I am writing a program to price American put options with binomial pricing model and to compare it with the market price.
When I used made-up numbers for $\sigma$ and $r$, the price by binomial pricing model is very close to its European counterpart by Black-Scholes equation. (And always a little bit higher which makes sense since American puts should be worth more than European puts). So my code should be correct in this sense.
And my question is what volatility and interest rate I should use?
1. If the duration of the option is 1 month, should I use $std(log(\frac{S_{i+1}}{S_i}))$ for the past month as the volatility $\sigma$ in the model?
2. For the interest rate $r$, I found that some people suggest to use the treasury rate for the corresponding duration of the option. However, I found no source saying whether the rates on https://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/TextView.aspx?data=yield are continuously compounded or annually compounded. I tried using both. But both of the resulted option prices are still way off from the market value.
Any other related suggestions are more than welcome. If anyone with practical experience can answer these naive questions, I'd really appreciate it.