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In the binomial tree options pricing literature, I see frequent reference to the definition that

$$ u = e^{\sigma \sqrt{n/t}} $$

I think I understand the model, but how do we derive this, i.e. how do I fit the model to data? I've tried to derive it myself but get a different answer, namely

$$ u = \exp(\sigma / 2\sqrt{np(1-p)}) $$

where $p$ is the risk neutral probability. The paper "A Synthesis of Binomial Option Pricing Models for Lognormally Distributed Assets" even argues that the CRR derivations admit arbitrage for discrete time periods.

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