2.2. Consider the Cox-Ingersoll-Ross short rate model dr(t) = [a br(t)]dt + or(t)dW+(t), where a, b,...

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2.2. Consider the Cox-Ingersoll-Ross short rate model dr(t) = [a br(t)]dt + or(t)dW+(t), where a, b, o> 0 and where W is a Brownian motion under the risk neutral (martingale) probability P. (a) Show in this model that zero-coupon bond prices are given by B(t,T) = m(t)-r(t)n(t) P whenever t T, where the functions m (t) and n(t) satisfy the Riccati differential equations dn (t) dt dm (t) _ an (t), = dt 1 = = bn^ (t) + zo? * (t)]? 1, 2 together with the final conditions (b) Verify that if c := m (t) are given by 1 (30 marks) Hint: Consider the term structure equation for FT(t, T(t, r): n(t)-rn(t). b +20 and k(t) := ec[Tt], then n(t) and := em n(t) = mT(t) = n(T) = 0, mT(T) = 0. k(t) - 1 c[k(t) + 1] + [k(t) 1]' =-a [ n(u)du. t 2.2. Consider the Cox-Ingersoll-Ross short rate model dr(t) = [a br(t)]dt + or(t)dW+(t), where a, b, o> 0 and where W is a Brownian motion under the risk neutral (martingale) probability P. (a) Show in this model that zero-coupon bond prices are given by B(t,T) = m(t)-r(t)n(t) P whenever t T, where the functions m (t) and n(t) satisfy the Riccati differential equations dn (t) dt dm (t) _ an (t), = dt 1 = = bn^ (t) + zo? * (t)]? 1, 2 together with the final conditions (b) Verify that if c := m (t) are given by 1 (30 marks) Hint: Consider the term structure equation for FT(t, T(t, r): n(t)-rn(t). b +20 and k(t) := ec[Tt], then n(t) and := em n(t) = mT(t) = n(T) = 0, mT(T) = 0. k(t) - 1 c[k(t) + 1] + [k(t) 1]' =-a [ n(u)du. t

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