Intuitive explanation of put option pricing based on put-call parity

Question

Assuming no dividends, the put-call parity equation says:

$c + Ke^{-rT} = p + S$

where $c$ is the price of the European call, $p$ is the price of the European put, $S$ is the current stock price, $K$ is the option strike price, $r$ is the risk-free rate, $T$ is the time to expiry.

In You Can Be a Stock Market Genius by Joel Greenblatt, a basic explanation of call option pricing appears in chapter 6:

The bottom line is that buying calls is like borrowing money to buy stock, but with protection. The price of the call includes your borrowing costs and and the cost of your “protection” — so you’re not getting anything for free [...]

Intuitively, there is a borrowing cost because the owner of the call does not have to tie up $\\\$K$ (which is effectively "borrowed") until the exercise of the call option.

I instantly recognized this as an excellent intuitive interpretation of a rearrangement of the put-call parity equation:

$c = \overbrace{S - K}^\text{intrinsic value} + \overbrace{\underbrace{K - Ke^{-rT}}_\text{borrowing cost} + \underbrace{p}_\text{downside protection cost}}^\text{time value}$

The book doesn't explain put options, so I tried to rearrange the equation to similarly explain the price of put options:

$p = \overbrace{K - S}^\text{intrinsic value} + \overbrace{\underbrace{Ke^{-rT} - K}_\text{?} + \underbrace{c}_\text{upside protection cost}}^\text{time value}$

However, I am unable to find an intuitive interpretation of this equation. Can someone help me out?

I tried: "buying puts is like short-selling a stock, but with protection ...", but I don't know how to intuitively explain $Ke^{-rT} - K$, which looks like a "borrowing rebate".

Answer 1

If you sold a stock short, you receive cash at $t=0$, so there is a negative borrowing cost. If you buy a put, you don't receive this funding.

So while a call would be too cheap if it didn't include a funding cost incurred when buying the stock, similarly a put would be too expensive if it didn't include the "borrowing rebate" that you mention, for the funding received from the short selling cash.

Since in reality it costs something to borrow a stock (in order to short), I wonder if in the real world, this 'borrowing rebate' might be slightly reduced do to the stock-borrow-cost... but that's another question.