Generally when the risk free rate of return increases, stock prices fall. In that case, how is that the value of a call option increases when risk free rate rises??
What you say is practically correct. In the general pricing formula for options, the risk free rate is assumed a deterministic constant - not correlated with stock prices. To model your case, a joint stochastic model for stock prices and interest rates is needed - which makes things extremely complex and difficult to calibrate - but the right way to go about.
If the risk-free rate increases then all other things being equal the value of the call option will rise. This is because the hedging strategy will comprise the same amount of the underlying asset but a smaller cash loan, since the latter will accrue interest faster. That means the buyer will have to pay more money up front in order that the seller can finance the hedging strategy. The situation is more complicated if the price of the underlying asset changes, as it probably will. But remember that the model we're looking at is simplified in several respects and it's not unreasonable to consider the effect of a change in the risk-free rate in isolation.
when we say that the rho of option is positive, we mean that keeping other parameters' values unchanged, the effect of increasing rate is to increase the option price. This is only a partial effect. to determine the full effect, one needs to take into account the negative correlation between the stock price and interest rate. Then we will have a +ve effect from interest rate change and -ve effect from stock proce decline: what you means here is probably the sum total of these two effect, and it is preety much possible that the -ve effect dominates overall change.
Hi, When the Core Reading says that an increase in the risk-free rate makes a call option more valuable it means all else being equal, ie assuming that all the other five factors that affect the option price (including the share price) do not change. In other words, we are talking about a partial effect on the option price. Remember that the Greeks are all partial derivatives telling us the effect on the option price of a small change in the relevant factor assuming that all the other factors remain unchanged. This is very important. The argument then as to why an increase in the risk-free rate increases the value of the call option (all else being equal) is that the call gives us the option to buy the share for a fixed strike price (K) in a year's time, say. So, the higher the 1-year risk-free rate, the lower the discounted present value of the strike price K payable with the call. Hence, the more valuable is the call.
