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7 Valuation of options, futures and swaps Options and futures are usually valued using techniques based on the principle of no arbitrage. The value taken is the cost of losing out the contract by buying an equal and opposite option or future on current terms. This is also true for swaps. Arbitrage is defined as the simultaneous buying and selling of two economically equivalent but differentially priced portfolios so as to make an instant and riskfree profit. In this context, economicallyequivalent means that the two portfolios yield exactly the same returns under every possible scenario and outcome. In an efficient market, the principle of no arbitrage implies that all such portfolios must be priced identically, otherwise it would be possible to buy the cheaper and sell the dearer portfolio so as to make an immediate, riskfree profit. Indeed, arbitrageurs would do just this, thereby forcing the portfolio prices back into line. Lets consider an interest rate swap, where payments based on a fixed interest rate are swapped for payments based on a floating rate, eg SONIA Sterling Overnight Index Average. Swaps can be valued by discounting the two component cashflows. At inception the value at market rates of interest of a swap to both parties will be zero, ignoring the market makers profit and expenses. In other words, the present value of an investors income from a swap minus the present value of their outgo is expected to be zero. Strictly, this also requires that we ignore tax and risk, and assume that both parties have the same view of future interest rates. As market interest rates change the value of the two cashflows will alter, leading to a positive net value for one party and a negative net value to the other. Even if rates dont differ from those expected, the value of a swap is likely to be positive for parts of its term and negative for others. The discount factors, and estimates of shortterm rates for the floating side of the swap, are extracted from the appropriate yield curve. This is equivalent to viewing a swap as a combination of bonds. For example, ignoring the risk of default, a party which has swapped dollar payments for sterling receipts is in an equivalent position to being long a sterling bond and short a dollar bond of the same maturity. An alternative way of viewing a swap contract is as a series of forward agreements. If each of these forward agreements can be valued, then so can the swap.