BDT model implementation

Question

I am looking for a nice and readable description of how to implement BDT model: $d log(r(t)) = [\theta(t)-\frac{\sigma'(t)}{\sigma(t)}log(r(t))]dt + \sigma(t) dW$.

I assume I already have steady-state IR curve $r^*(t)$ and volatility curve $\sigma^*(t)$.

It makes no difference whether it would be binomial tree or Monte-Carlo or FDM implementation. Monte-Carlo seems to be easy but I'm not sure whether I can use $\theta(t) = r^*(t)$ and $\sigma(t)=\sigma^*(t)$.

I went thru Derman's article and Haug's "Options pricing formulas" but found no answer.

Answer 1

All you need is to use the discretization to implement the MC approach. The following links should get you started:

http://www.lcy.net/files/BDT_Seminar_Paper.pdf

http://www-2.rotman.utoronto.ca/~hull/TechnicalNotes/TechnicalNote23.pdf

http://www.iorcf.unisg.ch/Forschung/~/media/Internet/Content/Dateien/InstituteUndCenters/IORCF/Abschlussarbeiten/Frey%202008%20MA%20Monte%20Carlo%20methods%20with%20application%20to%20the%20pricing%20of%20interest%20rate%20derivatives.ashx

In the last paper check from section 6.2

The following papers show examples of BDT model calibration:

http://belkcollegeofbusiness.uncc.edu/wtian1/bdt.pdf

http://www.csie.ntu.edu.tw/~lyuu/finance1/2012/20120530.pdf

http://www.iam.fmph.uniba.sk/studium/efm/phd/urbanova/urbanova-thesis.pdf

Answer 2

You can calibrate the model by discretizing in time, and using a forward induction method as originally proposed by Jamishidian in 1991:

F.Jamshidian, Forward Induction and Construction of Yield Curve Diffusion Models, J.Fixed Income 6, 62-74 (1991).

Although he formulated this induction in the language of the binomial tree, the method is more general, and can be applied for example by allowing the state variable to be continuous.