> [!tldr] > A **futures** (contract) is the agreement to buy/sell an **underlying** at the expiry/**maturity** date $T$ at a agreed-upon price of $K$ (the **strike price**). > > Both the holder (buyer of the underlying at expiry) and the writer have *the obligation to make the trade,* unlike options (which gives the holder the right, but not the obligation). > > If the risk-free rate (both borrowing and saving) is known to be a constant $r$ till maturity, and the underlying is currently priced at $S_{t}$ with no holding cost/benefits, then the fair price of the contract is $F(S, t;r, K, T)=S -Ke^{-r\tau}$where $\tau:= T-t$ is the time till maturity. ### Deriving the Fair Price Under the conditions above, consider the following portfolio (at time $t$): - Long the futures $F(\cdot~;r,K, T)$ and fulfill the contract at $T$. - Short-selling the underlying at price $S_{t}$ and save the sum into a risk-free account till maturity. Then at maturity, this portfolio has no outstanding position and a PnL of $Se^{r\tau}-K,$i.e. bank account minus the cost of fulfilling the contract. Therefore, by no-arbitrage, [[Time Value of Money|discounting]] this PnL gives the (present) fair price of the portfolio, i.e. that of the future: $F(S,t;r,K,T)=e^{-r\tau}(Se^{r\tau}- K)=S - Ke^{-r\tau}.$ > [!warning] Notes on the no-arb argument > - The above is only true since the portfolio is deterministic (the final PnL and position is determined at time $t$). > - Otherwise, we can argue that that $Se^{r\tau}-K \le F$: the futures contract can be made to turn at least this much profit, so it must worth this, if not more. > - Now make the same argument for shorting the same portfolio (short futures, borrow to long underlying) gives the reversed inequality: $K-Se^{r\tau} \le -F,$so the two sides must equal. ### Alternative Portfolio An alternative portfolio (or more like a position) is: - Longing the futures $F$, - Shorting a stock (not short-sell, just a position of $-S$), - And a savings of $Ke^{-rT}$ (which will grow to be $K$ at expiry). Then at expiry, the futures is fulfilled with the savings, and the stock bought is used to close the short position, leaving $0$ PnL and $0$ position, therefore the portfolio is worth exactly $0$ by no-arb.