# Up and Down Multiplicative Factors of the Binomial Option Pricing Model

When computing these factors, according to some sources, $$u=e^{r\Delta t+\sigma \sqrt{\Delta t}}$$, where $$r$$ is the risk-free interest rate, $$T$$ is the time for maturity, and $$\sigma$$ is the volatility. However, some sources suggest that $$u=e^{\sigma \sqrt{\Delta t}}$$.
(By thinking about the time value of money, I think the first one is more accurate)

Another discrepancy in the literature concerns computing volatility. I'm not sure whether to use the standard deviation of the logarithmic return or that of the ordinary return. In case it's merely a matter of computational preference, I'm unsure of the effect on the ultimate result.

Can somebody clarify these for me?

• Thank you very much for your detailed answer. I was a bit confused about this. To make it more confusing in $u=e^{r\Delta t+\sigma\sqrt{\Delta t}}$ some say $r$ is the risk-free interest rate while some say it is the expected return of the stock. So, all we care about is whether the model converges to Black-Scholes in the long run? Commented May 19 at 3:42