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.

  • $\begingroup$ 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. $\endgroup$
    – A.Oreo
    Commented Nov 1, 2017 at 6:13
  • $\begingroup$ @A.Oreo: thanks for the comment. I am thinking of writing to John Hull. $\endgroup$
    – pinaki
    Commented Nov 2, 2017 at 18:50
  • $\begingroup$ Why don't you tweet Emanuel Derman? He is incredibly kind and generous. $\endgroup$ Commented Nov 4, 2017 at 3:33

1 Answer 1


From the gentleman and scholar Emanuel Derman. Emanuel states "the last two pages answer the question asked".


Please thank him directly on Twitter.


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