I'm having trouble figuring out how to value the following option:The option is a standard American call, however, it only vests upon the underlying company achieving a certain predetermined level of earnings. For example, if the company achieves $100 of earnings at the end of the first year, 20% of the options vest as normal call options (albeit with a 9 year remaining life); if they achieve $110 of earnings for year 2, another 20% vests, etc. for 5 years. There is also a "catch-up" provision whereby if the company misses its earnings target in the first year but achieves the year 2 target, 40% of the options vest.The earnings targets are based on company budgets/forecasts. One thought I had for this was a monte carlo simulation based on earnings, but I don't know what type of distribution their earnings will have; even if I assume a normal distribution, I don't know how to estimate standard deviation. Presumably, the projected earnings level is based on management expectations and they would likely say that there is a greater than 50% probability of achieving/exceeding the targeted level.Thanks in advance for your suggestions!

One "option" is to assume a Brownian process for the stock's price (you need the stock volatility anyway so you must have some estimate of that) and then assume some model between the price and the underlying earnings. The model that maps price to earnings can be as simple or as realistic as you want. At the simple end, you could use a fixed P/E ratio, but that would be a horribly crude approximation. If you have the time (and the data), I'm sure you could do a quicky empirical correlation between earnings-per-share and share price an use that correlation as to generate a stochastic function that maps the stock price to a earnings level. A more sophisticated analysis would analyze lagged covariance between earnings and price to intercouple the price at the end of years 1, 2, 3, ..., 10 with the earnings at the ends of years 1, 2, 3, ..., 10.The point is that the value of the option vesting program is a function of both price and earnings and that price and earnings aren't independent variables on the timescale of years.

Thanks. I had considered an approach similar to the one you outlined, but the company has very limited history as a public company (less than 6 months) and it's P/E multiple has varied significantly during that period. So I thought it might be more accurate to run a simulation based on earnings.

I'd not worry too much about the "limited history" issue because you can model the company against a set of peers that have longer histories. Even if the company is in some totally new industry (e.g., a peer-to-peer web 2.0 nanotechnology start-up), you can still model it terms of tech start-ups.Modelling the earnings won't be easy, but you might try using sigmoid or logistic curves for revenue and costs where the plateau of the sigmoid is defined by random variable (with a constant profit margin between the cost and revenue curve) and the slope of the sigmoid are two correlated random variables for cost and revenue. This models the fact that as a company matures it will reach a plateau at some reasonable profit margin. It also models the early stage of the company where the rate of climb of revenues and costs may be decoupled and costs, in particular, might rise faster than revenues. A more sophisticated model would separately model the costs as variable/direct/cost-of-goods (which is directly proportional to revenues) and fixed/indirect/sunk costs that have a lagged effect on revenues (e.g., marketing, new product development, and depreciation of capex). The numbers for this model will need to come from the company's financial statements and growth projections. The scatter for the variables would need to come from a similar analysis of other analogous companies. This won't be easy and I'm sure you will find companies that vary by multiple orders of magnitude in terms of long-term outcomes (e.g., compare Amazon.com to eToys.com)Regardless of how you model it, you will need to model the link between earnings and prices.