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Given a initial discount bond $P^M(0, T)$ curve, the expression for $\theta(t)$ in the Hull White Short Rate model is a know result given by: $$ \theta(t) = \frac{1}{\kappa} \cdot f'(0, t) + f(0, t) + \frac{1}{2} \cdot \left( \frac{\sigma}{\kappa} \right)^2 \cdot \left( 1 - e^{-2 \kappa t} \right). $$ I have used a notation where the spot rate dynamics is ...

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