I was wondering if anyone could help me with the instantaneous forward rate equation for a Black-Karasinski interest rate model?
I was also after the Black-Karasinski Bond Option Pricing Formula.
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1$\begingroup$ Hi Ian, welcome to QuantSE. In order to maximize your chances to get an answer, please provide a link to a description of the model you are mentioning, or, even better, add the dynamics $dr$ to question. This will also make the site more readable for other users. $\endgroup$– SRKXCommented Oct 19, 2011 at 7:42
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$\begingroup$ Also, you should provide us with what you've come up so far. $\endgroup$– SRKXCommented Oct 19, 2011 at 7:50
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1 Answer
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Hi the forward rate equation is not dependent on the model it is calculated upon the prices of zero coupon bonds by the following equation :
$$ P(t,T)=exp{-\int_t^T f_t(u).du} $$
If you have a continuum of zero coupon bond prices which are sufficiently smooth then you can deduce from it that :
$$f_0(T)=-\frac{\partial Ln(P(0,T))}{\partial T}$$
Anyway, I think that what you are really asking for, is what is the set of SDEs followed by those instantaneous forward rates under proper measure. I haven't done the calculations (it really bothers me) but I can indicae the following procedure,to wit you have to reconcile HJM with BK, which are respectively given by :
$$d (ln r_t)= \kappa(\theta(t) - ln(r_t))dt +\sigma dW_t$$
and :
$$r_t=f(0,t)+\int_0^t\sigma'(u,t)[\int_u^t\sigma'(u,s)ds]du+\int_0^t\sigma'(u,t)dW_u$$
where
$$df_t(u)=\sigma'(t,u)[\int_t^u\sigma'(t,s)ds]dt+\sigma'(t,u)dW_t$$ and $r_t=f(t,t)$.
Anyway there is no analytical (to my knowledge) Bond and Bond Option prices in this model.
By the way that there are finite explosion time problems in this model. You should try Hull & white model or CIR model if you want your rates to stay positive.
Best Regards
answered Oct 20, 2011 at 11:59
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$\begingroup$ "there are finite explosion time problems in this model." In my simulations I noticed this. But wondering if there is a reference on this. This explosion seems to be time related ie r(t_2)>r(t_1), even when I ran 300,000 simulations. I wonder if this is present in LMM, also. $\endgroup$ Commented May 18, 2014 at 20:39