How does interest rate affect option price?

(post from old sites)

not as simple as many web sites suggest.

interest rate affects option price in 2 folds:

1. forward price of the spot stock
• as interest rate increases, forward price of the stock underlying, aka no arbitrage price of the stock underlying at option expiry, increases, and in turns, increases call value and decreases put value
2. cost of carrying of the option
• as interest rate increases, cost of carrying the option increases and in turns, decreases the option value
• a small amount, as option price is a fraction of the underlying price

FAQ:

how about future underlying?

stock option prices on the spot contract as the underlying.  Spot underlying and the interest rate determines the forward price, aka no arbitrage price of the stock underlying at option expiry.

Future options prices uses the FUTURE contract as the underlying.  Future price is effectively the equivalent of the forward price of the spot underlying and that's why when interest rate changes, the forward of the future doesn't change.  In fact and again, we use future as the underlying, not the forward.

Increase in interest does not drive up the forward price of future, but it should drive up future price because of no arbitrage, right?

Yes, but the change in future price is in fact a change in UNDERLYING price.   Perhaps another way to think of it is that the change in interest rate directly affects the underlying price of future option, but not affect the the future option itself.

As opposed to change in interest rate does not directly affect the underlying of a stock option, aka spot price of stock (economist may beg to differ), but directly affects the option itself as it changes the ATM forward.

How about far month future option?

Future option uses simple offset mode instead of forward price.  Far month future options still use front-month future as the base contract, which is most liquid.  To account for the different in expiry, traders put in offset for different months, which embeds the interest and dividends?

Reference:

Option volatility & pricing by Natenberg

Note:
confirmed using Orc.  push up interest rate, then

• both index call and put TV go down
• stock call TV goes up and put TV goes down