Chapter 22: Valuing Options: Binomial & Black-Scholes

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Welcome to the Deep Dive.

Today we're tackling a problem that, honestly,

it's stumped

experts for generations valuing options.

It sounds simple, but it's one of the trickiest things to get right.

It is.

And when a company decides to, say, invest in a new factory or figure out its capital structure or even just pay its top executives, they're dealing with options all the time.

They're everywhere.

These embedded options, the right but not the obligation to make some future decision.

So our mission today is to walk you through the really elegant and powerful mathematics that finally cracked this problem.

We're talking about the work of giants, people like Fischer Black and Myron Scholes.

And what they did was give us a way to put a precise dollar value on that corporate flexibility.

It was a revolution.

It really was.

Now, last time, we sort of established the five key variables that drive an option's value.

Right.

You've got the stock price, the exercise price, the interest rate, and the time until it expires.

And the big one, volatility.

Volatility is huge.

So you intuitively know, okay, if volatility goes up, the option's probably worth more.

More time, also worth more.

But that intuition only gets you so far.

You need a formula, a precise model.

Exactly.

But before we can even get to that formula, we have to confront the central mystery.

The reason this field was just stuck for decades.

Why do our standard valuation techniques, you know, the ones we use for everything else?

Like discounted cash flow,

DCF.

Right.

Why does DCF completely fail the moment you try to apply it to an option?

This is where the whole standard financial toolkit, I mean, the absolute bedrock of capital budgeting, just, it completely breaks down.

How so?

Well, DCF relies on one critical assumption.

A stable, known opportunity cost of capital.

A single discount rate.

Yeah.

Your WACC or something similar that reflects the asset's risk.

Precisely.

But the risk of an option is, well, it's fundamentally perpetually unstable.

It's always changing.

Just think about what a call option really is.

It gives you this incredible leverage.

Right.

You can control shares worth thousands of dollars for maybe just a hundred bucks.

And that's why an option is always riskier than the underlying stock.

It has to be.

So it'll have a higher beta, a higher standard deviation of return.

That's higher.

It just amplifies every single movement, both the gains and the losses.

Okay.

Let's unpack that a little for the listener.

If we know the option is riskier, why can't we just, I don't know, use the capital asset pricing model, find its higher beta, and just use that higher discount rate forever?

Great question.

And the answer is because that beta, that measure of risk, it changes every time the stock price blinks.

It's not a constant.

Not even close.

The option's whole risk profile shifts dynamically based on one thing.

How close the

imagine an option that's deep in the money, meaning the stock price is way above the exercise price.

Right.

That option is almost certainly going to be exercised.

Its payoff is almost a sure thing.

So it starts to behave just like owning the stock itself.

Exactly.

And it's risk.

And therefore the return investors demand for it is relatively low.

It's almost as safe as the stock.

Okay.

But what if the stock price tanks and that same option is suddenly far out of the money?

Now it's a completely different animal.

Its survival is a long shot.

It's highly speculative.

It could be worthless.

It's a binary outcome.

You get a massive gain or a total loss.

So its risk just skyrockets.

And here's the crazy contradiction, the thing that kills the DCF model.

What's that?

When the stock price rises, the options expected payoff goes up, but at the same time, its risk actually goes down.

Because it's becoming more of a sure thing.

Right.

And conversely, when the stock price falls, the payoff you expect drops, but its risk increases.

Wow.

Okay.

So you have this asset where the required rate of return that investors demand is changing literally moment by moment.

With every single movement of the underlying stock.

So if the discount rate is a moving target, you can't use a single constant rate.

The whole formula for discounting expected cash flows just collapses.

You're stuck.

The standard corporate finance tools won't work.

You need a completely different way of thinking about it.

And that new mechanism, that new way of thinking is truly one of the most brilliant breakthroughs in finance.

It's the whole origin of the black skulls model.

It's a huge eureka moment.

It is.

The big realization was this.

If you can't value the option directly, maybe you can value it indirectly.

By replicating its exact payoff structure using assets that you do know how to value.

So this is the trick, right?

The one that was pioneered by black skulls and Merton, you set up an option equivalent.

And this equivalent is just a simple portfolio.

It's a combination of a position in the stock.

And a risk -free borrowing or linting position, a loan basically.

That's it.

And the law of one price demands that the cost of creating this perfectly matched portfolio.

Has to be the value of the option itself.

They have to be the same.

Let's make this tangible with a really simple example, a one -step world.

Okay.

Let's imagine we're looking at a six -month call option on, say, a fictional Amazon stock.

The exercise price is $1 ,830.

And let's say the current stock price is also $1 ,830.

It's at the money.

Perfect.

And the risk -free interest rate for those six months is a simple 2%.

Now, to make the world simple, we're going to assume the stock can only do one of two things in the next six months.

Right.

It can either go up by 20 % to $2 ,196.

Or it can fall by about 16 .7 % down to $1 ,525.

Just two possible outcomes.

Okay.

So let's figure out the option's payoffs.

If the price jumps to $2 ,196.

Well, you exercise.

You buy at $1 ,830 and the stock's worth $2 ,196.

The option is worth $366.

Simple subtraction.

And if the price falls to $1 ,525?

It's out of the money.

The right to buy at $1 ,830 is worthless.

So the payoff is zero.

$366 or zero.

Those are our two goalposts.

Exactly.

Now, the magic is in constructing a portfolio of stock and debt that guarantees those two outcomes.

This is our replicating portfolio.

And the key, the absolute linchpin in building this portfolio is a concept called the option delta.

Delta, yes.

Can you explain what delta is for us?

Absolutely.

Delta, which we symbolize with the Greek letter delta, is basically the hedge ratio.

The hedge ratio.

It tells us the precise number of shares we need to buy to replicate the payoffs of one single option.

It's a measure of the option's sensitivity to the stock price.

So how do you calculate it?

Mathematically, it's just a ratio.

You take the spread of the possible option prices.

So $366 minus zero.

Right.

And you divide that by the spread of the possible share prices.

Which is $2196 minus $1 ,525.

That's a spread of $671.

You run that ratio, $366 divided by $671, and you find that you need to own

four or five shares of the stock.

Okay, so just over half a share.

That's the stock part of our downside.

It can't go below zero.

So we need the borrowing piece, the leverage, to mimic that specific risk return profile.

Exactly.

We need a second risk -free position.

We'll call it B.

And it represents the present value of the loan we need to take out.

And this loan is what guarantees the perfect replication in both scenarios.

It's what makes the math work perfectly.

And if we do the algebra, we find that to make the payoffs match, we have to borrow the present value of $831 .82.

Okay, let's just walk through the genius of this.

We buy four or five shares, and we take out this specific loan.

Let's check it.

Scenario one.

The stock soars to $2196.

Our shares are now worth .5, four or five times that, which is $1 ,197 .82.

And you have to repay your loan.

$831 .82 plus 2 % interest.

Wait, I did the math wrong.

The loan amount is what you owe in the future.

So the PD is what you borrow.

Let's start over.

The future value of the loan is $831 .82.

Correct.

So in the upstate, your shares are worth $1 ,197 .82.

You repay the loan of $831 .82.

And the net payoff is exactly $366.

It matches.

It's perfect.

Now scenario two.

The stock falls to $1 ,525.

Your shares are now worth .545 times $1 ,525, which is exactly $831 .82.

And you still have to repay that same loan, which is exactly $831 .82.

So your net payoff is zero.

Zero.

It's a perfect match in both states of the world.

The replication is flawless.

It's a mathematical twin.

So because the law of one price has to hold, the value of our option today must be the net cost of creating this portfolio today.

That's the conclusion.

So what's the cost today?

Well, we have to buy the shares.

That's .545 times today's price of $1 ,830.

That costs us $998 .18.

And from that, we subtract the money we received from the loan.

The loan's future value is $831 .82, so we need its present value.

We discount that back at the 2 % risk -free rate.

So that's $815 .51.

So the cost of the shares minus the cash from the loan.

That net cost, which is the value of our call, is $182 .67.

And the most profound thing about this is that we found the value without ever once having to guess or calculate a risk -adjusted discount rate.

We completely bypass the whole DCF problem.

Completely.

And the economic insight here is fascinating too.

Because B is negative, we borrowed money.

Yes.

It confirms that buying a call option is financially the same thing as owning a highly levered investment in the stock.

That's where the risk comes from.

That leverage is the source of the option's massive risk profile we talked about at the start.

Okay, so that value,

$182 .67,

it's locked in.

It has to be.

If the option traded on the market for, say, a dollar more or a dollar less, a smart trader could make a guaranteed risk -free profit.

An arbitrage opportunity.

A pure arbitrage.

They'd just trade the mispriced option against the replicating portfolio and pocket the difference.

Which means the option's price can't possibly depend on how risk -averse investors are.

It doesn't matter.

It doesn't matter if the market is full of aggressive lions or cautious chickens, as the textbook likes to say.

Right.

The price has to stay pinned at $182 .67 just by the threat of arbitrage.

And this.

This is the crucial leap in logic that opens up a whole new way to solve the problem.

Okay.

Since investor risk attitudes don't matter for the price, we are free to make the most convenient assumption possible.

Which is?

We can pretend that all investors are risk -neutral.

We just assume nobody cares about risk.

We pretend they're indifferent to it.

And the convenience of that assumption is just immense.

Why?

What does that unlock?

Well, if investors are truly risk -neutral, then the expected return on every asset.

Even a risky stock.

Must equal the risk -free rate of interest.

There's no such thing as a risk premium in that world.

I see.

So you use that principle to work backwards.

Exactly.

We use it to calculate a hypothetical, what we call a risk -neutral probability.

We call it P -star.

And P -star is the probability of the stock going up that would force the stock's expected return to equal that risk -free rate of 2%.

That's all it is.

There's a formula that connects the risk -free rate R to the upside move U and the downside move D.

So we plug in our numbers for the Amazon scenario.

And we find that P -star equals 0 .50909.

Okay.

So in this hypothetical, mathematically convenient world, the stock has a 50 .9 % chance of rising.

Right.

But we need to be crystal clear here.

Why is that not the true probability that Amazon's stock will rise in the real world?

Because in the real world, investors are definitely not risk -neutral.

They're highly risk -averse.

They demand to be paid for taking on risk.

A risk premium.

And because of that, a risky stock like Amazon must have a real expected return that is much higher than the 2 % risk -free rate.

So for it to have that higher real return, the real probability of it going up must be higher than 50 .9%.

Has to be.

The difference between the true probability and our P -star.

That's basically the market's risk premium captured in probability form.

Okay.

So P -star is just a mathematical tool.

It's a construct.

It lets us create a world without risk.

A certainty equivalent world.

Precisely.

And if we operate in this risk -neutral world, we can now use P -star to calculate the expected future value of the call option.

Right.

The options were 366 if the stock goes up 0 if it falls.

So we just multiply the payoffs by their risk -neutral probabilities.

So that's 50 .9 % times 366 plus 49 .1 % times 0.

Which gives us an expected future value of $186 .33.

And since we calculated that expected value under this assumption of risk neutrality.

It's like a certain cash flow in that hypothetical world.

Which means we can discount it back to today using the simplest rate there is.

The risk -free rate.

2%.

So 186 .33 divided by 1 .02.

Gives us $182 .67.

It's the exact same number.

That's incredible.

It is.

We have two completely different paths.

The replication method in the real world and this risk -neutral method in a hypothetical world.

And they lead to the exact same answer.

That's the real genius of option pricing theory.

Both methods work because they're both finding the certainty equivalent cash flow.

They just take different routes to get there.

And for anyone focused on corporate decisions, this is the core insight.

When you price an option, you're not really valuing the expected outcome.

You're valuing the cost of creating a perfect risk -free hedge against that outcome.

That's it.

It's a totally different mindset.

Okay, so let's apply this powerful framework to the other side of the coin.

The put option.

Right.

Let's look at the six -month Amazon put.

Same exercise price.

$18 .30.

So the payoffs here are inverted.

If the price rises to $21 .96,

the put is worthless.

Zero.

But if the price falls to $15 .25, the put is deep in the money.

It's worth $305.

$18 .30 minus $15 .25.

So now we calculate the put's delta using that same spread formula.

Okay, the option spread is now 0 minus 305.

The negative number.

And the stock spread is the same 671.

Right.

And that gives us a put delta of negative 0 .45455.

The negative sign is the key here, isn't it?

It's crucial.

It confirms that inverse relationship.

When the stock price goes up, the value of the put goes down.

And from a structural standpoint,

it means that to replicate a put option, you actually have to short the underlying asset.

You have to sell it.

That's exactly right.

To build this replicating portfolio, you would short 0 .45 Amazon shares.

And when you short a stock, you get cash today.

So the other component of the portfolio has to be lending that cash back out.

You lend the present value of $998 .18 to the bank.

A completely inverted portfolio.

And when you calculate the net cost of setting up that short stock, long bond portfolio, you find it to be $146 .79.

Let's just double check with the risk neutral method using the same P star.

The expected future value of the put is 50 .9 % of 0 plus 49 .1 % of $305.

Which comes out to about $149 .73.

Discount that back at the 2 % risk -free rate.

And you get precisely $186 .79.

The models align perfectly again.

They have to.

And since we're dealing with European options, so no early exercise,

we have this fantastic way to check our work.

Put call parity.

Right.

This ensures there's no arbitrage possible between the call and the put.

It's a fundamental identity that has to hold for European options with the same strike immaturity.

It basically says that a certain combination of a put, a call, and the stock creates a risk -free position.

It guarantees you receive the exercise price at maturity.

Which means there's a fixed relationship between their prices today.

The value of the put must equal the value of the call.

Plus the present value of the exercise price.

Minus the current stock price.

If we plug in our numbers.

Call value of $182 .67 plus the PV of $18 .30 minus the current stock price of $18 .30.

You get $146 .79.

It's perfect.

A perfect check.

It acts as the great economic anchor for the European options market.

It ties everything together.

This one -step model was brilliant for getting the concept down.

Replication, risk neutrality.

But it's just not realistic.

Hopelessly unrealistic.

The idea that a stock can only go to one of two places in six months is not how markets work.

So this is where the binomial method comes in.

The multi -step model.

Right.

Instead of one giant six -month leap, we just chop the options life into many smaller sub -periods.

So maybe two three -month steps.

Or 52 weekly steps.

The more the better, really.

And as you add more and more steps, you're creating this much more realistic tree of possibilities.

What does that do to the final picture of outcomes?

It allows the possible stock prices at maturity to start looking like a continuous log -normal distribution.

Which is how stock prices actually behave.

Exactly.

A stock price can't fall below zero, but it can, in theory, rise forever.

The log -normal distribution captures that skewed reality perfectly.

So the binomial method, if you chop it finely enough, it just mimics that smooth, realistic curve.

That's the goal.

And that's what makes the model so practical.

Let's look at the mechanics with a simple two -step example.

We'll split our six -month option into two three -month intervals.

Which means we need new, smaller inputs.

A new quarterly risk -free rate, say 1%, and smaller up -and -down moves for the stock.

And we also need a new risk -neutral probability.

A new P star for this shorter interval.

Right.

We run the same formula, but with the smaller quarterly numbers.

Let's say it gives us a new P star of 0 .5063.

And this is where the process gets really interesting.

We don't start today and move forward.

No, you do the opposite.

You have to start at the end and work your way backward through the tree.

So we start at month six.

We look at all the possible final stock prices.

In a two -step model, there are three possibilities.

Up, down, and down, down.

And we calculate the options value at each of those final nodes.

So $538, zero, and zero.

Now we take one step back in time to month three.

To those intermediate nodes.

Let's look at the high node where the stock has already gone up once.

From that point, we look forward one period.

The option can either go up to $538 or down to zero.

So we use our new three -month P star to find the expected value of the option from that month three node.

Exactly.

We find that expected future value is $272 .52.

Then we discount that single value back for one quarter using our 1 % risk -free rate.

Which gives us an option value of $269 .84 at that high intermediate node.

Right.

And if the stock fell in the first three months, well, the price at that low intermediate node is so low that the option is guaranteed to expire worthless.

So its value there is just zero.

So now we have the two possible values at month three to $69 .84 or zero.

And we just repeat the process.

Yeah.

We step back to today to month zero.

We treat those two values as the payoffs for the first period.

We use P star one more time, find the expected value, and discount it back one final time.

And the result for this two -step model is a call value of $135 .27.

Which is a lot lower than the 182 .67 we got from the crude one -step model.

And that really illustrates the need for this added realism.

The single -step model was just overstating the chance of hitting that one really high upside price.

It was too simplistic.

Now, you mentioned that the up and down moves have to be chosen carefully to be consistent with the stock's annual volatility.

How does that link work?

It's fundamental.

There's a specific formula that ensures that whether you use one long step or 52 tiny little weekly steps, the total amount of uncertainty over the year stays the same.

So it's grounded in the real observed volatility of the stock.

It has to be.

The formula links the up move factor directly to the annual standard deviation sigma and the length of the time interval.

As the intervals get shorter, the up and down moves have to get smaller.

Which makes sense.

A stock doesn't jump 20 % in a day.

Exactly.

This is what ensures the model converges correctly and reflects reality.

Okay, and this is where the binomial method goes from just being a tool for option traders to an absolutely essential tool for corporate managers.

This is the jump to real options analysis.

So a company is evaluating a big project, say a 10 -year mine development.

That project is full of choices.

The option to expand the mine if prices are high.

The option to temporarily shut it down if prices crash.

These are valuable options.

And if you're using a standard decision tree to analyze that project?

You need a discount rate.

But as we've established, since these future decisions are options, their risk changes dynamically.

So you can't just plug in the company's WACC.

You absolutely cannot.

If a manager applies a constant DCF rate to a project that has this valuable, embedded flexibility, they are guaranteed to underestimate the project's true value.

And they might end up rejecting a really valuable investment.

It happens all the time.

The binomial method becomes the only practical way to solve those complex managerial decision trees correctly.

You have to use option pricing theory.

Okay, so now we push this logic to its absolute limit.

To the theoretical extreme.

What happens if we keep subdividing the options life into smaller and smaller pieces?

An infinite number of intervals.

Where the stock price is just constantly, instantaneously changing.

What does our binomial tree become?

It converges.

It smooths out into the elegant, continuous time solution that we know as the Black School's formula.

This sounds impossibly complex, but you're saying the core idea is actually pretty simple.

The core intuition is identical to what we've already done.

The Black School's formula just states that the value of the call, C, is equivalent to that same replicating portfolio.

So it's the investment in the stock minus the present value of the loan.

That's all it is.

Formally, we write it as C equals N of D1 times S minus N of D2 times the present value of the exercise price.

That's a powerful structure.

You can see the two pieces right there.

N of D1 times the stock price, S.

That's your investment in the stock.

And N of D2 times the PV of the exercise price.

That's the loan you have to take out.

Which means that N of D1 term, that's just the option delta in continuous time.

It's the hedge ratio.

So if N of D1 is, say, 0 .57,

it means the replicating portfolio requires you to own 0 .57 shares at that exact moment.

At that instant, yes.

And it's constantly changing.

And the formula uses the same five inputs we started this whole journey with.

The same five.

Stock price, exercise price, time to maturity, the risk -free rate, and the stock's annual volatility, sigma.

Okay, but what about those strange functions?

N of D1 and N of D2.

They always look intimidating.

Can we break down what they mean conceptually?

Let's focus on their roles.

We already said N of D1 is the options delta.

It's how many shares you need in your hedge.

Got it.

What about N of D2?

N of D2 is related to the probability of exercising the option.

If N of D2 is high, like 90%, it means there's a very high probability that the option will finish in the money.

And since the loan component of our portfolio is there to cover the exercise money.

N of D2 basically determines the size of that loan.

It's the probability in that risk -neutral world that the option will expire in the money.

That makes so much more sense.

Okay, let's just quickly run the numbers for our Amazon example in this continuous model.

Just to see how close the two -step binomial got.

All right.

We feed our five inputs into the machine.

We run the numbers through the complex formulas for D1 and D2.

Which are just functions of volatility, time, and the ratio of the stock price to the exercise price.

And that calculation gives us two key intermediate numbers.

D1 is 0 .1998 and D2 is 0 .0175.

And we then just look these up in a standard normal distribution table or use a spreadsheet function.

Right.

And we find that N of D1, our delta, is 0 .5792.

So we need to own about 58 % of a share.

And N of D2 is 0 .5070.

Now we just plug those results back into that simple structural formula.

So it's 0 .5792 times the stock price of 1830.

Minus 0 .5070 times the present value of 1830.

The cost of the required stock position today is $1 ,059 .88.

And the present value of the loan we need is $909 .55.

The resulting black school's value is $150 .33.

And that's the exact number that our multi -step binomial process was getting closer and closer to.

It's the limit.

And what it means is that to perfectly hedge this option, an investor has to invest $1 ,059 in the stock and borrow $909.

And that combination perfectly replicates the options payoffs.

But only if you're continuously rebalancing the hedge.

And that's the big theoretical leap.

It assumes continuous adjustment of the delta.

OK, I have to challenge that point.

Continuous adjustment.

I mean, in the real world, no one can trade every nanosecond to perfectly rebalance.

There are transaction costs, frictions.

How does the theory handle that?

That is the single greatest practical limitation of the pure black school's theory.

It is predicated on this idea of continuous, costless hedging.

Which is impossible.

It is.

In reality, investors have to rebalance periodically.

Maybe daily, maybe hourly.

And every time the stock moves a lot, the hedge ratio changes and you incur costs to rebalance.

So does that invalidate the model?

Not really.

The theory remains incredibly robust because even with imperfect periodic rebalancing, the risk of mishedging tends towards zero as your trading frequency goes up.

So for big institutional investors with super low transaction costs.

For them, the black school's formula provides an astonishingly accurate price.

The theoretical elegance is, for all intents and purposes, practically valuable.

Let's go back to this concept of risk.

We've said options are risky.

Now that we have the replicating portfolio, can we actually quantify the leverage?

Can we calculate the options beta?

We can.

The beta of the option is just the weighted average of the betas of its two parts.

The stock and the risk -free loan.

And the beta of a risk -free loan is zero.

It's zero.

So the formula simplifies.

You just take the ratio of the investment in the stock to total value of the option.

And multiply that by the stock's own beta.

That ratio is the leverage factor.

That's the multiplier.

OK.

So in our Amazon example, we invested about $1 ,060 in the stock.

And the option itself was only worth about $150.

The ratio, the leverage factor, is about 7 .05.

So if we assume the underlying Amazon stock has a market beta of, say, 1 .5.

We multiply that by our leverage factor of 7 .05.

And you get an option beta of 10 .58.

An incredible number.

A beta of over 10 is.

It's almost unimaginable in the stock market.

That means this option is over seven times as risky as the stock and over 10 times as risky as the market itself.

It just confirms what we've been saying.

Options are leverage machines.

And for a portfolio manager, this tells you that if the market moves by 1%.

You should expect your option position to move by over 10%.

It's why volatility, sigma, is such a massive driver of the option's value.

It is the fuel for this leverage.

We can actually see this on a graph, right?

The option value curve.

Yes.

If you plot the option price against the stock price, you get this upward sloping curve.

And the slope of that curve at any given point, that slope, is the delta.

So if the stock price is really low, the option is way out of the money.

The curve is almost flat.

The delta is near zero.

A small change in the stock price does almost nothing to the option's value.

But if the stock price is really high, deep in the money.

The curve gets steeper and steeper, approaching a 45 -degree angle.

The delta approaches one.

So the option starts behaving almost exactly like owning the share itself.

Exactly.

The key takeaway is that delta measures your risk exposure, and that exposure is never, ever constant.

Okay, let's pivot to some practical applications in corporate finance.

A classic one is executive stock options.

Why do companies need black skulls just to pay their people?

Because they have to.

Accounting rules require companies to record the value of stock options they grant as an expense on their income statement.

Just like Seller.

And to calculate that expense, they need an accepted valuation model.

And black skulls is the industry standard.

This forces the company to actually quantify the impact of its own volatility.

How so?

Well, look at the textbook's example of two companies.

You have Establishment Industries, which is a safe, stable company with a low volatility, say, 24%.

And then you have Digital Organics, a risky volatile company with a sigma of 36%.

If both companies grant an option with the exact same term, same strike price, same maturity,

the option issued by the riskier company, Digital Organics, is worth significantly more, maybe $7 .40 versus $5 .26.

So the company with the riskier stock has to book a bigger expense for the same option grant.

Yes, because volatility is a gift to the option holder.

The downside is capped at zero, but the upside is unlimited.

So the riskier the stock, the higher the chance of a massive upward swing and the more valuable the right becomes.

A company with volatile stock is, in effect, paying its executives more, even if the options look the same on paper.

Another application is invaluing warrants.

What are those?

Warrants are basically just long -term call options that are issued by the company itself, often to raise capital.

Long -term meaning?

Seven, 10 years sometimes, which makes them far more sensitive to volatility and time than a typical exchange -traded option.

Right.

And the textbook gives that example of Owens Corning.

A great example.

When they came out of bankruptcy, they issued warrants.

The stock was trading at $30, but the exercise price was over $45.

They were way out of the money.

So you'd think they were worthless.

But they traded for $6 each.

Why $6?

Because black schools could capture the immense value of that long, seven -year time horizon and the company's inherent volatility.

It recognized that over seven years, there was a very real chance the stock could soar past $45.

Without a formal model,

valuing that long -shot bet would just be a wild guess.

Pure guesswork.

And what about portfolio insurance?

How do options play a role there?

Portfolio insurance is just a fancy term for buying protection against a market crash.

So you're buying a put option.

Exactly.

A big pension fund that owns a diversified portfolio can buy a long -dated put option on the entire market index like the S &P 500.

And that put guarantees them a floor, a price at which they can sell the index, protecting them from a crash.

Right.

And the black schools formula is what's used to calculate the price of that protection, the cost of that insurance premium.

Finally, let's talk about the VIX.

This is the ultimate example of using the model in reverse.

It is.

Instead of starting with volatility to find the price, you start with the market price to find the volatility.

It's called implied volatility.

Exactly.

It's the value for Sigma that when you plug it into the black schools formula makes the formula's price equal the price you actually see trading on the exchange.

So if an S &P 500 call option is trading at,

say, $195 and we know all the other inputs.

We can reverse engineer the formula to find the one value of Sigma that makes the equation work.

If the answer is 19%, it means the market collectively is expecting 19 % annual volatility.

And this measure based on S &P 500 options is the famous VIX, the fear index.

That's it.

Why does it spike so dramatically during a crisis?

Because the VIX is a direct measure of the cost of portfolio insurance.

When investors get scared and anticipate uncertainty, the demand for those protective put options, the insurance, goes through the roof.

Higher demand means higher prices.

And since volatility is the main driver of an options price, the only way the formula can justify that higher price is for the implied volatility, the VIX, to spike.

So during the 2008 credit crunch or the start of the COVID pandemic.

You see the VIX just explode.

It's a real time indicator of the market's collective expectation for future turbulence.

It's incredibly powerful.

Okay.

So all of this elegant math we've discussed so far really assumes two things.

European options and no dividends.

Right.

European options can only be exercised at maturity.

And we've assumed the stock pays no dividends.

We have to introduce these two real world complications now.

Early exercise and dividend payments.

They can break the simple black skull solution.

Let's start with American calls on stocks that don't pay dividends.

This one is actually very straightforward.

The conclusion is absolute.

You should never exercise an American call early if the stock pays no dividend.

Never.

Why not?

If the call is deep in the money, why wouldn't I just take the profit now?

Because if you exercise early, you have to pay the exercise price today.

That cash, if you had just held onto it, could have been earning interest for you risk -free until the option's maturity date.

You're giving up the time value of money.

Exactly.

And since the option itself always has some time value on top of its intrinsic value, it's always better to sell the option than to exercise it.

So its value is the same as a European call and black skulls works perfectly.

It does.

But now consider American puts.

It's the complete opposite.

Early exercise can be rational for a put.

It can be.

Imagine you buy a put with a strike of $100 and the company's stock immediately crashes to $5.

The put is worth $95.

And the stock can't fall below zero, so the most you can possibly gain from here is another $5.

I see.

Why wait six months to get your $100?

If you exercise now, you get that $100 exercise price today and can immediately start earning interest on it.

So for American puts, early exercise can be optimal.

Which means an American put is always worth more than or equal to its European twin.

And since black skulls assumes no early exercise,

it's only an approximation for an American put.

It's just a lower bound.

If you need the true value, you have to go back to the binomial method.

Why does that work?

Because it's step by step.

At every single node in the binomial tree, you can check.

Is this option worth more if I exercise it right now or if I hold it for one more period?

You take the maximum of the two choices at every step.

And then you work backward.

That's how you find the true optimal value for an American option.

Okay, now let's add the final layer of complexity.

Dividends.

Cash payments that the option holder misses out on.

For a European option on a dividend paying stock, the fix is actually pretty simple.

How do you adjust for it?

You just have to adjust the stock price input in the black skulls formula.

Since the option holder doesn't get the dividend, the stock is effectively worth less to them.

So you reduce the stock price S by the present value of all the dividends you expect to be paid before the option expires.

You just use that lower adjusted stock price in the formula and you're good to go.

And this logic applies to other assets too, right?

It does.

If you're valuing an option on a foreign currency, you have to deduct the present value of the foreign interest you miss out on.

That interest acts just like a dividend.

And now for the most complex case of all.

An American call on a stock that does pay a dividend.

Now the dividend creates a real trade -off.

It creates an incentive for early exercise.

Well, the only time it would ever be rational to exercise is right before the stock goes ex -dividend.

By exercising, you get the stock and you capture the upcoming dividend.

But you lose the time value of money on the exercise price.

Exactly.

You lose the interest you could have earned by paying that money later.

So it's a trade -off.

Is the dividend I gain bigger than the interest I lose?

And if the dividend is high and interest rates are low?

It might be optimal to exercise,

but only at those specific moments right before an ex -dividend date.

So again, we have an optimal decision that has to be evaluated at specific future points in time.

Which means, once again, the flexible step -by -step binomial method is the only tool that can solve it exactly.

It lets you check at every ex -dividend date on the tree, whether to exercise or hold, and then you work backward to find the true highest possible value today.

It's the workhorse for any option with complex real -world features.

Well, this has been an incredible deep dive, taking us all the way from the basic failure of DCF to the ultimate elegance of the Black Skulls model.

We've seen that valuing options is less about predicting the future and more about building a perfect hedge.

It's a different way of looking at value.

Let's quickly recap the powerful financial principles we've covered today.

First, you just can't value options with a constant discount rate.

Their risk is always changing.

So you have to use replication or the risk -neutral method.

No way around it.

Second, the binomial model is that flexible step -by -step framework that's essential for any option with real -world features like early exercise or dividends.

It allows for that optimal decision -making at every point in the future.

Third, the Black Skulls formula is the continuous time benchmark.

It's built on those five key inputs and is structured as an investment in the stock minus a loan.

And that N of D1 term is the continuous delta, the instantaneous hedge ratio you need.

And fourth, because of that inherent leverage in the replicating portfolio, options are just substantially riskier than the underlying stock.

They carry those extremely high betas, which is exactly why volatility is such a core driver of their value.

And we saw throughout our discussion that volatility isn't just risk to be avoided.

For an option holder, it's value to be captured.

That flexibility is where the value comes from.

Which leads to a final thought for all the managers and investors out there.

Remember that the Black Skulls model doesn't just put a number on a right to trade some stock.

It actually quantifies the value of uncertainty itself.

That's a powerful idea.

It is.

The next time you're evaluating a corporate investment, a factory expansion, an R &D project, whatever it is, and you realize that project has flexibility,

the option to abandon, the option to expand,

you have to remember you are not valuing a sure thing.

You're valuing an option.

You're valuing a strategic bet on future volatility.

You're valuing uncertainty.

And understanding that distinction is absolutely central to making sound financial decisions in the modern corporate world.

We hope you feel thoroughly well informed.

We'll catch you next time for the Deep Dive.