Consider again the Hull-White model with constant mean reversion = 1 %. Suppose also interest rates

Question:

Consider again the Hull-White model with constant mean reversion κ = 1 %. Suppose also interest rates are flat at 4% so that discount bond prices are given by D(0,T) = e^_0.04T First find the SDE for the instantaneous forward rate ƒ_t,_T = E_t^T[ry]. Take the forward rate as a proxy for the Libor rate. Suppose we choose σ(t) such that the term vol of the forward rate (i.e. Λ where Λ²T = var(f_0,T)) is constant at 0.7% for annual expiries 1y, 2y,..., 15y. Find the prices of swaptions with expiry ly and tenors 2y, 5y, and 10y. Also, find the prices of swaptions with expiries 2y and 5y and the same combination of tenors. Hence, determine their implied vols. What happens if mean reversion increases to κ = 10%?