Input five: the risk-free rate

The fifth input is the risk-free interest rate, and it is the one that does the least on the contracts most people look at. Understanding why it is there at all matters more than its size.

It enters through discounting. The strike is a cash amount paid or received in the future, so its present value is the strike discounted at the interest rate. A higher rate means a lower present value of the strike, which makes a call worth more — you are committing to pay a fixed amount later, and later money is worth less — and a put worth less, since a put's holder receives that fixed amount.

The size of the effect depends almost entirely on how much time remains, because discounting compounds over time. In the base case at 30 days, moving the rate from 4 percent to 8 percent raises the call from $3.02 to $3.19, seventeen cents. On a two-year contract the same change moves the call from $17.68 to $21.68 — four dollars, twenty-four times the effect. The input is the same; the leverage on it is time.

One consistency check that shows the mechanism cleanly: with the rate set to zero, the at-the-money call and put in the base case are both worth $2.86, exactly equal. That equality is put-call parity with no discounting to separate them. Every cent of the usual gap between an at-the-money call and put is the interest rate at work.

Dividends belong in the same slot conceptually. A dividend paid before expiry reduces the price the holder of a call would eventually acquire at, so it acts in the opposite direction to the interest rate: it lowers calls and raises puts.