# Black Derman Toy model: from tree to differential equation

The Black Derman Toy model of interest rates is usually introduced as the model governed by the stochastic differential equation: $$d \ln r = \left[\theta(t) + \cfrac{\sigma'(t)}{\sigma(t)}\ln r \right]dt + \sigma(t) dz$$ However, originally the model was developed using binomial trees. The Black-Derman-Toy paper does not contain any differential equation. How does one show that the model constructed using binomial trees corresponds to the above differential equation in the continuous limit?

In a number of articles, for example the Black-Karasinski paper, the derivation is attributed to Hull and White's 1990 article "Pricing interest-rate derivative securities", but it does not give the derivation, only states it as a fact. I have looked into other papers of Hull and White available from John Hull's website, but did not find anything.

• I asked the same question before, I can't deduce the relation between the drift and diffusion terms through the BDT tree method. But no one answered me. – A.Oreo Nov 1 '17 at 6:13
• @A.Oreo: thanks for the comment. I am thinking of writing to John Hull. – auniket Nov 2 '17 at 18:50
• Why don't you tweet Emanuel Derman? He is incredibly kind and generous. – Paul Portesi Nov 4 '17 at 3:33