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?


1 Answer 1


For the first question (form of up/down factors), CRR uses the second form you described. This answer has some discussion on the matter -- a takeaway from that is that there are various choices with different forms that are valid and converge to Black-Scholes in the limit.

For volatility, the volatility you really want is the forward volatility of the log return. If there are traded options on the market, you can observe (or estimate) it. If there aren't, you (probably) can't. Often, we'll use historical volatility as a proxy -- this is a big assumption, but sometimes it's the best we can do.

The decision to use log vs simple returns for computing historical volatility is small in comparison with the decision to use historical volatility at all. There will be some quantitative differences depending on the exact distribution of the returns, and if I had to choose I suppose I'd choose the log returns, but the difference should be small for most cases.

  • 1
    $\begingroup$ 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? $\endgroup$
    – Bumblebee
    Commented May 19 at 3:42
  • $\begingroup$ Convergence to Black-Scholes is usually a desirable property (especially for a stock option). I am not enough of an expert to confidently say it's the only thing we care about. As an experiment, you can build a binomial tree and try different forms of the up/down factors, different ways of computing the volatility, etc and see how much of an impact they have on your prices and deltas -- my guess is these "technical" considerations have a relatively small impact compared with, say, replacing the historical vol with an estimate of forward vol from the options market $\endgroup$
    – Rylan
    Commented May 20 at 7:46

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