### Financial derivatives, payoff functions and portfolios: motivation

Key ideas in this post
• Payoff functions are central, derivatives are just ways to achieve specified payoff functions. Your entire portfolio is also a derivative. We are interested in payoff functions that maximise certain combinations of expectation, risk, and other moments (depending on the investor's preferences).
• Shorting is just "investing in the rest of the market" and is the natural way to get a payoff function of $-x$.

When I first saw the definitions of several financial assets, I found them completely arbitrary -- it's not that I didn't get the reason one would have them, but rather that I saw no way to immediately understand them or a starting point for reasoning about them mathematically. Other than what was perhaps the most basic asset -- stocks (and also bonds, physical assets, etc.) and their baskets -- all the derivatives (and things that aren't called derivatives) based on them seemed really artificial in their construction.

But this isn't exactly unfamiliar territory, is it? You've seen unmotivated definitions in mathematics, and you've seen that you need to put in quite a bit of effort to really motivate them and understand why they make perfect sense -- you've seen that, e.g. here.

So let's do the same thing with finance.

Let's start with a simple one: shorting.

There is a certain asymmetry in the definitions of longing and shorting, isn't there? It's the "borrowing a stock" part of the definition of shorting that introduces this asymmetry.

But if you've spent any time thinking about economics, the idea of borrowing something you don't have should be familiar -- it's what you do when you don't have any investment capital to start with, but you think you can grow the value of what you've borrowed by e.g. investing it in a stock. Let's phrase this in a slightly different (and by "slightly different", I mean "take the buying-selling dual of") way:

How to invest in a stock without money at hand: Borrow some money, immediately "sell" the money for some stocks -- after some time has passed, "buy" back the money by returning the stocks. If the value of the stocks have increased, you'll get more money in return and be able to repay the loan.

This is precisely symmetric to the situation of shorting -- longing an asset just means shorting money -- or more precisely, shorting the rest of the market.

The apparent asymmetry between longing and shorting comes back from the fact that you are much more likely to already own some of "the rest of the market" than to own a particular stock -- for example, the unbounded losses of shorting arise from the fact that it's much easier for a single stock's value to skyrocket than for money's -- so in longing, there may still be ways for you to earn the money to repay it even if the value of the stock drops, i.e. the value of your other assets (e.g. your labour or property) relative to money would not have dropped.

One advantage of this approach is that it is conceptually interesting -- and will hopefully allow us to transfer insights and ideas between stocks and shorts (except when certain approximations may be involved) -- another is that it immediately nullifies "moral" criticism of shorting, from e.g. Elon Musk, as it is really just the same as investing in the "rest of the market".

Wait a minute -- but what if you actually just invested in "the rest of the market"? That would clearly have a much lower return than shorting the stock directly, right? Except you're thinking about investing in the rest of the market by paying money, not by paying the stock you're betting against -- that's a bet for the rest of the market against money, not against said stock.

Well, shorting was an example where we wanted to bet that the price of an asset goes down. But in general, we may have any sort of weird prediction on the price of an asset -- maybe that it will "fluctuate a lot", or that it "won't exceed a certain level", or that it "will go up but only to a point", or that it "will reach a certain range". You may have any sort of elaborate probability distribution $\rho(x)$ on the value $x$ of the asset after a period of time. Given such a distribution, what you'd want to do (ignoring risk) is to maximise your expected return (minus the cost of buying the contract, of course):

$$\chi=\int {\rho (x)f(x)dx}$$
Where $f(x)$ is the payoff you get if the asset reaches the price $x$ -- this is called the payoff function.

Well, why not just take $f(x)$ to be arbitrarily high? Because the contract will be really expensive, of course. How expensive? Predicting that would require:
• not only the $\rho$ distribution on this asset as believed by each seller and buyer in the market
• but also the amount of capital they have and their beliefs about the future behavior of other assets in the market contracts on which they could buy instead
And that is still not to mention the fact that people do not maximise the expected value of profit per say, but have varying levels of risk aversion.

But that's alright -- we don't need to predict that. That price is crunched for us by the market and is the market price of the contract -- it is the market price. What's more important is to estimate $\chi = E_{\rho}[f(x)]$. Well, in fact, if we're concerned with risk, then we'd also be interested in the variance of the distribution -- and in general, an individual may also have a skewness or kurtosis preference (an example of a kurtosis preference would be among gamblers, who want heavy tails for the "big win").

In fact, $\chi$ can depend on multiple underlying assets:

$$\chi=E_\rho[f(\mathbf{x})]$$
Where $\mathbf{x}$ is the vector of prices of each underlying asset. In fact, this multivariate $f$ can represent your entire portfolio of derivatives on assets. If $f(\mathbf{x})$ can be written as a sum of functions of each component, this can be considered as some number of separate univariate derivatives -- the reason such a portfolio is still useful is that of risk management, especially if we use a $\rho$ that has some correlations (even otherwise, one may use a portfolio to mitigate risk but correlations allow us to target specific risks).

There is an alternative definition of the payoff function, where it is $f(x)$ minus the contract price, i.e. a profit/loss function. The problem with this is that not every function can be a profit/loss function. But it often does make sense, and in general, a profit/loss function is more versatile than a payoff function (i.e. can be defined sensibly for any asset, which may not be possible with the payoff function with assets that have buying/selling at various points in time).

(Think about how one may define a payoff function for shorting (shorting traditionally isn't considered a derivative because it isn't a contract, but I think that's an arbitrary distinction) -- the analog of the "contract price" is then the negative price you "buy" it at (i.e. the negative of the price you initially sell the stock you borrowed), and the negative value that you eventually "get" (i.e. the negative of the price you eventually sell it at) is the payoff function. So the payoff function is $-x$, and is indeed the reflection in the asset value axis of the payoff for a long. Check that the profit/loss functions are also reflections, albeit the interest on the stock you borrowed.)

It's crucial to get some practice constructing various financial derivatives, i.e. constructing derivatives that have a given payoff function (using the first definition).

$$f(x)=(a-x)I(x<a)$$
Such a function would be a useful alternative to shorting, as it doesn't allow arbitrary losses.

The whole discontinuity of the function really suggests to me a fundamental change in behaviour at the point $x=a$ -- like you just don't make the trade if $x\ge a$. This decision can only be made once the final price is discovered, so you must have bought a contract that gave you the option to make a transaction: that transaction must be selling, it must be executed after the price is realised, but it must be at price $a$, which is initially fixed.

This is called a put option -- you buy the option to sell a stock at a pre-decided price. To exercise the option, you instantly buy the stock and sell it at that pre-decided price. Obviously, this price matters -- otherwise, you would be getting a guaranteed nonnegative profit. This is really equivalent to insurance.

(Verify that the payoff diagram of the seller of the put option is the negative of that of what's above.)

There's a natural analog of this notion that reduces risks with longing.

Once again, we see that there's a fundamental change of behavior if the price drops below $x=a$ -- you just don't complete the transaction. So you've bought an option to do something. Well, you need to sell something to make money, but the intercept of the graph suggests that you're also buying the asset, albeit at a fixed price. So this is a call option -- you buy the option to buy a stock at a pre-decided price. To exercise the option, you exercise it, then immediately sell the stock you bought to make your profit.

(Once again, the payoff diagram is a bit misleading and suggests that this is strictly worse than just buying a stock -- remember that the cost of a stock is the entire original stock price, while the cost of the call option is much smaller. These costs are not integrated into the payoff diagrams, but are into the profit/loss diagrams.)

Essentially, call and put options allow you to work on hindsight.

One might wonder that a call option is perhaps not as useful as a put option -- there's not much to insure with longing, right? (compared to shorting) Perhaps, but there are certain other uses of call options that work together with put options in an interesting way, as we will soon see.