles such as dust from
lava and meteorite Rock again he saw
them moving around in the same way so
Brown discovered that any particles if
they were small enough exhibited this
random movement which came to be known
as Brownian
motion but what caused it remained a
mystery 80 years later in 1905 Einstein
figured out the
[Music]
answer over the previous couple hundred
years the idea that gases and liquids
were made up of molecules became more
and more popular but not everyone was
convinced that molecules were real in a
physical sense just that the theory
explained a lot of
observations The Idea LED Einstein to
hypothesize that Brownian motion is
caused by the trillions of molecules
hitting the particle from every
direction every instant occasionally
more will hit from one side than the
other and the particle will momentarily
jump to derive the mathematics Einstein
supposed that as an observer we can't
see or predict these collisions with any
certainty so at any time we have to
assume that the particle is just as
likely to move in One Direction as
another so just like stock prices
microscopic particles move like a ball
falling down a gton board the expected
location of a particle is described by a
normal distribution which broadens with
time it's why even in completely Still
Water microscopic particles spread out
this is
diffusion by solving the browni and
motion mystery Einstein had found
definitive evidence that atoms and
molecules exist of course he had no idea
that bellier had uncovered the random
walk 5 years
earlier by the time bellier finished his
PhD he had finally figured out a
mathematical way to price an option
remember that with a call option if the
future price of if the stock is less
than the strike price then you lose the
premium paid for the option but if the
stock price is greater than the strike
price you pocket that difference and you
make a net profit if the stock has gone
up by more than you paid for the option
so the probability that an option buyer
makes a profit is the probability that
the price increases by more than the
price paid for it which is the green
shaded area and the probability that the
seller makes money is just the
probability that the price stays low
enough that the buyer doesn't earn more
than they paid for it this is the red
shaded area multiplying the profit or
loss by the probability of each outcome
the Shier calculated the expected return
of an
option now how much should it cost if
the price of an option is too high no
one will want to buy it conversely if
the price is too low everyone will want
to buy it bellier argued that the fair
price is what makes the expected return
for buyers and sellers equal both
parties should stand to gain or lose the
same amount that was B's insight into
how to accurately price an
option when bellier finished his thesis
he had beaten Einstein to inventing the
random walk and solved a problem that
had eluded options Traders for hundreds
of years but no one noticed the
physicists were uninterested and Traders
weren't ready the key thing missing was
a way to make a ton of
money hey so I'm not sure how stock
Traders sleep at night with billions of
dollars riding on The Madness of people
but I have been sleep sleeping just fine
thanks to the sponsor of this video
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video in the 1950s a young physics
graduate Ed Thorp was doing his PhD in
Los Angeles but a few hours drive away
Las Vegas was quickly becoming the
gambling capital of the world and Thorp
saw a way to make a fortune he headed to
Vegas and sat down at the blackjack
table back then the dealer only used a
single deck of cards so Thorp could keep
a mental note of all the cards that had
been played as he saw them this allowed
him to work out if he he had an
advantage he would bet a bigger portion
of his funds when the odds were in his
favor and less when they weren't he had
invented card counting this is a
remarkable Innovation considering
Blackjack had been around in various
forms for hundreds of years and for a
while this made him a lot of money but
the casinos got wise to his strategy and
they added more decks of cards to the
game to reduce the benefit of card
counting so Thorp took his winnings to
what he called the biggest casino on
Earth the stock
market he started a hedge fund that
would go on to make a 20% return every
year for 20 years the best performance
ever seen at that time and he did it by
transferring the skills he honed at the
blackjack table to the stock market
Thorp pioneered a type of hedging a way
to protect against losses with balancing
or compensating transactions Thorp did
it mathematically he looked at the odds
of winning and losing and decided that
under certain conditions you can
actually tilt the odds in your favor by
using certain patterns to be able to
make
bets suppose Bob sells Alice a call
option on a stock and let's say the
stock has gone up so now it's in the
money for Alice well now for every
additional $1 the stock price goes up
Bob will lose
$1 but he can eliminate this risk by
owning one unit of stock then if the
price goes up he would lose $1 from the
option but gain gain that dollar back
from the stock and if the stock drops
back out of the money for Alice he sells
the stock so he doesn't risk losing any
money from that either this is called
Dynamic hedging it means Bob can make a
profit with minimal risk from
fluctuating stock prices a hedged
portfolio Pi at any one time will offset
the option V with some amount of stock
Delta it basically means I can sell you
something without having to take the
opposite side of the trade and the way
to think about it is I have
synthetically manufactured an option for
you I've created it out of nothing by
doing Dynamic trading Dynamic hedging as
we saw with Bob's example Delta the
amount of stock he has to hold changes
depending on current prices
mathematically it represents how much
the current option price changes with a
change in the stock price but Thorp
wasn't satisfied with Bell's model for
pricing options mean I mean for one
thing stock prices aren't entirely
random they can increase over time if
the business is doing well or fall if it
isn't Bell's model ignored this so Thorp
came up with a more accurate model for
pricing options which took this drift
into account I actually figured out what
this model was back in the middle of
1967 and I decided that I would just use
it for myself and then later I kept it
quiet for my own investors the idea was
to basically make a lot of money out of
it for everybody his strategy was if the
option was going cheap according to his
model buy it if it was overvalued short
sell it that is bet against it and that
way more often than not he would end up
on the winning side of the
trade this lasted until
1973 in that year fiser black and Myon
scholes came up with an equation that
changed the industry Robert Merton
independently published his own own
version which was based on the
mathematics of stochastic calculus so he
is also credited I thought I'd have the
field to myself but unfortunately Fisher
black and Myon scholes published the
idea and they did a better job of the
model than I did because they had uh
very tight mathematics behind their
derivation like bellier they thought
that option prices should offer a fair
bet to both buyers and sellers but their
approach was totally new they said if it
was possible to construct a risk-free
portfolio of options and stocks just
like Thorp was doing with his Delta
hedging then in an efficient market a
fair market this portfolio should return
nothing more than the risk-free rate
what the same money would earn if
invested in the safest asset US Treasury
bonds the Assumption was that if you're
not taking on any additional risk then
it shouldn't be possible to receive any
extra
returns to describe how stock prices
change over time black scholes and meron
used an improved version of bachelier's
model just like Thorp this says that at
any time we expect the stock price to
move randomly plus a general Trend up or
down the drift by combining these two
equations black scholes and Merton came
up with the most famous equation in
finance it relates the price of any kind
of contract to any asset stocks bonds
you name it the same year they published
this equation the Chicago Board options
exchange was
founded why is that equation
so important like for finance how did
that change the game well because when
you solve that partial differential
equation you get an explicit formula for
the price of the option as a function of
a bunch of these input parameters and
for the very first time you now have an
explicit expression where you plug in
the parameters and out pops this number
so that people can actually use it to
trade on this led to one of the fastest
adoptions by industry of an academic
idea in all of the social Sciences
within just a couple of years the black
schs formula was adopted as The
Benchmark for Wall Street for trading
options the exchange traded options
Market has exploded and it's now a
multi-trillion dollar industry the
volume in this market has been doubling
roughly every 5 years so this is the
financial equivalent of Moore's Law
there are other businesses that have
grown just as quickly like credit
default swaps Market
the OTC derivatives Market the
securitized debt Market all of these are
multi-trillion dollar industries that in
one form or another make use of the idea
of black shes meron option pricing this
opened up a whole new way to hedge
against anything and not just for hedge
funds nowadays pretty much every large
company governments and even individual
investors use options to hedge against
their own specific risks suppose you're
running an airl line and you're worried
that an increase in oil prices would eat
into your profits well using the black
schols meron equation there's a way to
accurately and efficiently hedge that
risk you price an option to buy
something that tracks the price of oil
and that option will pay off if oil
prices go up and that will help
compensate you for the higher cost of
fuel you have to pay so black schs
Merton can help reduce risk but it can
also provide leverage an ongoing battle
between bullish day Traders and hedge
fund short sellers that have bet against
the stock gamestock shares have now
risen some
700% well GameStop is a really
interesting example for all sorts of
reasons but options figured prominently
in that example because a small Cadre of
users on this Reddit subchannel Wall
Street bets decided that the hedge fund
managers that were shorting the stock
and betting that the company would go
out of business needed to be punished
and so they bought shares of GameStop
stock to try to drive up the price turns
out that buying the stock was not enough
because with a dollar's worth of cash
you can buy a dollar's worth of stock
but with a dollar's worth of cash you
can buy options that affected many more
than a dollar worth of stock perhaps in
some cases 10 or $20 worth of stock for
a dollar's worth of options and so
there's natural leverage embedded in
these Securities and so the combination
of buying both the stock and the options
caused the prices to rise very quickly
and what that did was to cause these
hedge fund managers to lose a lot of
money quickly how big is this market for
derivatives how big is this whole area
that kind of comes out of black schles
Maron there are estimates of how large
derivatives markets are and first let's
be clear what a derivative is a
derivative is a Financial Security whose
value derives from another Financial
Security
so an option is an example of a
derivative in general the size of
derivative markets globally is on on the
order of several hundred trillion
dollars how does that compare to the
size of the underlying Securities
they're based on it's multiples of the
underlying Securities I just have to
interrupt because it seems kind of crazy
that you have more money riding on the
things that are based on the thing than
the thing itself that's right so so tell
me how that makes any sense because what
options allow you to do is to take the
underlying thing and turn it into 5 10
20 50 things so these pieces of paper
that we call options and derivatives
they basically allow us to create many
many different versions of the
underlying asset versions that
individuals find more palatable because
of their own risk reward
preferences does this make the markets
and the global economy more stable or
less stable or no
effect all three so it turns out that
during normal
times these markets are a very
significant source of liquidity and
therefore stability during abnormal
Times by that I mean when there are
periods of Market
stress all of these Securities can go in
One Direction typically down
and when they go down together that
creates a really big market crash so in
those circumstances derivatives markets
can exacerbate these kinds of Market
dislocations in 1997 Merton and scholes
were awarded the Nobel prize in
economics black was acknowledged for his
contributions but unfortunately he had
passed away just two years earlier we're
going to make a lot of money in options
but now black and shs have told
everybody what the secret is with the
option pricing formula now out for
everyone to see hedge funds would need
to discover better ways to find market
inefficiencies enter Jim
Simons before Simons had any exposure to
the stock market he was a mathematician
his work on rean Geometry was
instrumental in many areas of
mathematics and physics including knot
Theory Quantum field Theory and Quantum
Computing churn Simon's Theory laid the
mathematical foundation for string
theory in 19 1976 the American
mathematical Society presented him with
the Oswald velin prize in Geometry but
at the top of his academic career Simons
went looking for a new challenge when he
founded Renaissance Technologies in 1978
his strategy was to use machine learning
to find patterns in the stock market
patterns provide opportunities to make
money the real thing was to gather a
tremendous amount of data and and uh we
had to get it by hand in the early days
we went down to the Federal Reserve and
copied interest rate histories and stuff
like that cuz it didn't exist on
computers his rationale was that the
market is far too complex for anyone to
be able to make predictions with
certainty but Simons had worked for the
US Institute for defense analysis during
the Cold War breaking Russian codes by
extracting patterns from masses of data
Simons was convinced that a similar
approach could beat the market he then
used his academic contacts to hire a
bunch of the best scientists he could
find what was your employment criteria
then if they knew nothing about Finance
what were you looking for in I someone
with a PhD in physics and who'd had uh 5
years out and had written a few good
papers and was obviously a smart guy or
in astronomy uh or in mathematics or in
statistics uh someone who had done
science and done it well it's not
surprising that mathematicians and
physicists are involved in this field
first of all Finance pays a lot better
than you know being an assistant
professor of mathematics matics uh and
for a number of mathematicians the the
the beauty of option pricing is equally
compelling to anything else that they're
doing in their professions one of these
was Leonard bomb a pioneer of hidden
marov models just as Einstein realized
that although we can't directly observe
atoms we can infer their existence
through their effect on pollen grains
hidden marov models aim to find factors
that are not directly observable but do
have an effect on what we can observe
and soon after that Renaissance launched
their now famous Medallion fund using
hidden marov models and other data
driven strategies The Medallion fund
became the highest returning investment
fund of all time this led Bradford
Cornell of UCLA and his paper Medallion
fund the ultimate counter example to
conclude that maybe the efficient market
hypothesis itself is wrong in 1988 I
published a paper testing it for the US
Stock Market and what I found was that
the the hypothesis is false
you can actually reject the hypothesis
in the data and so there are
predictabilities in the stock market so
it's possible to beat the market is what
you're saying it's possible to beat the
market if you have the right models the
right training the uh resources the
computational power and so on and so
forth
yes the people who have found the
patterns in the stock market and the
randomness for that matter have often
been physicists and mathematicians but
their impact has gone beyond just making
them Rich by modeling market dynamics
they've provided new insight into risk
and opened up whole new markets they've
determined what the accurate price of
derivatives should be and in doing so
they have helped eliminate Market
inefficiencies ironically if we are ever
able to discover all the patterns in the
stock market knowing what they are will
allow us to eliminate them and then we
will finally have a perfectly efficient
market where all price movements are
truly
random