let's say that I have simulated the interest rate using the Rendleman-Barttermodel, (which is not the best for rates I know) and then I want to simulate paths for the bond paying 1 at maturity:

$$dr_t = r_t \left( \theta dt + \sigma dW_t \right)$$

Is there a closed form formula for ZCB under this model? Any recommendations?

PS: I am not experienced in Interest Rate Modelling. Besides Vasicek and CIR, are there any other easy to understand and calibrate-simulate models for the interest rate?



1 Answer 1


Under the Rendleman-Bartter model, a closed-form formula exists for the zero-coupon bond price. However, it is very complex involving Bessel functions and complex numbers...

Deriving the formula is actually the purpose of a paper by Uri Dothan called "On the term structure of interest rates" that you can find here: https://www.sciencedirect.com/science/article/pii/0304405X7890020X

The solution is also given in "The Lognormal Interest Rate Model and Eurodollar Futures" by Michael Hogan and Keith Weintraub that you can download here: https://www.researchgate.net/publication/269111948_The_Lognormal_Interest_Rate_Model_and_Eurodollar_Futures

It gives the price of the zero coupon $P(0, T)$ conditionally on the terminal short rate value $r(t)$: $$ \mathbb{E} \left[ e^{-\int_0^t r(u)du} | r(t) = x \right] $$

You can integrate over $r(t)$ whose density function you know to get the (unconditional) zero-coupon value: $$ \mathbb{E} \left[ e^{-\int_0^t r(u)du}\right] $$ You can add to your models list Ho Lee, CIR++, Hull-White, that enable to match the market's zero curve at t = 0. Then you can make the parameters piecewise constant and calibrate on other instruments as well such as swaptions, etc. For interest rates modelling, I would recommend Andersen and Piterbarg's excellent book: Interest Rate Modelling (Part III - Term structure models): https://www.amazon.com/Interest-Rate-Modeling-Structure-Models/dp/0984422110/

It is generally not recommended to work with lognormal short rate models (such as Rendleman-Bartter, Black-Karasinski, Black-Derman-Toy), as they give an infinite expectation for the inverse of future zero coupon bond prices: $$ \mathbb{E} \left[ \frac{1}{P(s, T)} | \mathcal{F}_t \right] = +\infty, t < s < T $$

Furthermore, Rendleman-Bartter doesn't have the mean-reversion property empirically observed in interest rates.


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