Firstly, what are risk free interest rates? The risk free interest rate is the theoretical interest rate that would be returned on an investment completely free of risk, generally taken to be the yield on 3 month Treasury Bills.
At first glance it is a bit hard to imagine why this would affect option prices. However, it is in the presumption that one side of the trade may be holding stock for no possible return and that the stockholder is due a risk free return for having their capital tied up and not earning interest.
Firstly, let’s have a look at call options. As we know, call options give the owner of the option the right to buy the underlying stock at the strike price. And we know that the writer of the option is obligated to sell shares to the holder of the option at the strike price. The option-pricing model therefore assumes that the option writer is actually holding the stock to be able to sell it to the buyer, if the buyer exercises.
This means the stockholder is incurring carrying costs in the form of loss of risk free interest income on cash and is due some compensation in the option price. The option buyer is in effect, paying the option writer risk free interest rates, in advance, for the life of the option. The effect however, is incremental according to the moneyness of the option. The risk free rate is fully priced in for deep ITM call options, whereas it is not priced in at all in far OTM call options.
The fact that you are paying the risk free rate “up front” means that the longer to option expiry, the higher the interest rate component of the call option premium.
Of course it also means that higher risk free interest rates mean higher call option prices, all things being equal.
The effect is quite obvious on our risk graph of call options. Below is an image for comparison of the same option at differing risk free rates. Both graphs are of a $50 ATM call option with 145 days to expiry. To make the comparison visually obvious, I have used radically different risk free rates. It is most obvious in the deep ITM side of the graph, as marked. The top ATM call option at 1% has a premium of $3.85, whereas the bottom image at 12% has a premium of $4.95.
With put options the story is somewhat different. It is not the option writer who is presumed to be holding stock, as is with the case with call options; it is the option holder who is presumed to be holding the stock. The holder of a put option has the right to sell the underlying to the writer, so it is the put option holder who is compensated for loss of interest income on cash before putting to the option writer.
Therefore the rule of thumb for put options is that higher risk free rates mean cheaper put prices, all things being equal.
With put options it is not so graphically obvious, but we can see it in the option premium. Both graphs are of a $50 ATM put option with 145 days to expiry. The top ATM put option at 1% has a premium of $3.65, whereas the bottom image at 12% has a premium of $2.87.
It should be noted that the above only applies where there is real cost of carry considerations, i.e. when the stockholder is required to finance the stockholding with cash and/or interest paid on margin. With options on futures, the cost of carry is prices into the price of the futures contract; therefore risk free rates have no significant effect on option prices.