Alan Cring Productions in association with the Emergent Light Studio presents
the Illinois State Collegiate Compendium, academic lectures in business and economics.
This is business finance, FIL 341 for autumn semester 2024.
Today, the Black-Scholes Options Pricing Model.
Before we do that, let's have a look at the numbers for the day.
And this is one of the weirdest things.
We're all asking the same question.
Why did the Fed cut rates yesterday and nothing happened,
and then today the markets just went cuckoo bananas?
Were they all stoned last night or something?
We can't figure out what the heck, how this happened.
But there's no reasonable way because the markets were expecting
a quarter percent cut in the discount rate.
Okay, so if the Fed says, well, we cut the discount rate by a quarter percent,
we wouldn't see much of a reaction because it was already impounded in the expectations.
But when the Fed cuts the discount rate by a full 50 basis point,
that should have sent the markets.
Traders should have started making monkey sounds and swinging from trees, and they didn't.
And then this morning, all of a sudden, look, and the Dow is climbing like gangbusters.
You've got the Dow is up, oh, holy cow, more than a percent and a half.
The S&P is up 2 percent, and the NASDAQ is closing in on a 3 percent increase.
That almost qualifies as a white swan.
So, I mean, okay, we'll take it for what it's worth.
That the markets waited to see if it was real or if they were dreaming or something like that.
So we got a great day, a really hard day up.
And it's almost more than we would have expected out of this.
And notice that it spiked. At the bell, boom, it was up.
And then, interestingly enough, as the day has gone along, it has continued a slow grind upward,
mostly as traders are beginning to see positive impact on more and more industries.
And that this is really strong good news.
So the initial jump was just the cut in the discount rate.
The slow growth of price, rise of prices after that, is assessing impact industry by industry, company by company.
And it's all favorable. There's not much of a downside.
And you'll see there are sub-sectors that are just going crazy.
The darker part of it is crude oil is showing some signs of life upward.
And that is primarily because of just a little bit more concern about the Middle East,
because Israel is blowing up people with pagers and now with walkie-talkies.
And there's a concern that this is going to piss off Hezbollah more.
And they may start some more serious attacks out of Lebanon.
And then Israel will respond with more attacks on Lebanon.
And then Iran will get involved in it and it'll be all kinds of fun.
Not much chance of that. As you can see, oil isn't spiking.
But there is a little bit more of a war premium beginning to slip in to the prices.
Nothing really to worry too much about.
And you notice that gold did a bouncy bounce there, and it's up a little bit too.
War kind of gets gold bugs excited sometimes.
But moving over here.
Now here's the interesting one.
Ten-year bond yield is going up.
One of the perverse effects is you see now bond yields going up, that means bond prices are going down.
Bond prices are going down would mean that investors are selling bonds.
Well, actually, that makes all the sense in the world.
The world is making sense right now because investors are grabbing equities.
So they're selling bonds and jumping toward equities.
And of course what that does is buying the equities drives equity prices up.
And at the same time, though, selling the bonds drives bond prices down and bond yields up.
So you're seeing the other side of that.
Even though the discount rate was cut, that should cause yields to go down.
In fact, the opposite is happening because there's so much excitement about equities.
Investors are pulling out of bonds to buy equities.
And of course pulling out of the bonds drives bond prices down and the bond yields up.
So the counteractive effect on the Fed's discount rate cut is that it can have that backlash,
the backwash of causing bond yields to actually increase.
Fortunately, that should be a temporary phenomenon.
We see it happen.
And eventually the sell-off in bonds will abate and we'll be back to the bond yields going down.
But for now, that's not good news.
Bonds yields are going up, so we've got higher interest rates.
Although right now in the consumer market, if I'm not mistaken, mortgage interest rates on mortgages just hit a...
they have gone below what they were in 2022.
So bond mortgage rates are going down.
That should stimulate the construction industry, big sector of the economy, and all that good stuff.
And loans on auto loans, I have tracked those over the last couple of days and they are sliding too.
I ran into...I did something kind of evil yesterday.
I talked to this car dealer I know, former student from years and years ago,
and I said, what's your best rate right now?
And he said, for you, 4.39%.
That is lower than it has been in a long time.
I mean, you had car loans at 7.29%, 6.99%, and he's trying to wheel and deal with me at 4.39%.
I feel bad about making him think I was going to buy a car, but he'll get over it.
Okay, so you see, interestingly enough, that Tokyo and London also responded favorably to what's happening here.
Interestingly, though, over in London, the exchequer of the currency said,
we're not going to lower interest rates right now.
So he was putting the kibosh on...the Americans did it, so we're going to do it too.
He said, no, we're not. They're still fighting inflation.
They got...they jumped into the fight later than we did.
So did the Europeans.
And so they're going to have to hold their interest rates higher for a little longer than we will,
just because they've got...they started dealing with the inflation problem later than we did.
And so that is credit to the Fed, and I've criticized the Fed to the point where I've described them in fairly harsh terms.
I think at one point I described them as the Antichrist.
That's a little harsh, but right now they've done a pretty hella good job of managing rates.
Now, look at it.
You saw this effect yesterday and the day before, but look at, for example, Citi.
Well, let's try that again.
Citi.
Oh, I kicked the plug out again.
I've got to stop doing that, because they're going to start fussing, I'm ruining their USB port here.
Okay, Citi.
Look at that.
5% up on the bank.
Of course, interest...and so that was one that you could almost bet would have happened.
But if we had known...if I'd known that the Fed was going to cut as hard as it did,
I would have gone long some call options on Citi.
But look at Bank of America.
3% up.
I wonder if a dog like Wells Fargo...I didn't look at Wells before, WFC.
Even Wells Fargo is up 3%.
I mean, the banking sector is just rolling...they're in hog heaven while we're in interest rate slop right now.
So they're happy with this.
This is good news for them.
Other sectors of the economy.
Telecom had had a good run for a little bit, and then they sort of petered out.
I wonder what they're doing.
They're down.
Interesting. Verizon.
Down. So the telecoms are not happy about this, what's happening here.
But that's them going over consumer goods.
Let's try...let me try Kroger.
Just...yep.
Walmart.
Ooh, look at Walmart.
Walmart got spanked.
Let's have a look at Target.
Oh yeah.
Walmart.
I'm not sure what's ticking off the investors about Walmart, but I mean the rest...a lot of it's looking really good right now.
Looking at consumer basics.
Colgate-Palmolive.
Down.
Why is it that interesting?
I wonder where that is.
Medical and...medical.
Down.
So this is not all the way across the board.
It looks like...what am I thinking?
Come on, come on, come on.
Let's look at Palantir.
Whoa, Palantir.
That is one of those toy stocks.
You can put money into that one time and make a lot of money out of it, and throw it in a couple of months later, and you will lose your shirt.
It's a nasty, unfriendly kind of stock.
Moderna.
Well, pay attention.
MRNA.
Shouldn't...yeah, it shouldn't react too much.
Moderna.
Not showing much signs of life.
Johnson & Johnson.
J&J.
Huh, isn't that interesting?
So this is not all ships rise in the rising tide.
There are some that are not taking this well at all.
Which, you know, that's their thing.
It kind of surprises me, though, that Johnson & Johnson and those are not doing so well right now.
Coming across here, going back to a couple of old friends.
The Bull Play.
It's not on a stock.
It's on the general sentiment.
TQQ is Bull.
Should be...holy cow.
I mean, that is just stupid good.
Just happy good.
And of course, it's Dark Cousin SQQ, the Bear.
Pretty much about the opposite.
So the bears are being spanked and the bulls are being fed an extra bale of hay.
So this is one of those things, that 9%.
That is...now, again, we are talking about financial options here.
So the whole thing about this is that if I were to look here, TQQQ...
Let me go back to it.
And I'm going to look at the options.
So let's take it out.
Expiration on the 26th...27th, sorry.
On the 27th.
Now, keep in mind, the 7169.
We're going to look at one that's just out of the money at 72.
Just for sharts and giggles, 72 on it.
See how it's done today.
Oh, quit.
72. Look at this options change.
Right there. See, just out of the money at 72.
And we run over here.
Hello.
Percent change from yesterday on the 72 option.
Why does it keep doing that?
Percent change on the 72 option.
317%.
Now, these are highly leveraged, highly risky instruments.
I am guessing that if you had thrown in...that was probably...
God, I don't know what that would have been to cause that.
If you had bought yesterday at a dollar...
It probably was less than a dollar yesterday.
Today, it is a dollar 96.
So you would have, on the swing, for one option, have made 96 times 100.
So you would have made $96 on a single option on this.
Just in one day.
If you had thrown 10...if you had gotten 10 contracts, you would have made a really lot of money in a single day on this.
And if you...look...
Okay. I might as well say, bid and ask.
191, 196. You would have bought it at...
Yesterday, it would have been a lot less than this.
It's too late to buy this now because the information is already impounded.
You're not going to make anything off it at this point. It's not worth it.
But today, you'd buy at a dollar 96.
If you wanted to write one of these options, you would write it at 191.
I would not recommend that because there still could be some push upward in it.
Now, over here, I'm not familiar with options chains on Yahoo.
Open interest volume. There.
This number...see this implied volatility?
This is one of our most important numbers.
And I have a spreadsheet where you can calculate implied volatilities.
I have an Excel sheet. I'll upload it. You won't have to need it for this course.
But you can calculate them yourself.
And a service like Yahoo, it's...I mean, they're usually pretty close.
But you can do it yourself in Excel, finding implied volatilities.
It's just a matter of putting in the data.
What you'll do is you'll put in historical data.
And then Excel will make the adjustment to turn it into a forward-looking volatility.
Now, oftentimes, a service like Yahoo, it would take the market data
and then back into the implied volatility that would match that data.
That's where the term implied comes in.
So they're not really giving you an implied volatility.
They're not calculating one like we figured out that it exists.
They're just taking the market numbers and solving for the sigma
so that that's what they're giving you, is what is already known.
Now, let me take you on a little bit of a journey here for just a few minutes.
And we're going to walk that sheet.
And here it is.
Now, let me explain...well, let me do something here.
Something...let me see if there's a way that I can put up this slide for you to see here.
That is a...how shall I put it?
A pleasant version of the Black-Scholes options pricing model.
These two functions, these are functions, N.
N evaluated at D1 and N evaluated at D2.
And there are the formulas for what you plug in to the N formula.
For D1, you would actually calculate this incredible mess right here.
Plug it in to the N function.
And then you would get the D2, which is D1 minus the implied volatility
times the square root of the time remaining on the option.
And you would plug that into this N formula.
The N formula, those are integrals.
They are integrals.
They are actually that long S with a mess inside of it.
Those integrals are the normal distribution function.
And another part about that is that in mathematics,
you are often, almost always in undergrad math,
you are taught what are called closed form solutions.
In other words, you are shown an integral and the integral...
Well, let me do it this way.
I'm giving you a little bit of background here.
For example, the integral of x squared dx is one third of x to the third
plus the constant of integration C.
You saw that somewhere in your dark past.
That's a closed form solution.
In other words, we know what the result is as an equation.
The problem is that when you get into functions like the integral of e to the rt sigma dt,
that doesn't have a solution.
It does not have a closed form solution.
The way you find those numbers mathematically on paper in Excel
is you have to take a guess.
And then you keep taking guesses that get closer and closer to...
The difference between this one guess and the next one gets smaller and smaller and smaller.
That's called iterative solutions.
And that's what Black-Scholes does.
It has these functions.
That's related to a normal distribution function.
There are no closed form solutions.
If you want to calculate the normal distribution at some value,
you have to do an iterative calculation.
Little calculators as well as big calculators, Excel,
they do that for you.
So let me take you back over here.
As you can see, it is actually...
You plug in all of these numbers here, and it does all of that there.
In other words, those equations that I just showed you, it does those for you.
Instead of you having to do it, it does it for you.
And these actually, if you look, this is actually the formula for the normal distribution
evaluated at.3616.
This is the normal distribution evaluated at.2505.
I'm telling you the background to convince you do not ever touch those numbers on that side.
They're done for you.
And this, my good people, these are your Greeks that I talked about on Tuesday.
The Greeks.
And each of them tells you something, and I'm not going to go into it in deep detail in this class.
Take.347, and we will use these like a boss.
But for now, all we're going to do is just walk through so you can see some of the processes that go on.
Now, what comes out of here, this is your goal line.
Finding the arbitrage-free prices of calls and puts.
And then if you're a trader, you look at what the actual market says it is.
They should be very, very close.
If they deviate somewhat more, like for example, I might look at this option and I see that a call with these parameters is right now trading at 10.40.
Well, the black shoals, the arbitrage-free equilibrium is 10.4577.
So I know that briefly this option is underpriced.
So I would buy these and I would buy and others would, doing the calculations, would buy them.
And the price would rapidly move back toward, through the buying pressure, move back toward $10.45, $0.46 per option.
That's what's called arbitrage.
It is the discipline of the market.
If prices deviate more than a little bit, traders are going to come in and move.
Now, suppose on the other hand that you see that right now this option is priced at $10.50.
The market is pricing at 10.50.
That means that right now it is above equilibrium.
So in this case, I would write options, write call options, because I would sell them at a price higher than what they should be.
And as I flooded the market, as traders wrote options, they'd flood the market, supply of options would go up, and that would push the price back down toward equilibrium.
That's what the black shoals does.
It gives us where it should be.
Now, suppose that the price right now, aha, I see that the price is $10.44 and I know it should be $10.46.
You're not going to do anything with that.
In other words, there's going to be some rattle.
Now, if you're a ginormous trader doing high volume, high frequency, you can take advantage of it.
But as a small time trader, it's not worth even trying.
So for us, we need to see enough of a deviation to be able to move.
I have a famous story that's of my own.
Once I saw a deviation of about $0.07 on a call option from black shoals.
And what did I do?
I saw my numbers, I saw this, and so what I did was I just stopped and I checked my numbers all again.
By the time I got back to trade it, it was right back on that number right there.
It just pissed me off.
So it's going to happen very quickly because these computers that are doing this, they can do this really fast.
So it's almost like it's partly, yeah, just academic.
Okay, that's great.
But also, where you can still do some arbitrage is on small cap stocks,
where there might be a delay in big dogs seeing them, they do not watch those.
So yeah, it still can be used.
But if you're talking about ginormous stocks, Target, Ford, Pfizer, Tesla, you're not going to get anything.
You see a deviation, you see an arbitrage opportunity.
By the time your neurons fire here and get to your paws, it'll be gone.
But do appreciate that this still is useful to us.
And one of the nicest things is that we can play and see how options behave, see what they do.
So suppose that I have an option that I buy today.
So I'm going to put in 76, I'm sorry, 76, so that we have, let's say, nine, I hope no one's played with this.
What's today, the 20th? No, the 19th.
And we want one that is two weeks later.
October the 4th, 10, 4, 2024.
It's not going to work for me. It's not going to work for me.
There we go.
Option explorations. I see I was bitching about that.
So anyway, now let's do one here.
Let's say we have an asset price.
In other words, the underlying is a stock that's right now selling at 74.20.
And we'll grab one that's at 75, has a strike price of 75.
Now the time to expiration, in this case, that's.04195 years.
Actually, it's using this one. Isn't that interesting?
I don't know why it's doing that, but I could fix that or you could probably even fix that.
Let me just put in 10, 4, 2024.
So this is giving you, I think something's weird about this.
You'll want to do it down here, I guess.
But anyway, okay, so that's.0411 of a year.
And if you wanted to, you could just put in that cell C13 equals however many days divided by 365.
You could do it that way if this is getting wonky.
Now, this is the sigma, the volatility, the implied volatility.
You can look it up, you can calculate it yourself.
But whatever it is, let's say that we have an implied volatility of 8.96%.
Now we'll put in the risk-free rate.
Usually we put in the current price of a short-term T-bill.
In this case, well, I'm going to guess that it's, let's say a short-term T-bill right now is yielding 3.89%.
Now, the Black Scholes Options Pricing Model works as long as you can express the dividend as a percent of the underlying asset price.
If it can't be done that way, then you have to go to the binomial model over here to do it.
But in this case, we're going to say this stock pays no dividend.
So once we put in all these numbers, you can see the call price and the put price.
Now notice in this one that this exercise price, the stock is out of the money,
and there's not a lot of time left for it to get into the money.
So you're going to see the put, there's more likely that it's going to finish, expire out of the money.
That's why the put is going to be higher value than the call.
Because right now the put is in the money, and so it's going to have a nice in the money little piece to it.
Now, all of these, these, not much I can say, but you can see more about it if I change some of the parameters.
But before I change too many parameters, let us watch what happens as you look at a higher strike price.
Let's just change the strike price.
Remember that the underlying is 74.20.
So the first strike we looked at was 75.
Now let's take it to 76.
Now notice that makes the call further out of the money and the put deeper into the money.
So we should see the call go down and the put go up.
Spank me Jesus, look.
You see it?
Let's take it even further.
Let's say that we're, the underlying is 74.20 and the exercise price is 77.
Well, the call sucks.
The put is deep enough in the money, it's going to be worth a lot, it's going to be worth a lot at this point.
Because there is a very good chance that the stock is not going to make it to 77.
So the call is going to be pennies.
But there is a very good chance that the stock will finish below 77.
So the put is going to be almost a sure bet.
That's why it will cost you more to buy one of those.
Do you follow it?
Okay, cool beans.
Let's take it back here to 75.
Now let's take the time to expiration and let's increase that we are going to do this one on the 11th.
10, 11, 2024.
Did you see what happened?
Let me take it again.
Let me go back.
Look at the call and the put.
When there were two weeks to expiration.
This was two weeks to expiration.
Okay.
Now let's take it forward back to the...
Do you see them?
Both the call and the put become more valuable because there is more theta premium.
Do you follow it?
Let's take it even further out.
Let's say that we got this one finishes on the 18th.
10, 18, 2024.
Watch the call and the put.
They should both increase in value because there is more time for that gas diffusion to hit either high or low.
And so in this one...
Look at that.
This is why I like to do it with these especially in a course like this.
Not so much so you know how you've memorized the equations and you can do them rapidly on your little calculator so you can see what the dynamics are.
Now let's take this out.
Let's just leave it where it is for the time being.
And let us change the standard deviation.
In other words, how widely the gas is diffusing.
Is it going like this, like this, or is it diffusing like this from a physics point of view?
And if we do that, let's change this.
Watch what happens to the call and the put prices if I change this to let's say 15%.
See how they both become more valuable?
Because there is more chance that both of them will finish in the money.
Because there is a wider spread of possibilities on it.
Take it another...
Ow, that hurt like hell.
Now let's look at the risk free rate.
Now this is a subtler one.
Let me take this one...
No, I'll leave it where it is. That's okay.
Let me take the risk free rate and let me raise the discount rate to 5%.
And let's see what happens to the call and the put price.
Call 99, put 156.
Do you see something weird that happened there?
Let me go back to the first ones.
The call was 99 and the put was 156.
Now let's bring the discount rate up.
The call and the put didn't do the same thing.
Let me back it up.
The call goes up, the put goes down.
Isn't that an interesting effect?
The interest rate dynamic for a call.
Now you can see it in the book or the PowerPoint presentations,
the put call parity theorem.
Puts and calls are related by an equation.
But the practical implication, which is what I'm trying to teach you here,
is that calls actually respond positively to increases in interest rates.
Puts respond negatively to increases in interest rates.
And vice versa.
So if you reduce interest rates, that tends to bring down call prices,
but it tends to improve put prices.
So what happened with the Fed cutting that discount rate more than we expected?
I can't show it to you.
I should have captured some data from a few days ago.
But what it did was it bumped the calls and the puts in opposite directions.
So when the interest discount rate went down, call prices eased up a bit.
But put prices went up a bit.
And if you think of the logic of it, just the financial logic of it,
higher interest rates tend to suppress business activity
because the present value of future expected cash flows goes down.
Well, wouldn't that be adverse to a stock price?
So that's what's going on here.
It's simply the mathematics, universe's mathematical way of taking that into account.
So whatever we think we're doing in business,
we're really just the complete slaves of processes that happen way beyond finance.
I just thought I'd bring that one up.
It's kind of an interesting little thing.
But with respect to the Black-Scholes options pricing model, the equation itself,
I mean, you can do the equation and all these formulas I've shown you,
they're just doing it for you.
So in the book's homework, they give you all the,
find all of these formulas, the risk-free rate, the standard deviation,
the natural log of P of the underlying price over the strike price, and all of that.
This, all you have to do is put in the numbers and it will come out for you.
You'll be just fine using the Black-Scholes, this model right here.
The only caution I have is, it looks like there's some kind of a,
it's getting pissy about this little piece over here.
So when you put in your expiration date,
put it in this slot right here instead of up there at the top.
I'm not sure what's going on with that.
But once you put it there, it will calculate the number of years,
the number of days divided by 365, and then it will reflect it over here.
This is where the action happens on it, is right in here.
And again, this is the Black-Scholes options pricing model.
Oh, I did want to show you one more thing.
Sorry about that.
This is geek stuff, the numbers.
Okay, we've got all these parameters put in here.
What happens if there's a dividend?
Now remember, a dividend wouldn't matter unless it occurs during the life of the option.
It wouldn't matter because it's not going to affect, well, tiny bit maybe,
because of the expectation of the dividend, but realistically.
But watch what happens if there is a dividend that is paid during the life of the option.
Watch the call in the put, 0.99 and 1.56.
Let's say we have a 1% dividend.
Again, you have an odd effect.
Why did the call price reduce?
Because that dividend is actually going to decrease the price of the stock.
Because the dividend, as the dividend is approaching,
that dividend begins to impound in the price you would pay.
So if you buy the stock and the dividend is going to be there, you get gravy.
So in this case, the dividend as it's building, if the dividend, after it is paid, the stock price will drop.
That's the way dividends work.
It's an underlying, there's so much noise that you won't really see much of it.
But if these are dividend dates, ex-dividend dates, stock prices will do that.
See, because the dividend, if you're holding the stock, you get the dividend.
So as the dividend is approaching, the present value of it out here is small.
But as you get closer to the time the dividend is paid,
it begins to get closer and closer to the dividend itself.
And then when the dividend is paid, flub, and then flub, and all that.
So what's happening with the call option is that it is literally, it is, if you have a call option,
buy it here, expiration here, the price of the stock is going to drop during the life,
during the time you have the call option.
So you get kind of this little surprise effect in here.
And so call options tend to shave, be a little bit cheaper,
simply because they are going to, by definition, lose some value during the life of the option.
You're running toward a goal, and then you suddenly get on a hill and you fall backward.
Puts are the other way, though.
Let me go back, watch the put.
Puts would be the opposite way, simply because you're on the other side of the transaction.
You don't have to pay the dividend.
It's kind of a way you get the dividend.
But anyway, now one last thing just to show you something.
Let me take this option out.
Look at the theta.
That's the one that is the premium for length of time.
Let's get this option in closer.
Let's do 10, 11, 24, and watch the theta.
What happened to it?
Let's take it in.
10, 4.
The theta decay is getting stronger, because you're getting closer and there's not as much time for something great to happen.
That is saying that for every dollar the option moves, the underlying moves,
the call option will lose about 11.25 cents of its value.
That's one way of interpreting it.
There are other ways of interpreting it, too.
Now, where are we starting?
Let's take it in one more, 19.
Let's say 10, 4, 2020.
No, no, 10, no, 9, 26.
9, 26, 2024.
Watch the theta.
It's decaying at a very strong rate now.
14, almost 15% value lost for every change, $1 change in the underlying.
So that just shows you theta decay is real, and that's why as you get closer and closer to the expiration date,
the options will get closer and closer to either zero or just the stock price minus the strike price,
because the theta premium is just vanishing.
And every stock and options chain is going to be different.
There's no all of them do the same thing.
It has to do with the theta uses every last one of these parameters.
Some of them use only a couple of them.
But I do want to point out, what was I going to say?
Oh, your book will have you calculate some D1s and D2s in that formula.
For God's sake, try to do it this way instead of trying to, because if you're like me,
once I get about three numbers in and then I have to hit a divided by, I know I'm going to get the wrong answer.
But you might be better than I am at it.
But anyway, this is all for you.
And it's again, I strongly encourage you not to share this.
This is this is pro stuff.
This isn't the toys that you can share on to get likes on TikTok.
This is the real stuff.
But anyway, that's all I have for you today.
I thank you.