Single Factor Model Implied Term Structure
Overview
The generic model is a generic functional form of various short rate models listed below.
{% dr = u(r, t) dt + v(r, t) dW %}
Recovering the Term Structure from the Short Rate
Assuming we have some payoff {% f_T %} at some future date, we can value the payoff using the following expectation.
{% \mathbb{E}_Q[e ^{-\int_t ^T r(\tau) d \tau } ] %}
The expectation is also taken with respect to the risk neutral measure
(
wilmott 16.5)
Using this equation, and setting the payoff to a dollar, we get the discount rate at the given maturity.
{% R(t,T) = - \frac{1}{T-t} ln \, \mathbb{E}_Q [e ^{- \bar{r}(T-t)}] %}
{% \bar{r} = \frac{1}{T-t} \times \int_t^T r(t) dt %}
(
hull pg. 682)
