Choosing Volatility, Rates, and Steps for American Option Trees

Summary

The document discusses practical inputs for valuing American options with a Cox–Ross–Rubinstein binomial tree. It recommends using a fitted implied volatility curve or surface where available, since volatility varies across strikes and expiries; historical volatility is an alternative but may not reflect current option market pricing. The answer does not prescribe a separate volatility for every tree node, instead pointing toward market-calibrated volatility inputs.

For discounting, it recommends using the yield curve to present-value tree cash flows, while observing that a single rate may have a limited effect on the result. The number of steps should be chosen adaptively so prices converge within a chosen tolerance, since the required resolution depends on expiry, volatility, and the option. One step per day may be reasonable for many liquid exchange-traded options around a quarter from expiry, but it is not a universal rule. Coarser-than-daily steps can require attention to overnight price gaps. The document offers practical guidance rather than a calibrated procedure or evidence from comparative tests.

Key ideas

• A fitted implied volatility curve or surface better reflects market pricing than a single historical volatility estimate.
• Use the yield curve to discount cash flows at the relevant points in the tree.
• Choose the tree resolution by checking price convergence for the option being valued.
• Daily steps may be a reasonable starting point for some liquid options, but not a universal standard.
• Coarse time steps may need to account for overnight gaps.

Tags

# Volatility input for American options


# Volatility input for American options












I have to price an american option on a daily basis and I have some questions regarding the CRR binomial tree model:

- Is it correct to use implied volatility as an input? Or is it better to use historical vol? If I were to use implied vol, should I calculate one vol per node or just one vol using the time-to-expiry?

- Is it correct to use a yield curve to price each node of the tree or should I discount all the nodes just with a single interest rate?

- Is there a standard number of steps to build the tree? I'd like to use one step per day, but I don't know if it is correct.

Thanks for your help!

## Answer by kurtosis (score 1, accepted)

https://quant.stackexchange.com/a/57901

Usually, you would use the volatility from a fitted volatility curve or surface. Those are based on implied volatilities. You can use historical volatility, but then your valuation is likely to be off because the volatility curve/surface is not constant and at historical vol.

You should use a yield curve to present value nodes. This is unlikely to make a big difference in pricing, but you should do it since it is the right thing to do.

The steps will vary with the time to expiry, volatility, etc. You want to have an adaptive number of steps that gives an answer convergent to within some tolerance. That will vary by option and maybe even in other ways. For most exchange-traded options which are most liquid one quarter out, it probably makes sense to have one step per day. Going below one step per day means you may need to consider overnight gap effects.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.