How does the Greenspan/Bernanke Put semantically relate to put options? If rational investors reckon that &quot;the Fed, under Alan Greenspan, its longtime chairman, would always bail stock investors out of their losing positions&quot;, then rational investors would buy call options or sell put options, not buy puts. So how can a Greenspan or Bernanke Put be sensibly analogised to a Put Option?
Greenspan and Bernanke Put and Other Central Bank Tools

The term Bernanke put has become almost as ubiquitous as the Greenspan put was during the late-1980s and 1990s. Derived from the concept of a put option, these terms refer to central bank policies that effectively set a floor for equity valuations. For instance, Alan Greenspan was known for lowering the Fed Funds rate whenever the stock market dropped below a certain value, which resulted in a negative yield and encouraged movement into equities.
In these situations, investors have been given a put option of sorts by central banks, since they have a price floor in place. For example, an investor holding shares of a broad market index may have a sort of guarantee from the central bank that the stock won’t drop below 20% since if it did, the central bank would intervene with low-interest rates to boost equity valuations. There was no actual guarantee by the central bank, but the precedent was enough for many investors.

Economist's View: The &quot;Greenspan Put&quot; and Excessive Risk Taking

This article argues that many people view the Fed as an insurance policy against financial market risk and believe that the Fed would not allow substantial asset devaluation.  My own view is that people ascribe all sorts of motives to the Fed that aren’t there.  The Fed is trying to tell us that it’s really simple (as are Kash at Angry Bear and Tim Duy in a Fed Watch).  Watch output and inflation relative to target and adjust accordingly.  We can argue about how to measure output, the target, and how to parameterize “accordingly” in the Taylor rule, but those are details, the essence is simple.  The Fed’s behavior is explained very well with a Taylor rule and reinterpreting it as an insurance program or something else is an entertaining exercise, but we probably shouldn’t take it much beyond there.

Understanding the Greenspan, Bernanke, Now Trump &quot;Put&quot; - Halbert Wealth Management - Commentaries - Advisor Perspectives

The term Greenspan Put was coined in the late 1980s as a result of certain policies implemented by then Federal Reserve Chairman Alan Greenspan. He was Fed Chairman from 1987 to 2006, followed by Ben Bernanke from 2006 to 2014 when the term changed to the Bernanke Put.
Throughout both men’s terms as Fed Chair, they routinely lowered the Fed Funds rate to help support the economy, which in turn helped the stock markets. As this practice went on, investors came to view the Fed’s policy of lowering interest rates to shore up the economy and risk assets as something akin to a “put” option.
then rational investors would buy call options or sell put options, not buy puts. So how can a Greenspan or Bernanke Put be sensibly analogised to a Put Option?

Investors do not need to buy or sell anything, the story is that the Greenspan Fed was giving out free puts to investors.
It doesn’t literally work that way, but it is directionally close  to making sense.