> [!tldr] > *If the interest rate is positive, then money worth $P_{T}$ at some future time $T$ is worth less than that at present (time $t < T$).* > > Intuitively, this is because that having money now is better than having the same sum in the future, because I can always put it into the bank and collect extra interest between now and the future date. > > If the interest rate is (for simplicity) a constant $r$, then if the price of product that (deterministically) worths $P_{T}$ at some time $T$, then it is worth $P_{t}=P_{T} \cdot e^{-r(T-t)}$at time $t < T$. - Another way of putting this is that *a risk-free portfolio must grow at the risk-free rate.* This assumes that the product can only be held till $T$ (if not longer), with no strings attached like converting it to another product like early exercising an American option. To see why, consider the portfolio of borrowing a sum of $P_{t}$ and buying the product at time $t$. Then at time $T$, sell the product for $P_{T}$ and repay the loan. This leaves no positions and a PnL of $P_{T}-P_{t}e^{r\tau},$where $\tau:= T-t$ is the time-till-expiry at the time when the portfolio was created. By no-arbitrage, this PnL should be (deterministically) $0$, and $P_{t}=P_{T} \cdot e^{-r\tau}$. - If it is above $0$, then people will create the portfolio with no upfront cost or any risk, receiving a gt;0$ profit at expiry -- this demand will drive up the price of the product $P_{t}$ and/or increase the interest rate, thereby driving the profit to $0$. - Similarly, if it is below $0$, people will sell the portfolio (i.e. selling $P_{t}$ and putting money into the bank) and drive its price back up.