What are the standard methods to price american call/put options on illiquid underlyings?
For a non-listed company you usually have a poor idea of the share value. If you do not know the underlying price it is impossible to accurately estimate the option price.
So, for cases like this, people typically analyze the options on a risk/reward basis, using similar scenario analyses to those used to analyze the cashflows, assets and other components of the firm valuation.
Here are some practical application for trading illiquid names:
For pricing/forecasting: You still still calculate fair volatility using stock print data. In an illiquid stock and there's a large open interest in the market where professional traders are long, vol might diminish since when stock goes up(down), all the vol traders would be selling(buying) stock to gamma scalp and thus diminish vol.
For quoting: You quote width for options should be correlated with the average width of the stock market. For example: for a 50d call, you would quote 0.3 - 0.32 when stock is: 1.00 - 1.01. In the case where market is wider for stock say 1.00 - 1.05 you should only be willing to quote 0.3 - 0.34. This is because in the first case, when someone lifts your .32 offer, you can buy 1.01 to hedge but in the latter case, you can only buy 1.05.