How the risk-free rate enters option values: it raises calls and lowers puts, because holding a call defers the cash needed to own the underlying.
A call is a deferred purchase, so its holder keeps the strike money earning interest until exercise — worth more when rates are high. A put is a deferred sale, so its holder waits to receive cash, which costs them interest. The effect is captured by rho and grows with both time and rate level.
The consequences show up in places that look unrelated. put-call-parity shifts, box-spreads become a real financing instrument, deep in-the-money puts become candidates for early-exercise, and the fair price of a conversion changes. In a zero-rate decade all of this was invisible; at 5% it is arithmetic that matters.
Example: XYZ at $50 with a one-year $50 call. At 0% rates the call is worth about $4.98; at 5% it is worth about $6.15. The stock did not move and volatility did not change — the extra $1.17 is the value of deferring a $5,000 payment for a year.
Original diagrams for the ideas on this page. Illustrative, not real market data.
Buying a call: payoff at expiry. A 105-strike call bought for 3 loses that whole 3 if the price finishes at or below 105, breaks even at 108, then gains a dollar for every dollar higher. The loss is capped at the premium; the upside is not capped.The bid-ask spread. Buy orders sit below, sell orders above, and the gap between the best bid (50.01) and best ask (50.04) is the spread you pay to cross. Bar length shows the size resting at each price.
Educational only, not advice. Spotted an error? Post in Site Feedback.