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I'm reading about Binomial Model on "Arbitrage Theory in Continuous Time" by Tomas Bjork. I found an important result which allow us to state that in a one period model $q_u$ and $q_d$ are actually probabilities. The following theorem holds:

"The model is arbitrage-free if and only if $$d\le(1+R)\le u."$$ At the same time I'm reading "The Bible" by Hull where the author deals with the model but using continuous compounding of interest rates. I'm wondering if an analogous of Bjork'sresult holds, such as

The model is arbitrage-free if and only if $$d\le e^{rt} \le u,$$ where $t$ is the option maturity (I'm taking into account maturities which are different from 1). Thanks in advance.

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    $\begingroup$ If the first statement is true then the second is also as you essentially just convert between the different compounding frequencies - i.e. from the one-period rate $R$ to the continuously compounded rate $r$. If the second result didn't hold then the up-probability $p = \left( e^{r \Delta t} - d \right) / (u - d)$ would not be in $[0, 1]$. $\endgroup$
    – LocalVolatility
    Feb 28, 2017 at 20:14
  • $\begingroup$ @LocalVolatility could you please add this as an answer? $\endgroup$
    – SRKX
    Mar 2, 2017 at 1:24

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If the first statement is true then the second is also as you essentially just convert between the different compounding frequencies - i.e. from the one-period rate $R$ to the continuously compounded rate $r$. If the second result didn't hold then the up-probability

$$ p = \frac{e^{r \Delta t} − d}{u − d} $$

would not be in $[0,1]$.

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