Why would a lower stock price leads to higher value of a call option?

Question

Currently I am reading Basic Black Scholes: Option Pricing and Trading by Timothy Falcon Crack.

At page $47,$ the author mentions the following.

Higher interest rates decrease the present value of the strike price. Other things being equal, this increases the value of a call because the strike price you potentially give up has lower present value; conversely for a put.

I do not understand the bold sentence.

Intuitively, I thought that if interest rate increases, then stock price decreases (which is the same as above). But wouldn't this decrease the value of a call option as the difference between terminal stock price and strike price is smaller.

Can someone verify whether my reasoning is correct and explain the bold sentence?

Answer 1

In the B&S world, interest rates are constant and thus deterministic. In particular, they are not correlated to the stock price whatsoever. Thus, firstly interest rates don’t change in the first place in the B&S world.

You can generalise the model and allow for time dependent (but still deterministic) interest rates. The interest rates are then still uncorrelated with the stock price.

A different generalisation is to allow for stochastic interest rates. So you have a multifactor model. Possible choices are the models from Hull-White (1990) and Cox-Ross-Rubinstein (1985). Here, you can explicitly introduce a correlation between stock prices and interest rates. Due to the short time of expiry of many options and the low influence of interest rates on option prices, it is questionable how much better your model is after this generalisation.

It may be true or not that in real life stock prices react to interest rate changes (there are many empirical papers investigating this relationship), but the standard B&S model simply does not incorporate this feature.