Binomial Option Factors and Volatility Input Choices
Summary
The note addresses two choices in a binomial option model: how to specify the up and down multipliers, and whether historical volatility should be calculated from log returns or simple returns. It identifies the Cox-Ross-Rubinstein convention as using an up factor based on volatility and the time step without including the risk-free rate in that multiplier. Other formulations can also be valid, and the answer says different choices may converge to Black-Scholes as the step size shrinks.
For volatility, the conceptually relevant input is forward volatility of log returns, which may be observed or estimated from traded options. When options are unavailable, historical volatility is often used as a proxy, but that substitution is a substantial assumption. The answer expects the difference between log-return and simple-return historical estimates to be relatively small in many cases, while emphasizing that choosing historical volatility at all is the larger modeling decision. No numerical comparison or derivation is supplied.
Key ideas
- The Cox-Ross-Rubinstein up factor excludes the risk-free rate term from its multiplier.
- Different valid binomial specifications can converge to the Black-Scholes model as the time step becomes small.
- Forward volatility of log returns is the volatility concept relevant to option pricing.
- Historical volatility is a proxy when option-implied information is unavailable and requires an assumption.
- The return convention can affect historical volatility estimates, though the answer expects a modest difference in many cases.
Tags
Full text
# Up and Down Multiplicative Factors of the Binomial Option Pricing Model
# 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?
## Answer by Rylan (score 2, accepted)
https://quant.stackexchange.com/a/79417
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.