Thread by @LilyWrites4

Okay, a thread.
So one of the things that interests me is options theory applied to more macroscopic phenomena, and one of the more interesting and salient ways is the potential existence of the Fed Put (formerly known as the Greenspan Put).
en.wikipedia.org/wiki/Greenspan…

There's a lot of hoopla about asset price inflation, and it's pretty accurate by any metric that there's an acceleration in beta, especially over the past few years. What this means in plainer English is that the market isn't just increasing in value (the market being well

everything, down to Pokemon cards), but that it's increasing *faster* in value than before. This is puzzling of course, because the general/old-fashioned mode of understanding equity returns is as a function of the risk free rate, usually proxied to the 10 year Treasury.

This has interesting implications in multiple markets (including the bond spread puzzle), but the general popular consensus is that it's related to a rapid decrease in the value of the dollar itself. However, this is problematic in many ways.

First and foremost, as @AOC noted recently, inflation should impact across sectors fairly equally, and it's clear it hasn't been doing that. Secondly, since the 2008 era, we've often been in a period of deflation, which doesn't jive with the theory that inflation is caused by QE.

Okay, so let's pretend inflation isn't a thing here. How could the Fed's actions be impacting the markets and causing acceleration in returns? Well, let's turn to option theory and find out.
In the most general mechanism, we can model "the market" as an Ito process of the form:

dX_t = mu(x,t)dt + sigma(x,t)*dWt
What this means in non-math is that the rate at which the spot price of the market changes (dX_t) is a function of the normal drift rate (the expected rate of return of the market) and the volatility of the market (sigma(x,t)*dWt).

This is pretty well established as a model, and although it has some moderately significant flaws in practice (absence of jumps, for example), it works *well enough* over long periods of time.
What this means is that we expect, in the absence of *any* volatility, that the market

should increase over time by about mu(x,t), which is classically some function of the risk-free rate + the amount of risk we're taking by holding an asset.
Again, this has issues, but works well enough for now.

However, in a real market we have volatility, and during crashes, we get lots of volatility.
We can turn to option theory to understand more in depth what the spot price means and how we expect it to change in response to the existence (or belief of existence) in the Fed Put.

For argument's sake, we can assume that the Fed steps in at some fuzzy interval when the equity market (our market) is some fraction off all-time-highs. During the COVID pandemic, for example, we can place "the Fed Put" around 30% off ATH. While in practice this fluctuates

every time depending on the action of the Fed and severity, it's a good enough model to understand some theory without delving into minutiae.
So we can more formally look at the Fed Put as, in essence, a real put, that has infinite duration (it lasts forever), and resets

constantly. In practice, there is no "exact" match for some put where the strike price resets constantly, but we have a pretty similar real world exotic option called the cliquet.
en.wikipedia.org/wiki/Cliquet_o…
In our case, it's a bit more exotic - our cliquet resets constantly when

a new ATH is reached, and exists not at-the-money, but at some fraction of it (in mathese, we can say that the put is struck at max([S0,S1...S_t-1])*delta).

So what happens over time to the price when we have this "Fed cliquet put", which unlike a real cliquet has no cost?

Well, intuitively, you'd assume that the existence of the cliquet causes a barrier on the downside. In a normal Brownian motion, there is roughly equal chance of upward/downward movement (except if a drift is present, of course). But at the point where the cliquet is reached

we except that the probability of further loss is exactly 0. What this means is interesting. In an intuitive sense (again intuitive, lol), the spot price buying an asset is precisely the point where the upside risk and downside risk of the asset are equal (that is, we expect

no-arbitrage). If we argue that the Fed Put exists even without a firm delta (we have a fuzzy idea where the Fed *should* step in), we've chopped off much of the downside risk. Therefore, we should expect that the spot price should *increase* overall to compensate sellers for

the embedded optionality.

But this is well, theoretical. Let's look at it in simulation.
I coded up a nice little Colab, probably poorly, to simulate the behavior of the Fed Put (per the criteria above) versus a scenario where the Fed Put does not exist (the normal Ito process).

If you want to check out the code, here's the link!
colab.research.google.com/drive/1d30_nmg…

As expected, as the delta of the Fed Put increases, so does our expected return. We see an acceleration of beta simply because well, JPow got our back.

Fin.

We can see this in a simple scatter over 96 simulations, each of around 10,000 steps and 5,000 subsimulations of the mean returns achieved based on the mean delta (of ATH) of the Fed Put:
As anticipated, the closer to 100% (in this case up to 75% mean delta), the higher return.