We all know what a bond is: it's a promise to pay a certain sum of money over a set amount of time, typically in the form of biannual interest payments, plus the price paid for that bond when it was created. And we all know what a stock is: it’s a share of the total profits a company will earn. One fun surprise in finance is that stocks, bonds, preferred stock, etc. can all be expressed as a combination of options, much like how any program can be expressed as an elaborate set of NAND gates.

(As a quick refresher, an option confers the right, but not the obligation, to buy (“call”) or sell (“put”) something at a specific time ("expiration") and price ("strike"). So a call option, which is effectively a bet between two parties that takes the form of a derivative contract, starts with someone making a bet: an option can only come into existence if someone sells it, which is usually called writing an option. That seller might be selling an option on something they own—you might own 100 shares of Ford and decide to sell someone the option to buy those shares from you next month for $10 apiece—but the seller doesn't need to own those shares in order to sell the right to buy them in the future. It’s actually pretty normal: when a stock price spikes wildly, and it's expensive to borrow shares to short it, some traders will write call options instead.¹ )

Let's consider a company that, for simplicity's sake, decided to issue a single bond that pays zero interest and matures in one year, at which point the bondholder gets $1m. The company has no other debts. You can think of buying this bond as the equivalent of buying a one-year call option on the company's assets with a strike price of zero, and simultaneously selling a one-year call option on those same assets with a strike price of $1 million: the bondholder is entitled to everything the company is worth up to $1m, but none of the value past that point. And who is the buyer of that second option? It's whoever owns the equity: if, a year from now, the company's assets are worth $900k, the bond is worth $900k and the stock is worth zero.

We can complicate this as much as we want; an interest-paying bond, for example, is a series of options on each slice of assets that can satisfy the interest obligation. A more complicated stack of obligations—maybe a mortgage, some senior debt, some junior debt, and then common stock—is just a big stack of options (the junior debt is a call option on the company's assets, struck at the value of the senior debt, paired with a short position in a call option at whatever the payout of the junior debt is). In other words, the financial obligations of a company can be described as "you own all of the company's assets past the point where more senior creditors have gotten paid, and you owe someone all of the appreciation of those assets past the point at which your own obligation is satisfied.”

In one sense, it's needlessly complicated to model things this way. Most of the time, it's pointless. But once you're locked in on the idea that every claim on a company's value can be modeled as a set of call options, you get to apply some options math, or at least options heuristics. Options traders will look at options' sensitivity to different factors, like the price of the underlying assets, the volatility, the passage of time, and the interactions between them. For our purposes, let's look at two useful heuristics:

1. An option's sensitivity to volatility is highest when the asset price is close to the strike price.

2. The sensitivity of a call option to changes in the underlying asset, the "delta," is 1 when the strike price is zero—an option to buy something for $0 is economically equivalent to owning it. The delta is .5 when the option is at-the-money: if you own a call option with a $50 strike on something worth $50, your option will go up by roughly $0.50 when the underlying asset moves up by $1.

So a stockholder, as the theoretical owner of the last call option in the stack, tends to like volatility. And they like volatility a lot more when their option is right at the money. Consider a company that is teetering on the edge of insolvency; its assets are worth about $1m and it owes its creditors $1m, too. Any weird thing they can do will either make the company worth less (the stockholder doesn't get harmed by this, because they were already going to get nothing) or make it worth more (all of which accrues to the stockholder). The bondholder in this scenario is long a call option that's in-the-money (i.e. they'll probably get something if the company goes bankrupt) but short a call option at-the-money (anything past $1m goes to the stockholders; the bondholder gets the $1m they’re owed, but nothing beyond that). An increase in volatility has a bigger effect on the value of the option the bondholder is short than the one that they're long.

Let's suppose that whatever volatility-inducing thing the company did—perhaps a pivot to AI!—has, in fact, increased the value of the underlying business. As that value goes up, the equity's "delta" also rises: the more in-the-money the stock is, the relatively closer the strike price is to zero.

One fun result of this is when we imagine a more complicated capital stack: $1m of senior debt, $1m of junior debt, and equity. Let's start with the same scenario: the company's assets are worth $1m. Suppose you think the next thing the company does will increase its value by, say, $100k. Do you buy the stock? The junior bond actually has a higher delta, because it's at the money; roughly half of that $50k accrues to that bond (and some goes to the senior bond, since a bond backed by a company with just enough collateral to cover it so long as nothing goes wrong is also not going to trade at 100 cents on the dollar). The equity gets what little is left. In percentage terms, the equity can do better, but in terms of pure dollar capture, the bond wins. In this scenario, where the junior bond is at the money and the equity is out of the money, the bond behaves like a stock and the actual stock behaves like an option! The legal label for the securities doesn't correspond to their economic reality.

Let's take another example, this time inverting it: when does a stock behave more like a bond? In other words, when is it the case that shareholder's gains are somewhat capped, and the residual claimant on the company's assets is somebody else? This happens in any company or industry where there's someone who has high pricing power and knows their work is essential. Which means it shows up in a surprising variety of places:

• At publicly-traded investment banks, the senior bankers have some ownership of the client relationship, and what the company sells is, fundamentally, their time. If Smith feels mistreated by Jones & Co., the natural result is the future existence of Smith & Co. instead. (You can view the increasing institutionalization of the hedge fund business as an industry-wide effort to fight against this: a trader who goes independent is offering a less diversified stream of profits than what a big firm can offer by bundling that trader's strategy with many unrelated ones, so the economically rational thing to do is to stick around even if the firm is taking a big cut of fees.)

• At unionized, capital-intensive companies, unions can capture at least some share of the company's future profits. An airline that can't fly or fix its planes isn't much of an airline. So higher profits tend to lead to higher wages. (Strictly speaking, the workers own something more akin to a convertible bond rather than equity, since they don't usually get an automatic pay cut if profits nosedive. But, per the discussion above, when the company's survival is more tenuous, that bond's value behaves more like a stock.) If you want to own actual equity in the airline industry, rather than a hybrid fixed income like product with uncertain terms, you should pressure the Air Line Pilots Association to go public.

• A company that's captive to management, where management is focused on enriching itself, is also essentially a business where shareholders are junior claimants. When times are good, they'll get enough money that they don't sue (or that if they sue they don't win), but management will probably reap most of the benefits. This doesn't take the form of managers literally paying themselves a bonus equal to all of the company's above-normal profits every good year. Instead, they get paid in status and leisure: a controlling shareholder can use company funds to buy a fancy headquarters, a private jet, to supply good jobs to in-laws, or to engage in low-return empire-building.

• And in the most literal case: companies with high stock-based compensation that are limited by their ability to hire well-paid employees often end up effectively owned by those employees, not by shareholders. This is most visible in acqui-hires, where the present value of jobs the acqui-hired employees get can exceed the return investors get. But it also shows up when companies struggle expensively to enter a tough market. Over its life as a public company, Lyft's cumulative stock-based compensation expense actually exceeds its market value. If you want something that legally counts as Lyft equity, you can buy it through your broker; if you want something that's economically Lyft equity, you'll have to work there.

This is all grim news for investors, but it should be a bracing source of clarity: the overwhelming default setup for businesses is that there is one owner, who is also a full-time employee and perhaps manager. The vast majority of the time, outside equity simply does not make sense, and the incentive alignment problem is so hard that it dwarfs the advantage of having access to more capital. And many of those companies look a lot more like a bond than equity: starting a barbershop or a lawn-care business is unlikely to lead to 100x growth in valuation over time, but it can be a nice business with repeatable profits.

The space of equity-like assets is in one sense broader than it looks, because there are so many economic roles that can have a similar payoff. And in another sense, it's narrower: only a handful of lucky or well-run companies will be able to consistently produce growth and reinvest their capital profitably. Legally, equity is ubiquitous, but finding something that economically behaves like equity, and in a good way rather than in a "take the first loss" way, is rare and special.

Read More in The Diff

The Diff has occasionally used this kind of thinking in describing companies and incentives:

• Monday’s post asked why so many more rich people made their money in equities than bonds even though bonds have the same risk-adjusted return. Bonds-as-options is part of the reason this is a fun puzzle.

• This piece on the difficulty of decentralizing finance also uses the “treat it as an option” framework to understand why automated margin lending and liquidity provision is so hard to pull off.

• Another related concept is the idea of equity as an option on future reinvestment.

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1. The price of the option reflects the cost to borrow the stock, since everyone else is trading off between shorting, selling call options, or buying put options, too. But options prices are set by the market-makers who trade them, and those market-makers, as a core part of their business, get very good at borrowing at the best rates possible. So the option market partly represents a way to tap into the financing ability of Citadel Securities, Susquehanna, and the like. Note that this is a very high-risk strategy, and it's entirely possible to save basis points from structuring a trade and then lose percentage points when the trade doesn't work out.

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