Let R(t,t+) denote the yield to maturity over the period (t, t+] of a discount bond maturing

Let R(t,t+δ) denote the yield to maturity over the period (t, t+δ] of a discount bond maturing at t + δ, and f (0,t,t + δ) be the forward rate observed at time zero over the period (t, t +δ]. Let σ_B (t, T) denote the volatility function of the discount bond price process B(t,T). Show that R(t,t +δ) can be expressed as (El-Karoui and Geman, 1994)

where Z(u) is a Brownian process under the t-forward measure Q_t . Furthermore, when the bond price volatilities are deterministic, show that

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1 - R(t, t + 8) = f(0, t, t+8) + [₁ [OB(U₁1 + 8) - OB(u. 1)]dZ(u) 8 + 12/15/10 [OB(u, t+8) - OB (u, t)]² du, 28