Session 21: Introduction to Real Options

This session covered the basics of options, starting with why real options are so attractive to analysts and investors: they allow you to add a premium to your DCF value. The two building blocks for real option value are learning (from what is going on around you or ongoing events) and adapting your behavior. There are three questions that underlie the use of real options. The first is recognizing when you are dealing with an option, with a payoff diagram being the give away. The second is looking for exclusivity which is what gives options value. The third is using an option pricing model, which is built on replication and arbitrage. We laid the foundations for all three questions today and will build on those.

Slides: http://www.stern.nyu.edu/~adamodar/po...

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Closed Caption:

couple of things before i start one is
up all the mystery projects have been
graded and returned back to you
hey and a couple of things about the
mystery project as you recognize this is
a project about dealing with a lot of
data it's very different from your
intrinsic valuation which is about
understanding the mechanics of value
this is about money ball with data how
do you deal with a lot of data and
obviously given the pushin and this is
understandable with that in class
towards regressions and statistics
that's where you all wet I think I think
there were two groups that did not use
regressions everybody else use
regressions and one of the things I saw
in the project is how much your
statistics class actually stuck and
simple things like you know how big a
sample so see when you don't regression
you start adding variable to the
regression it's almost by definition
you're r squared is going to go up so we
have 7249 instead of seven
there's a catch . if you have a sample
size of a hundred and eighty-three which
is what I think your entire sample size
was you have to be careful about not
getting carried away and it showed up in
some of the actually broke your sample
down into subgroups which actually makes
intuitive sense i'm going to look at it
in seven different sectors and i'm going
to run regressions and each one
kudos to you nice thinking but then if
you're and regression with five
variables and lemonade sector you're
gonna have a problem because of sample
size of $MONTH 27-30 and is a very
simple rule of done
here's my rule it for every 15
observations you're allowed one
independent variable to see where this
is going if you have 30 in your sample
stop with two and this is more about
your big project that about the mystery
project because some of you when you do
your relative valuation for companies
which you have to get done get it done
it out of the way it doesn't take much
time your sample size are often will be
15 16 17 companies so if you adopt my
rule your regression if you decide to
run one can have only one independent
variable don't throw in three or five it
will blow up your aggression you can get
great looking regret
since but they're mean absolutely
nothing the second part of the project
ask you to do what find at target for an
activist investor right first thing you
need to do anytime you're asked to find
a target with any kind of mission is
tell me what you are looking for before
you go looking
ok and one of the things again that came
through ways many of you were looking
for companies we could borrow a lot of
money
okay that's that's a piece of the up of
this process you want to borrow money
why because you want to take advantage
of the tax laws and make but there are
three words in an LBO there's an LLB and
i know i know this is obvious the l is
the leverage part the B is what by our
part we are taking a public company
private and the oh I have no idea so I
guess it's Abby always about there's a
leveraged part and why do you want to
take a public company private what do
you want to do with it as a private
company
I'm include both public and private
companies get the benefits of leverage
so the leverage reason is what not
what's making you go private it's what
you want to change something about the
company right break it up runner
differently and that's the piece i was
looking for in the second part is were
you looking for things in your company
that you could fix which means your
typical target company should be one
that's well done a badly run it should
be a badly run companies this is the
water and company what the hell are you
bringing to the table and do you have to
use leverage you don't have to i don't
think that's a requirement but if you
look for leverage and you look for
companies badly managed when you get
both things working for you but what I
am i arguing for a separate the two you
can have buyout for leverage might not
make sense you can have leveraged
buyouts don't make sense you can have
leveraged buyout to both makes sense but
let's not bundle them together and you
get a chance read that paper i have on
lbs because what I did was I took harm
and audio you've seen those speakers in
the high-end they were targeted by two
very big names for an LDL goldman sachs
and take care using those names they
must know what they're doing
so actually we take the harm and deal
apart basically what I look at is I look
at the confidence and does this company
fit the characteristics or what you go
looking for in an LBO can it carry that
the answer with harness not really the
company that has margins much higher
than the sector so what are you going to
fix the return on capital ten percent
above the cost of capital we're going to
fix so my conclusion the end of the
analysis is this company doesn't make
sense for me as a target an LBO because
none of the characteristics fit so the
next time you see a company target
didn't know beyond a new story take it
apart yourself don't assume that just
because they're big names involved in
the deal that somehow the deal makes
sense so talk about today squares is
will mention a couple of things
ok much of the quiz was actually like
pascuas right the first prom very much
like a problem from a previous quiz with
one small twist which was I through the
options in every item there they could
only question faces add subtract or
ignore add subtract or ignore add
subtract or ignore right three choices
and most of you didn't even think about
the ignore that I subtract a subtract
that's what I'm looking for did you add
something you should have subtracted
subtract up to somethin you should have
added and for the most part little bit
as i said this might open up wounds but
let's do it anyway
ok so you have share price duns number
of shares sir I assume that everybody
multiplied the two that your starting
point market cap you have to get from
market cap to enterprise value so I'm
going to item by item tell me what you
added subtract it didn't even see the
idea ok ready market gap debt add ok
everybody got it except for two people
there
cash
okay good we're on our own here and
minority holdings we're going to get
some I already know what have to look
for what you have to subtract that
because the income from those holdings
are not part of your everyday right you
subtract that minority interest add
subtract and now comes the one that I
know that I was looking at the answers
this was your weakest lick option value
add or subtract actually add
yeah i was so close how what the hell do
you mean you are so close your eyelids
this is the 01 prom and yes why you have
to support you have to add the way the
reason you I know why you subtract
because an intrinsic valuation where do
you come up with the value of the equity
then you subtract out to get to the
value of equity in the shares and you
divide by the number of shares in
innocence you're working backward
through that table right that's what
this whole processes the market cap is
the value backward in your common stock
so in the case of square you take share
price levels number of shares you know
whatever the price was you come up with
about 3 billion but there are a hundred
and twelve million option sir
right those options value the actual
market value of equity and squares about
thirty percent higher than the market
cap because that market value of options
is not reflect the market cap when you
buy these shares you're already
reflecting those options the rest of the
bronze I think what pretty much what
you've seen before the second one was a
little bit of mechanical stuff you had
to do the third and the fourth I thought
were pretty much working with if you got
the equation that you are working with
right the price-to-book equation for a
bank and the toll debate equation for
the last one the rest kind of flow so
the usual rules apply know none of yous
gonna pick up your quiz because you're
all offer Thanksgiving right after this
class is over some of you already left
and that's ok you know so i would leave
it out and I'll actually take it in
tomorrow because god only knows what
happens these buildings during
thanksgiving weekend i think that
party's here the janitors get together
and so I will bring it back outside next
week so if you want to pick it up it'll
be out tomorrow probably come and pick
it up if not not a big deal says I
promise what I'm going to do in these
last 37 minutes of class is take
everything i know about options but
thank God is not not a lot and fit into
37 minutes so think of this is a very
speedy introduction to the next topic on
our agenda
it's what's called real options it is in
fact one of the hardest areas in
corporate finance evaluation and minute
you're going to see what
to show you how hard real options are i
remember i had 15 years ago getting on i
think it was a diversion of amazon that
existed then doing a search for books
and real options and I think that was
one book maybe a drill options perhaps
to today if you type in the word real
options you see about 50 bucks pop-up
under options on any online bookstore
you going to any consulting firm there's
a real options group that's in there
some equity research department a real
option specialist real options have
become a bit of an obsession for people
to think of themselves as advanced
finance people and that scares the heck
out of me because in finance
we have this really bad habit of taking
the obvious and claiming the absurd
pushing it to a level where it becomes
outrages and real options in a sense
have been hijacked by the purest you
know the Paris are these are people who
believe in real option the point that
everything they look at is a real option
which reminds me over charlie munger one
set if the only tool you have is a
hammer after while everything starts to
look like a nail
that's what these real options people do
they look at everything that looks like
an option that looks like one so i'm
going to give you up front what I where
I'm coming from i believe in their
options in fact I'd chapter and real
options 20 years ago my first edition of
evaluation but but i'm a bit of a cynic
when I look at how real options get
applied in practice and hopefully i can
give you some ammunition if you run into
one of these real options people on what
questions to ask them to see if in fact
there is a real option embedded in some
argument so here are some of the places
where you see real options pop-up you
see an investment that doesn't look good
today and there we talk about the leg
that investment is a real option in
there if your production schedules in a
factory adjusting the schedule or it
could be in an energy company there's a
real option in there if you're investing
in a new market let's say China cloud
computing bigger the market the more
valuable the option is this talk of real
options in fact when people talk about
strategic considerations
he literally talk my drill option they
think we know we lose money in China but
it's okay because there's a strategic
consideration today we're actually are
not today but the sessions to come we're
actually going to look at whether there
is in fact an option and there is in
fact in some big markets and optionality
that might let you pay a bigger amount
for that particular investment so the
advantage of real options and this is
what makes them so attractive to people
who push them is you know how a
discounted cash flow valuation come up
with a value for an asset presently the
cash flows when you do capital budgeting
you take a net present value and the
rule in capital budgeting is the net
present value is negative don't take the
investment the value is less than the
price don't buy the stock if you buy
into a real options argument you can
break those rules
those rules are viewed as fundamental
rules and finance you can't get away
with breaking them if you're making the
real options argument what you're saying
is there's a premium on top of my DCF
survive value a biotechnology company i
come up with the value of three dollars
if there is a real options argument to
be made and it's made legitimately you
might pay a two-dollar premium or a
three-dollar premium over and above your
DC value and that is an immense
advantage if you're pushing an
investment right you do you take the
values that we should be a premium so
that's a way to think about your options
i'm going to start this process with a
very simplistic example it's not a
finance example it comes from
probability and statistics so let's
assume I came to you with an investment
well there's a fifty percent chance of
success and fifty percent chance of
failure if you succeed you make a
hundred million if you fail you lose a
hundred and twenty million what's
expected value of this investment minus
tan right . five times a hundred plus
port st. bad investment
now i'm going to take the same
investment and make it into a slightly
different kind of investment it's now
going to become a two-stage investment
of the first date is a seventy-five
percent chance of success and a
twenty-five percent chance of failure if
you fail at the first stage you stop and
you lose 20 minute
if you succeeded the first date you
continue and there's a two-thirds chance
of success where you make an 80-minute
and a one-third chance of failure where
you lose a hundred many you see the
resemblance within the two investments
my outside is still a hundred my
downside is still a hundred and twenty
there's a fifty percent chance of
success in a fifty percent chance of
failure at least a very peripheral level
these investments looked equivalent
right but if you compute the expected
value of this investment you can try it
if you want you will get a positive
number so what happens between the two
examples to make a negative investment
of positive invest what is the advantage
I game when I do this investment in two
stages
what happens the first stage i get an
early it's like a market testing right i
get an early snapshot or whether this is
an investment I should throw more money
in and if that early signal that I get
is a negative signal what do i do I stop
by get a positive signal i continue
there are two words on which real option
arguments are built on the first is what
I call learning which is you observe
what's happening around you and you
learn about your project he learned
something about your project and the
second is what I call adaptive behavior
which is based on that learning you
change the way you behave
who cares what I'm evaluation this looks
like some probability tree
let's you evaluate oil company in a
traditional discarded casual evaluation
what do we do we estimate the number of
barrels of all you produce your x an
expected oil price come up with expected
cash flows discount them back at a
risk-adjusted this country we come up
with the value for the oil company right
so let's say you're the manager of your
company so you've got these this 10-year
project i put an estimated price what
you get to observe each year as the
manager you get to observe the oil price
right and you think there's a feedback
effect from observing the oil price of
oil prices are high what do you want
likely to do do more exploration produce
more oil olive oil prices are low you
adjust to put in other words you
an autopilot you should not be this is
where I'm going next
if you do a discounted cash flow
valuation of a commodity company you
will tend to undervalue it because what
you're missing is the fact that the
managers of that company can adapt their
behavior to what they see in the market
that's a big change right because when I
valued valley and I came up with the DCF
value that was just the DC value i'm
saying there's a premium on top of it
and that premium is going to depend on
what not in the company but how
unpredictable oil prices are because
that uncertainty is what's going to
drive the value of the option so this is
something that's going to let us abandon
traditional intrinsic valuation for some
companies but you can already see how it
can very quickly get you into into
danger because we added premiums on you
got to be careful that you're adding it
on for companies where this learning and
adaptive behavior actually adds value to
your investment so that's going to be
what we're looking for so let's go back
two options 101 you you started
foundation to be going to have taken the
futures and options class
ok so you've seen this index it in in
extreme detail but let me cut to the
chase and the three questions I need to
get answered for real options to come
into play
the first is I'm going to ask when is
there an option in bed in action when
should I even be talking about option
pricing or real options in investments
i'm going to start with that
how do you know there's an option what
is it that gives away the optionality
the second question one else can this to
me is the key question where a lot of
things that look like options drop off
the table is when does this option of
significant economic value because the
answer is no to the second question i'm
just going to stop it why am I wasting
my time and third if I want to value
that option i need to use an option
pricing model right so I'm gonna ask
when can i use an option pricing model
black-scholes binomial whatever option
pricing model I want to value an option
so let's start with the first one when
is there an option embedded in a
decision to answer that question think
about the features that make an option
an option the first is
it's a derivative acid what does that
mean it cannot live on its own it has to
suck energy from something out there has
to be an underlying acid so far if you
tell me something is an option to what's
the underlying asset and he said I don't
know there's no option so the first
thing you need is an underlying is the
second is options have contingent
payoffs
what does that mean on a call option
what does happen for you to actually
make money the stock prices to exceed
the strike price that's what i mean by
continued something has to happen for
the payoff the kickin so it's going to
be an underlying I said there has to be
a contingent payoff and there has to be
a finite life in other words i can't
give you the right to this option for
the rest of eternity
it's got to be three years five years
ten years so those are the three things
you go looking for to call an option an
option and i'll give you an even simpler
way to identify an option and you all
seen this at some point in time and some
finance class you know what gives you an
option it's payoff diagram so what the
heck are you talking about
basically if i draw the payoffs on an
option
it looks very different from almost
every other investment and what makes it
very different is this kink around the
strike press so this is a call option if
the stock price is less than the $MONEY
strike price the most i'm going to lose
is what i paid for the call
that's what makes it an option as
opposed to futures agreement or a
forward agreement is I get the right not
the obligation to buy at a fixed price
if the stock price exceeds the strike
price of course I get a payoff
potentially unlimited because the stock
price go to infinity so the call option
this is what i'm looking for is a fake
basically a fixed laws and an unlimited
profit if you have a put option of
course it gets flipped her up because I
get the right to sell at a fixed price
if the stock price exceeds the strike
price i lose what i paid for the option
if it's less than the $MONEY strike
price i gain money but the prophets are
not are limited because the price cannot
go below zero so it's bounded but it's
bounded 0 so here's what I'm looking for
and this is what
going to see me use every time I make an
argument about a real option i'm going
to show you the payoff diagram and it
looks like this i would say we have an
option on our hands we have to think
about value it but i also have to
identify what the underlying asset this
and what the contingent pay office and
what the life of the option is for this
option to actually have been so let me
put this on the table there's lots of
options area
our lives are full of options vary
factor here means I mean you get a job
offer you are in fact exercising an
option right and if you're comparing two
job offers they might have the same pay
by one might have more so options are
all over the place so it's easy to see
options if you're looking for them so
the second question is where you really
have to spend your time and the second
question says basically asking when does
an option have significant economic
value i'm going to give you the one word
for me that allows me to separate when
an options argument substance from when
it doesn't the word I use is exclusivity
know what that means if you and only you
have the right to do something you have
exclusivity the further away you get
from that the more dangerous it becomes
to use a real options argument you have
no idea what I'm talking I let me throw
a few real options examples at you and
you can see exactly what I mean let's
say you value a young biotechnology
company and the only thing it has going
for it is a blockbuster drug working its
way through the pipeline this
optionality right because if their
continued my office if it gets approved
you could make a lot of money is their
exclusivity if so where does it come
from it comes from patent protection is
incomplete exclusivity it gives you
exclusively on this particular drug but
not every director treat so if you're
coming up with the new drug to treat
diabetes your particular compound that
you created might give you exclusivity
but if other people out so its
exclusivity with a qualifier which means
if you're a monopoly pharmaceutical
company
optionality works great right your
complete and total exclusivity the more
competitive this business becomes we
have 15 pharmaceutical companies all
working on drugs for the same disease
the exclusivity starts to wear down
let's take another argument and this is
something we'll talk about as well let's
say that I make the argument that we
should invest in China because of
optionality why because it's a big
market
what's a weakness of that argument
am I the only one who sees China's a big
market maybe I'm the only one with the
governor's a 1.2 billion people
everybody else in the world is
completely blind to it everybody sees it
right and unless i have exclusivity
that's not an option
the title for my paper and real options
on my website is opportunities are not
options because I see people using the
argument of hey this is a big market
there must be optionality it's a young
startup there must be optionally what
they're missing is the second part of
the question which is where is the
exclusivity
so what conditions can i use the options
argument China what type of company do i
have to be have to be a company with
some kind of exclusivity that might come
from a license if i told you that i have
an exclusive license from Beijing to
build hotels in southeast China there's
an option area argument time but let's
look there's a lot of people in
southeast China that's why I'm going to
build the hotels there there's no
optionality there so exclusivity is the
word that you're going to use and if you
can show me that exclusivity we're
almost home because then I know you have
an option the option is significant
economic value and for 45 years in
finance we've been building up option
pricing and we know the six variables in
there only six you're going to see why
they drive the value of an option three
of these variables related to the
underlying asset as the value of the
underlying acid goes up and down the
value of every option will also change
so that's the first very so i give you
the right to buy something and the value
of the asset goes up your call options
became more valuable because you got the
right to buy the fixed price if i gave
you the right to sell something at a
fixed price and the value the answer
goes up the value put options will
decrease the first variable value the
underlying asset
the second is the variance in that value
and this is where if you talk about real
option you go to bed and everything
we've talked about for the last 20
sessions after now whenever I've talked
about risk it's been a bad thing right
in discounted cash flow valuation risk .
pushed up my discount rate pushed on my
value in relative valuation risk made me
give pay lower multiple for earning so
book value of revenues or whatever
variable but the minute i talk about
options risk becomes my Ally we talked a
little bit about this at the very start
of the class let's make sure we affirm
why that is what is it about options
that makes risk your ally so it but
that's stating the same thing what is it
what kind of risky we'd usually worry
about upside risk a downside risk
downside risk right so what is it about
these payoff diagram that makes you feel
much better
the nature of options is that Florida
downside risk that's why risk is your
ally with options as simple as that you
don't need the algebraic proof if i make
your downside risk go away then risk is
now a good thing so in the case of
options increasing the variance of a
Lagonda lying i said will in fact make
options more valuable if there are any
dividends that I expect together get on
that acid it'll affect me as an optional
let's see why
let's make this very simple let's
support have a call option on a stock
that pays a big divot tomorrow's an
ex-dividend day you know what happened
to the ex-dividend direct you have a
fifty-dollar stock it pays a two-dollar
David on the ex-dividend Dale roughly
speaking what should happen your stock
press it should drop down by two dollars
otherwise people be making money hand
over fist
so you can see already that if you're a
call option holder on a stock that pays
dividends that call option will get less
valuable the bigger the dividends are
expected to be over the life of the
option because those dividends will
translate into lower prices think this
is actually an argument that some people
have used for wide
buybacks have become more popular than
evidence because you compensate managers
with options you see how you already
tilted the scales because when they
appeared in the stock price goes down
the argument is they buy back stock
because they have options and options
are hurt by dividends you're going to
pay less than $MONEY difference there
are two variables relating the option
one is of course a strike price it's at
the right to buy something at a lower
price is worth more than the right by
the same thing at a higher price and the
more time i give you to play this game
the more valuable options become the
only macro variable that determines the
value of options is what your screen
readers why because I give you the right
to buy something at a fixed price five
years from now when interest rates are
high the present value of what you have
to pay becomes much much lower so as
interest rates go up call options will
become more valuable because you're
going to have to pay less in present
value terms and put options will become
less valuable so those are the only six
variables and all option pricing models
try to build around those variables
now I know most of you just look at
option pricing models get intimidated
and move on but I don't want that to
happen because option pricing models are
not complicated taken over by Greek
alphabets but there's actually a very
simple rationale that drives all option
pricing models all option pricing models
are driven by two very simple finance
principles
the first is what the the principle of
replication you know that what that
means if i create something that looks
just like the option it has exactly the
same cash flows the option have
replicated the option so the first step
in option pricing is creating what's
called a replicating portfolio composed
of the underlying acid and either
borrowing and lending every single
option pricing model is built on
replication the second principle is
arbitrage where does that come in if i
create two investments which have
exactly the same cash flows what is
arbitrary require their prices to be
they have to be the same that said if
you get that you've got option pricing
and that's effectively what every option
pricing
auto tries to go let's do some very
basic option pricing and to see
replication arbitrage pop-up you're good
and the reason this matters if I want to
use an option pricing model that then
the replication the arbitrage
assumptions have to hold which means i'm
going to be able to trade the underlying
asset and trade the option easy enough
to do we have a call option on a stock
but remember that biotechnology example
that I gave you the underlying asset is
the drug that comes out of this approval
process you can trade that the option is
the right to the drug you can buy and
sell that but it's not widely traded you
can already see why real options become
more difficult to apply and price
because you don't beat those two
requirements the way i would start
replication was using what's called the
binomial model and you can see very
quickly
why binomial models bring home
replication much better than the
black-scholes model which is more widely
not here's the basic principle i can
create something that looks exactly like
the call option by borrowing money and
buying Delta shares of stock that's
telling you nothing recent what's Delta
that's one of the things you have to
answer the option pricing models what
the Delta is I can create something that
looks like the put option by selling
short the underlying I said taking the
cash and lending it out almost every
option pricing models about how much
they're bored i'm doing a call option
how much ya borrow and how many units of
the underlying share do i have to buy so
let me show you a binomial model because
it kind of illustrates this replication
arbitrage process so let's assume you
have a stock that's trading at 50 right
now so that's all you know it's today's
price let's assume over the next two
time periods
this is what I think will happen in the
next time . the stock and jumped only
one of two . 7135 that's what's called
the binomial model there are only two
possible outcomes at each stage and
you'd see in a minute why have to make
it only two if it goes to 17 the
following time . can go up to a hundred
or drop back to 50 if it goes to 35
conjunct 250 dropped 25 i give you a
call option with a strike price of 40
that expires at t equal to 2 so right
now this is where we are
stock prices 50 I've given you
call option with a strike price of 40
let's make this simple if you exercise
today how much money do you make
you're going to make that right you're
going to buy it for you so basically buy
it for t turn on but this is an option
of two time periods to play- will let
yourself play and what are the three
possible prices you can end up at $TIME
hundred 50 and 25 so let's see what the
cash flows on this call option are at
expiration because that's the only time
you know but certainly what that value
is going to be so the stock goes to a
hundred and your strike prices for do
you make 60 if it goes to 50 you make 10
if it goes to 25 you make minus 15y not
because you have the right to buy it for
the internet stalker the 25 why the hell
would be exercised that right so
basically become 0 60 10 0 what is my
objective i want to create a replicating
portfolio that is exactly the same cash
flows as a call composed of two
variables how much your borrow how many
shares of the underlying stock dua ban
so i'm going to make make this almost an
algebraic problem let's say the mountain
bar always be and the number of shares
of stock cube is Delta I know you think
this algebra left behind no 15 years ago
pull it up again it can maybe we'll come
in useful so let's say the stock is at
70 I want to combine Delta and be to
have exactly the same cash flows as the
call so if the stock goes to a hundred
hundred times Delta and i'm going to
repay the board because I borrowed the
money with eleven percent interest rate
hundred times delta minus the boring
with interest should be equal to 60 if
the stock goes to 50 50 times delta
minus 1 point 11 b has to be equal to 2
equations to our notes or two equations
and two unknowns and you sit and stare
long enough there sooner or later they
are known should be known so basically
time to take this equation solve it's a
very simple simultaneous equation the
Delta is one the B is thirty-six dollars
and force you what does that even mean
if the stock goes to 70 and I go out and
borrow 32
dollars in four cents and buy one share
of stock that replicating portfolio of
the stock goes to a hundred will give me
a cash flow of 60 is the goes to the
store close to 50 could be a cash flow
of dead
I've replicated my call option how much
is the cost me to replicate well i have
to buy one share of stock at 70 border
thirty-six dollars and four cents so it
cost me 33 dollars and ninety six heads
to create that position and because of
arbitrage the call option has to be work
3396 this you now wired to start with
the last branch and move backwards
because the last price is the only time
I know the cash flows
I'm essentially solving for Delta B sub
created the value for the call option i
do the same thing at 35 I come up with
the value for the call to 499 now that
the values for the calls i can work
backwards one more step and asked the
same question if I went out and bought
the shares today and the store close to
70 70 times delta minus the borrowing as
we could 3396 it goes to 35 35 times out
there- the borrowing has to be equal to
4 99 again i solve for Delta B and if
today I go out and borrow $21 in 61
cents and by . 8278 or even ask me how I
by . 8278 shares of stock if I can do
that I've created what's called a
self-financing replicating portfolio you
see what self-financing the stock goes
to 7021 3396 i can use the 3396 to
create the next replicating portfolio
and the value the call today has to be
equal to the cost of that replicating
portfolio it's ninety dollars and 42
search
I know it's a little tedious to go
through this but if you really want to
understand option pricing spend a few
minutes on it it's not really difficult
once you break through the numbers and
you see what we're trying to do here is
the replication and the arbitrage the
promise binomial trees is I give you a
chance to go to New now let me stop and
that's why did I have have to make it a
binomial tree how many unknowns do i
have to have you ever tried solving for
two are known for three equations
don't do it
you're going to get have nightmares
because in a sense if I three equations
you're going to head up with variables
that might not be identified because
you're three equations so i had to make
it a binomial tree because I've only two
unknowns Delta a and B which also means
that i'm a little restricted because for
to have any chance of having a binomial
tree in the real word i have to make
time really really really short training
let's make it a second maybe if i do it
in seconds and the next second IBM stock
price could go to only one of two points
it's plausible right I'll give you three
month call option and IBM and ask you to
draw binomial tree with time measured in
seconds
already some of the really big sheet of
paper really fine pencil start drawing
the tree tiny little branches you're
really really detail-oriented person and
you draw every branch I won't even ask
you how many branches there will be
industry there's been hundreds of
thousands of branches so let's say two
weeks later i come to New drawn this
really laboratory can almost visualize
this tree with tiny little branches to
hang in there with me
let's assume you take the sheet of paper
you drew and turn it on its side in
order to look like it look like a
Christmas tree with no branch right
because you've got the binomial tree
drawn now smooth out the outside of the
Christmas tree
what does it look like I know it's
getting close to Christmas come on you
can visualize the Christmas tree smooth
out the outside hey you got a normal
distribution if you take time and as
time gets smaller you make the price
changes smaller the limiting
distribution for the binomial is the
normal distribution the black-scholes
model is a special case of the binomial
model if you make time short and you
make price changes small called
continuous price distributions the
black-scholes model becomes the binomial
model for the binomial model becomes the
black-scholes model but is that
reasonable assumption that has time
short of the price changes also get
shorter what am i assuming will never
happen i'm assuming that prices will
never jump right there in the next
moment the price can go up ten dollars
and that's unrealistic we know prices
jump in short periods news items come
out price jumps scandal price drops we
are assuming that you can never have
those price changes when we do a black
Scholes model that's why black-scholes
model is consistently underestimate the
value of D part of the money options dcy
4d part of the money option to become
into my in the money in a black Scholes
were the price has to move into tiny
little pieces if I don't let prices job
and you're twenty percent below the
stock price it's going to take a lot of
work for you to get above so when you
take the black-scholes model and value d
part of the money options you will end
up with two lower value because of the
underlying assumptions you make with the
black shorts but the black-scholes does
simplify the world because it basically
says the value of your options is a
function of Phi variables the value of
the underlying asset the strike price
the life of the option the riskless
interest rate and the variance in the
log of the press little detail what do i
do the log of the price in fact whenever
you do a black scholes variants you
supposed to take the log of the press
sound like an inside option staying
what is the log of the price I'll give
you a clue what is the distribution that
I built this off the normal distribution
rate can prices ever be normally
distributed
why not the lowest value price can take
is 0 right what's the natural log of 0
minus infinity that's basically that's
why we use log prices is it makes life
convenient for us it's always a chance
it's going to be normally distributed
late so the next time you see somebody
using the log of the prices to come up
with the standard deviation ask them why
most of the time they'll have no idea
the reason is very simple to give you a
chance at the normal distribution but
yes
yeah I'm the only five variables in the
original black shorts when I listed
variable so I had six right
what is that what's missing variable in
the black
Schultz the original black-scholes 75
but I said there was six variables we
can't wait to go back and look at that
page you can
what's missing variable there are no
dividends in the black-scholes model you
know why there's a legend of how
official black and modern shows actually
derived this model I mean they were
bought the university of chicago and he
was struggling with coming up the closed
form solution to this equation that was
the black-scholes equation remember the
black-scholes predated the binomial so
this equation nobody's ever seen
anything like this in finance it sits on
their deaths four months until a physics
PhD student was walking by and Chicago I
guess pieces should wander all over the
place walked into the office don't ask
me why
look at the equation said we see this in
first year physics it's a stochastic
calculus equation you're the answer so
the Nobel Prize should actually have
gone to Fisher black myron scholes and
an unknown PhD student in the university
of chicago but they have to struggle so
much to get a closed-form solution that
they decided to get rid of dividends
away condiments get rid of problems
which is to assume it away so the
original back shoulder was designed for
what I called dividend protected options
and how the heck would that work
remember what I said have a stock price
of 50 you painted two-dollar dividend
stock price going to drop the roughly 48
with a dividend protected option i just
your exercise price accordingly so your
exercise price was 40 I make it 38 and
say you have been unfair to you because
it's do there is no dividend protected
option anywhere in the world but the
black-scholes was the desired for David
protected options and with a dividend
protected option i'm sorry about all
those question marks that should show up
as Sigma's in your in your lecture note
packets and here's what you do in the
black-scholes model the value of a call
shows up as the value of the underlying
asset today BS turns end of d 1 i'll
come back in and talk about what end of
d 1 is minus k e- RT lands end of d to
the way I describe the black-scholes it
makes you look use all those virgin
buttons on your calculator
Peter noone so you've never touched the
natural log through the excuse is that a
minus RT you have that's a crazy-looking
thing all those but if you take a look
at the calculated that brand-new because
you the pv buttons all war not the
actual is use it now that's why you have
to buy that calculator and a hundred
dollars for it but here's what you do in
the black-scholes you take the variable
to plug them into an equation people do
it mechanically come up the d1 and d2
you go to a table looks like this it's
called the cumulative normal
distribution table as opposed to what
in most statistics book there's a normal
distribution table but it gives you the
height of the distribution at every
point a cumulative normal distribution
get a table gives you the area that the
distribution and if you look at the
table the lowest number you can get for
this for this cumulative distribution is
0 and the highest number you can get is
1 and already I'm going to give you a
way which you can make an intuitive leap
from that number that's like a
probability and that's the best way to
think of anybody 1 and D 2 and
especially end of d 2 is it gives you
probably think probably what anybody to
roughly speaking is the risk-neutral
probability that your option will end up
in the money
repeat that again get an energy to a
point 7 1 i'm saying this is 71 percent
chance your option will end up in the
money in fact the best way to kind of
get an intuitive grasp of the black
shows is to think in terms of the
replicating portfolio the black-scholes
has the replicating portfolio embedded
in it in there in the black-scholes of
you asked me to replicate a call option
i'm going to go out and buy end of the
one shares of the underlying acid and
I'm going to borrow ke- RT and let me
explain what the e- RTS the strike price
you don't have to pay till the
expiration of the option right isn't
always true
can you exercise an option before
expiration the most options you can
those options accord American options
the black-scholes was designed to value
what are called European options got
nothing to do with geography so don't
think if you go to your of every option
is going to be your appear okay European
options you can exercise only on the
expiration day you think so what so
that's the expiration date is two years
from your the strike price of 50 k e-
RTS the present bad 450 brought back to
us because you don't have to pay for two
years so I don't use e minus RT to do
present value normally we don't because
we work in discrete time discrete time
and what says we need to DC evaluation
your cashflow 101 happened one year from
now the black-scholes continues time if
you don't have an exponential button on
your calculator just use a traditional
president you'd be surprised at how
close the numbers are going to be
because that's what the K minus RT does
that's called the replicating portfolio
and almost everything an option pricing
is built around that replicating
portfolio makes late the option Delta
anyone as the stock price changes do you
think the option Delta will change
yeah stoppers goes up and done I just as
the time to expiration changes in fact
that's called the option gamma as the
time to expiration changes with the
delta T yes that's called the option
theta every single Greek alphabet
associate with option pricing has
something to do with how that
replicating portfolio changes as the
various changes at vega there
I told you hijacked the Greek alphabet
entirely theta gamma everything has to
do with how will that replicating
portfolio change but it's built on the
notion of replicating put for you so i'm
going to close i know we're running one
minute ahead and put this in my life so
that there is a way in which you can
take the traditional black shorts with
did not allow offered for Ford evidence
and adapted to include evidence and all
we're going to do that is assumed that
the dividend yield over the life of the
options stays fixed so if you have a two
percent dividend deal so when we start
on Monday we'll start with this equation
because this is the equation that's
going to allow us to value real options
because almost every option has a
dividend component to it
yeah
yeah
yes
yeah
you through the industry dummies and
yeah anyways okay Isaac no study are
scratches we can make sure the pieces
and rather than pick every sector and
throw it in maybe you want to just pick
one or two and make it a dummy variable
so yeah because in a sense five of them
you might said not much difference
technology looks like it's crazy
somebody throw technology dummy into the
regression right right
no because the freaking you already
controlled for free cash flow near
aggression right so what the industry
variable is capturing is things are not
captured their fundamentals right so we
already have free capture equity in your
regression you don't need an industry
dominated capture what industry dummies
capturing I think you did not capture
the fundamentals you might throw in cash
flows growth and risk and you don't know
what people are crazy they see the name
tech in a company they push up the price
story percent that's what the dummy
variable is capturing is what the
fundamentals could not capture and
that's why there are little dangerous
because let's say that people are being
faddish that are attaching a 30-percent
dream to any company which is what
technology in its name then you're going
to get a great-looking piece statistical
great-looking r squared and your
predictions for tech companies that all
Christ your strength right but the price
is right because you've built into the
pricing there's a rational component you
got to be careful about drawing the line
at adding a variable that might improve
your r squared from recognizing that you
have a mission to take advantage of
people's mistakes right so when you add
those dummy variables that's what I'd
like you to think about is that dummy
variable bringing in something that I
should be worried about fundamental so
the reason to bring a technology
variability might say lock technology
companies have shorter lives maybe
because they're shorter lives that
multiple should be different from non
technology companies that's an intrinsic
value factor i'm capturing that the
technology done so
basically start with just regular one
see what the deal is if you get a lot of
skewed outliers that's really a signal
from the regression saying there's
something wrong with your data so for
instance we find a subgroup of all know
p/e ratio company's coming as
undervalued and all high p companies
comments / value we got a distribution
which basically is a curve around the
light right to fit a straight line
through something that is a good
relationship so that's what you're
looking for look at the output because
that kind of give your aggressions