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EXERCISE 8.2. Suppose the dynamics of the forward rate are given by (8.7) with $f_0^T = \lambda_0 + \lambda_1 e^{-\gamma T} - \frac{\eta^2}{2\gamma^2} (1 - e^{-\gamma T})^2$, $\sigma_t^T = \eta e^{-\gamma (T-t)}.$ Show that the short rate R admits the form $R_t = g(t, R_0) + \int_0^t h(s; t) d\widetilde{W}_s$, and identify the functions g and h.

          EXERCISE 8.2. Suppose the dynamics of the forward rate are given by (8.7) with
$f_0^T = \lambda_0 + \lambda_1 e^{-\gamma T} - \frac{\eta^2}{2\gamma^2} (1 - e^{-\gamma T})^2$,
$\sigma_t^T = \eta e^{-\gamma (T-t)}.$ 
Show that the short rate R admits the form
$R_t = g(t, R_0) + \int_0^t h(s; t) d\widetilde{W}_s$,
and identify the functions g and h.

EXERCISE 8.2. Suppose the dynamics of the forward rate are given by (8.7) with
f0^T = λ0 + λ1 e^-γ T - (η^2)/(2γ^2) (1 - e^-γ T)^2,
^T = η e^-γ (T-t).
Show that the short rate R admits the form
Rt = g(t, R0) + ∫0^t h(s; t) dWs,
and identify the functions g and h.

Added by Jennifer V.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Nick Johnson

10/08/2021

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The short rate is typically defined as the instantaneous interest rate at time t, denoted as R(t). Show more…

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Exercise 8.2. Suppose the dynamics of the forward rate are given by (8.7) with f_(0)^(T)=lambda _(0)+lambda _(1)e^(-gamma T)-(eta ^(2))/(2gamma ^(2))(1-e^(-gamma T))^(2), sigma _(t)^(T)=eta e^(-gamma (T-t)). Show that the short rate R admits the form R_(t)=g(t,R_(0))+int_0^t h(s;t)dW_tilde(W)_(s), and identify the functions g and h.