This might be another basic derivatives question. When interest rate rises, stock prices generally fall. Assuming an option's underlying is a stock, this should lower the option's price as well. However, according to many sources, when interest rate rises, options prices rise. What causes this and does this actually cancel out the effect of lower underlying prices?
Personally I think there is no easy answer to this question.
Economically a rise of interest rates often means an increased demand for capital. Banks need more money to lend to the industry thus they increase rates to entice consumers.
On the other hand a demand for capital on the side of the economy often means increased market activity - companies want to invest more. More market activity can translate into higher market volatility. And higher volatility translates to higher option prices.
Also, let us assume stock prices do fall because of rising interest rates. It has been observed that market volatility goes up in a bear market. (e.g. confer the following book - page 196) Once again you would have a higher volatility and thus higher option prices.
How an option reacts to interest rates depends on it's maturity and also on the type of option.
Generally: The higher the maturity the more sensitive the product is to changes in the interest rates (for you discount over a longer period of time)
I can recomend this mathematica tool for the B&S Model. You can play around with it to get a feel for how option prices react to changes in different parameters. If you crank up the maturity you will notice the price surface will shift significantly more if you change the interest rate. For a low maturity the changes will be minuscule
By derivating the Black-Scholes formula in function of r (ρ=∂C/∂r), you get
ρ_call=0.01TKe^(-rT) N(d_2 )=ρ_put+0.01TKe^(-rT)
You can see that call prices increase (and put prices decrease) if interest rates (risk-free) increase.