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Suppose that the price of interest rate risk is constant and that the spot rate r t follows the first - order autoregressive process under

Suppose that the price of interest rate risk is constant and that the spot rate rt follows the first-order autoregressive process under the risk-neutral measure:
rt+1=(1-N)N+Nrt+ut+1N,
where ut+1N i.i.d.N (0,2), so that the logarithm of a zero-coupon bond price with n periods to maturity )=(logDn,t if an affine function of the short rate:
-dn,t=an+bnrt.
(a) Derive a formula for the coefficients an and bn in terms of the parameters of the short rate dynamics.
(b) Evaluate:
EtN[dn-1,t+1]-dn,t.
(c) Assuming that -1N1, derive the process for a one-period forward rate fn,t in terms of the risk-neutral parameters for a very long maturity, n. What are the implications of the model for the long forward rate? [Hint: Note that fn,t=[:dn,t-dn+1,t*}
(d) Now assume that the short rate follows a random walk:
rt+1=rt+ut+1N.
Derive the process for a one-period forward rate fn,t for a very long maturity, n. What are the implications of the model for the long forward rate?
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