option value with risk-free rate?

CFA Level III
#1

on schweser" “Call option value increase when the risk-free rate is higher?” why? anyone can explain?

#2

I will try… RFR is 10% per annum, Suppose the stock is 100 and one year ATM call option is 15, now u buy the stock and sell the call, u r gaining 15 but sacrificing 10 of Risk free return, so basically 5 with downside risk now if the RFR is 20%, the call option has to be atleast 20

#3

You can assume that prices of assets go up at the risk-free rate. You might remember that formula for futures value is F = S0*(1+r)^T assuming there are no dividends, etc. A call option allows you to buy an asset at a fixed price K. For a given K, the higher the risk free rate, the more the price of the asset is expected to appreciate -> the more expensive a call option is going to be. I hope that helps.

#4

Thanks guys. If the statement is true, surely asset price should go up at the risk-free rate. Can we consider RFR is inflation rate+ some premium, or treasury notes rate? If nominal rate goes up, price of stock will go up, another kind of price parity?

#5

Back to level 1 and 2 basic. The value of a call option using Black-Scholes Model (or Binomial model) is Stock price (discounted by dividend yield multiply by a probability factor) less Exercise price (discounted by risk free rate multiply by a probability factor). In short, we assume one can borrow at the risk free rate and invest in the stock price to earn the dividend income (or call put parity). Hence if risk free rate increase, the discounted exercise price will be lower and the value of the call ie. S - X will be higher. Any comment?

#6

pmoonoi Wrote: ------------------------------------------------------- > Back to level 1 and 2 basic. The value of a call > option using Black-Scholes Model (or Binomial > model) is Stock price (discounted by dividend > yield multiply by a probability factor) less > Exercise price (discounted by risk free rate > multiply by a probability factor). In short, we > assume one can borrow at the risk free rate and > invest in the stock price to earn the dividend > income (or call put parity). Hence if risk free > rate increase, the discounted exercise price will > be lower and the value of the call ie. S - X will > be higher. Any comment? Yeah, the statement of Black-Scholes isn’t right (that’s a bound or something), Black-Scholes assumes all kinds of stuff that isn’t true and the statement that rf rate increase causes call options to increase is way more general than B-S. Put-call parity doesn’t have anything to do with Black Scholes and is almost all you need for this. Assume asset price is independent of risk-free rate (the statement isn’t true if that’s not true), the bond + call side is looking more appealing as the risk-free rate increases, so the call better increase in cost to compensate for the bond giving a better return (or the put could decrease and it’s not unreasonable to expect that both happen at approximately the same amount).