How do short stock positions lower the value of calls and raise the value of puts?

Question

I'm reading Option Volatility and Pricing by Sheldon Natenberg who in the chapter on Risk Management is trying to explain the effect of interest rates on options.

He says

The value of a stock option will also depend on whether the trader has a long or short stock position. If a trader's option also includes a short stock position, he is effectively reducing the interest rate by the borrowing costs required to sell the stock short. This will reduce the forward price, thereby lowering the value of calls and raising the value of puts.

How can holding a short stock position affect the interest rate on the option?

Answer 1

The short answer? In the absence of a dividend, call premium exceeds put premium by the carry cost (look at the explanation of conversions and reversals).

If you're short the stock, your receive interest on the proceeds. However, you pay a borrow cost to the lender of the stock, thereby "reducing the interest rate by the borrowing costs required to sell the stock short. "

I'm not a fan of Natenberg's explanations. I'd guess that his statement might be part of a larger story about a position he's explaining, perhaps even a conversion or reversal, hence his wording "the value of a stock option will also depend on whether the trader has a long or short stock position." If so, don't confuse value with premium.