New answers tagged short-rate
3
If we take your model literally (with the correction that I suggested as a comment), then there exists no (semi-)closed form, IMHO, that you can use for asset pricing. What you could do is then to make the model a bit simpler or to simulate.
Simulation
This is the nasty part. Based on your model, you simulate a very large number of the discount factor(s) and ...
3
Just an addendum to the above answers and comments:
The main decision is whether to use single or multiple factor dynamics.
LMM models term forward rates. HJM models instantaneous forward rates.
The main disadvantage of HJM, high-dimensional
stochastic process as underlying, was overcome by Cheyette, back in 1994, by restricting the general HJM model to a ...
6
I am not sure if you can classify it like that. Mind you, I never wrote a book. I'll write what I know below and you can decide if the classification makes sense or not.
1 ) STIR: as the term indicates - short term like Eurodollar frequently modelled with Black or Bachelier (normal) model. HW1F is also a short rate model.
2 ) HJM is a framework (M is not ...
1
Just adding my two cents. Without taking the logarithm of the price, the Ito's Lemma should result in:
$d p(t,T) = \left( \partial_t A(t,T) - \partial_t B(t,T) r + \frac{1}{2}\sigma^2B(t,T)^2 \right)p(t,T) dt - B(t,T) p(t,T) dr_t$
substituting now the partial derivatives and the differential $dr_t$, and simplifying the identical terms:
$d p(t,T) = r_t p(t,T) ...
1
You can simply use Ito's lemma under the risk neutral measure $Q$.For the log-bond price $p(t,T)$ this gives
$$dp(t,T)=(A_t(t,T)-B_t(t,T)r_t)dt-B(t,T)dr_t$$
$$=[A_t(t,T)-(B_t(t,T)+B(t,T)a)r_t]dt-B(t,T)\sigma dW_t$$
Here $A_t(t,T)$ and $B_t(t,T)$ are partial derivatives wrt $t$ and $W_t$ is Wiener process under $Q$.
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