Zero-Coupon Bond Pricing and Limitations of Rendleman–Bartter Rates
Summary
The document discusses pricing a zero-coupon bond when the short rate follows the Rendleman–Bartter lognormal diffusion. Its answer says a closed-form expression exists but is complicated, involving special functions, and points to published derivations. It describes obtaining an unconditional bond value by averaging the discounted payoff across the terminal short-rate distribution, after considering a conditional expectation.
The response also names Ho–Lee, CIR++, Hull–White, and other models that can fit an initial zero curve and be calibrated to additional instruments. It cautions that lognormal short-rate models can have problematic expectations involving inverse future bond prices and notes that Rendleman–Bartter lacks mean reversion. These are qualitative model-selection observations; the document gives no derivation, parameter calibration, or numerical comparison, and the original questioner’s experience level is acknowledged.
Key ideas
- Rendleman–Bartter bond pricing has a closed-form treatment, though the cited answer characterizes it as complex.
- An unconditional zero-coupon price can be obtained by averaging discounted payoffs over rate outcomes.
- The answer lists alternative short-rate models that can fit the initial curve.
- It cautions about inverse-bond-price expectations and the lack of mean reversion in Rendleman–Bartter.
Tags
Full text
# Zero-coupon bond price under Rendleman-Bartter Model
# Zero-coupon bond price under Rendleman-Bartter Model
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?
Thanks!
## Answer by byouness (score 1)
https://quant.stackexchange.com/a/46536
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.