How Interest Rates Affect European Call Option Values
Summary
The document explains why a higher risk-free rate can raise the value of a call on a non-dividend-paying stock, despite also increasing the discount applied to its future payoff. In the Black–Scholes framework, the same rate affects both the stock’s assumed risk-neutral drift and discounting. For a call whose payoff is uncertain, a higher drift increases the chance that the stock finishes above the strike, which can increase the option’s value.
A zero-strike call provides an intuition check: its payoff is always the stock value, so the option is equivalent in value to holding the stock. With a positive strike, exercise is uncertain, and the effect of rates works through the probability of finishing in the money. The answer is conceptual rather than a derivation and focuses on a non-dividend-paying stock; dividends and other model assumptions can change the rate sensitivity.
Key ideas
- In Black–Scholes, the risk-free rate enters both risk-neutral stock growth and discounting.
- A higher rate can increase the chance that a stock finishes above a call’s strike.
- For a non-dividend-paying stock, a zero-strike call has the value of the stock.
- The explanation is qualitative and does not cover dividend-paying underlyings.
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Full text
# Interest rates, effect on call price # Interest rates, effect on call price Generally, we assume that an interest rate increase makes the call price more expensive. From my understanding it is because the expected return on the stock price increases. However the interest rate increase makes also the discount factor lower, and we know that the price today is the discounted expected payoff. Does that mean that we assume that the expected return increase on the stock always outweigh the discount factor decrease, so that the price today increases? ## Answer by RandyF (score 1) https://quant.stackexchange.com/a/44820 No, they're actually the same, kind of. The risk-free rate that we assume the underlying stock price grows at is the same as the rate we discount once we have determined if that path exceeds the strike price (the discount factor we multiply the stock price by uses the same $r$ as the one used in $N(d_1)$. Accordingly, if you have a theoretical zero-strike call option on a non-dividend paying stock worth \$100 and the risk free rate is 3%, the value of that call option would be \$100 - the same as the price of the stock since you will always exceed the strike price and it's guaranteed to be worth the stock at the future point in time. Now, if the strike was \$100, the probability of exceeding the strike price goes down quite a bit. One thing that helps the stock hit the \$100 is that the stock is assumed to drift higher at a risk-free rate of 3%. So, the value of the call option is worth more because it increases the probability of receiving payment, even though it doesn't increase the value of that payment if the payment is certain to be made.
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