Both interest rates and underlying stock’s volatility have an influence on the option prices.
Impact of Interest Rates
When interest rates increase, the call option prices increase while the put option prices decrease.
Let’s look at the logic behind this. Let’s say you are interested in buying a stock which sells at $10 per share. You buy 1,000 shares at $10 each with a total investment of $10,000. Instead of directly buying the stock, you could also have purchases a call option selling for only $1, making a total investment of $1 x 1,000 = $1,000. If you choose to buy the call option instead of the underlying stock directly, you could have used the remaining $9,000 to earn some interest. The higher the interest rates, the higher your interest income would be. This makes the call option more attractive and more expensive.
For put options, the opposite holds true, that is, the higher the interest rates the lower the put option price. This is because if interest rates are high you will have to hold the asset for a longer time to deliver it under the put option. Simply selling the asset and using the proceeds to invest at a higher rate would be a better option. This makes the put option less attractive and hence less costly when interest rates are high.
This said, the impact of interest rates on option prices is minimal.
Impact of Volatility
Unlike interest rates, volatility significantly affects the option prices. The higher the volatility of the underlying asset, the higher is the price for both call options and put options. This happens because higher volatility increases both the up potential and down potential. The upside helps calls and downside helps put options.