Swap value (a) Prove that the value  (t) at time t ≤ T₀ of a swap from T₀ to T_n, where we pay fixed rate K and receive libor, is

(b) By expressing a swap as a difference between a floating rate bond and fixed rate bond, prove that, for a given K,   (t) is bounded, that is, there exists finite l and u independent of interest rates such that l ≤  (t) ≤ u. For t = T₀ = 0 (a spot starting swap), T_n = n and frequency α = 1, find bounds in terms of n and K.

(c) Is the value of a forward contract on a stock S_t necessarily bounded above and below? Explain the key difference between the value of a swap and a stock forward contract.