Black–Karasinski Forward Rates and Bond Option Pricing Limits

Summary

The note separates the definition of an instantaneous forward rate from the interest rate model used to describe rates. It derives the forward rate from zero coupon bond prices: when a sufficiently smooth continuum of such prices is available, the initial forward rate is the negative maturity derivative of the log bond price. It then contrasts the Black–Karasinski short rate specification with the Heath–Jarrow–Morton forward rate framework and sketches how their dynamics might be reconciled.

The response does not derive that reconciliation or provide a bond option pricing formula. It says that, to the respondent’s knowledge, the model lacks analytical bond and bond option prices, so numerical methods may be needed in practice. It also warns of finite time explosion problems and suggests other short rate models for users who require positive rates. These are brief forum remarks rather than a complete derivation or a detailed comparison of model behavior.

Key ideas

• Instantaneous forward rates can be inferred from the maturity slope of log zero coupon bond prices.
• The forward rate definition is independent of the chosen interest rate model.
• The note outlines a possible connection between Black–Karasinski and Heath–Jarrow–Morton dynamics but does not work through it.
• The response reports no known analytical bond or bond option prices for the model.
• It flags finite time explosion as a model limitation.

Tags

# What is the forward rate for a Black-Karasinski interest rate model?


# What is the forward rate for a Black-Karasinski interest rate model?












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.

## Answer by TheBridge (score 4)

https://quant.stackexchange.com/a/2204

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

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