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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.

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    $\begingroup$ The treasury publishes interest rates for periods less than 1 year in annualized form. So when they say 3 month bills 1.01% per year, it means you make 1.01*(3/12) percent in 3 months: you invest 1 now and you have 1.002525 three months from now. $\endgroup$ – noob2 Oct 19 '17 at 14:54
  • $\begingroup$ @noob2 Thank you very much! And do you know about the volatility and is my way a general way to estimate the volatility? $\endgroup$ – codeedoc Oct 19 '17 at 16:08
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    $\begingroup$ What you propose is sometimes called the "historical volatility" method. It is not that bad, but by now most people recognize that the market is forward looking and if for ex. some important event is coming up the market prices options with a higher vol than historical and vice versa if the market thinks that volatiliy will calm down a lower. And everyone is trying to figure out what volatility other people expect. So it is not just history, it is a dynamic process in the marketplace that detrmines vol. $\endgroup$ – noob2 Oct 19 '17 at 18:17
  • $\begingroup$ @noob2 OK, I see that. Thank you! $\endgroup$ – codeedoc Oct 19 '17 at 18:41

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