That has nothing to do with volatility and options pricing. But merely the financing.
The risk free return is used as an input for the interest component. To be fair, 'we' shouldn't really use the risk free rate, since not everyone can use that for financing.
This quora stuff has to do with arbitrage and put/call-parity... where prices of a put and a call with the same DTE and strike need to align with each other according to (roughly) C-P=Spot+interest-Strike.
So, say Spot = 100... interest = 1... Strike = 100... will mean when the call = 6, the put = 5
If the call is 6.50 instead... the arb would be to do a conversion, Sell the call, buy the put (which is a synthetic short stock) and buy the stock. You hold until expiration... and make 50 cents.
If you would use an expected rate of return in stead of the risk free rate... say 5%...
Than your pricing would be call = 10 and put = 5.
Which is fine by me... but I (and everyone else) would most certainly arb that by doing a conversion and make 4 bucks.
Remember, at expiration, the short call and long put are cancelled out by the long stock. I would have paid 1 in interest... and end up making 4.
Expiry at 100 means call = 0 and put = 0
Expiry at 105 (your rate of return!) means call = 5, put = 0... but I was long the stock that went from 100 to 105...
If you're still not seeing the light, and think that you wouldn't have lost money since the call is worth 5 vs put of 0 (exactly what it was when you trade)...
If you think the stock is going up to 105 by applying a rate of return of 5%, wouldn't you rather buy the stock than the calls?