, using prices instead of rates, as prices are more readily available. Rational pricing also applies to interest rate modeling more generally: yield curves must be arbitrage-free with respect to the prices of individual instruments. See Bootstrapping (finance) and Multi-curve framework. Pricing derivatives A derivative is an instrument that allows for buying and selling of the same asset on two markets – the spot market and the derivatives market. Mathematical finance assumes that any imbalance between the two markets will be arbitraged away. Thus, in a correctly priced derivative contract, the derivative price, the strike price (or reference rate), and the spot price will be related such that arbitrage is not possible. See Fundamental theorem of arbitrage-free pricing. Futures In a futures contract, for no arbitrage to be possible, the price paid on delivery (the forward price) must be the same as the cost (including interest) of buying and storing the asset. In other words, the rational forward price represents the expected future value of the underlying discounted at the risk free rate (the "asset with a known future-price", as above); see Spot–future parity. Thus, for a simple, non-dividend paying asset, the value of the future/forward, , will be found by accumulating the present value at time to maturity by the rate of risk-free return . This relationship may be modified for storage costs, dividends, dividend yields, and convenience yields; see futures contract pricing. Any deviation from this equality allows for arbitrage as follows. In the case where the forward price is higher: The arbitrageur sells the futures contract and buys the underlying today (on the spot market) with borrowed money. On the delivery date, the arbitrageur hands over the underlying, and receives the agreed forward price. He then repays the lender the borrowed amount plus interest. The difference between the two amounts is the arbitrage profit. In the case where the forward price is lower: The arbitrageur buys the futures contract and sells the underlying today (on the spot market); he invests the proceeds. On the delivery date, he cashes in the matured investment, which has appreciated at the risk free rate. He then receives the underlying and pays the agreed forward price using the matured investment. [If he was short the underlying, he returns it now.] The difference between the two amounts is the arbitrage profit. Swaps Rational pricing underpins the logic of swap valuation. Here, two counterparties