The task is to determine the value of restricted stock grants. In this program, a person can earn up to 1,000 shares of common stock. The number of shares depends on the Company's 30-trading day average stock price (so if this average stock price is $100 at any point during the next 5 years, person gets 250, if it is $110 he gets an additional 250, $120 an addition 250 and $130 an addition 250). Assume a company's expected volatility is 30%. I can do this via a binomial model, but was wondering whether or not the volatility of 30% is appropriate to use when vesting is based on a 30-day average. If not, is there an accepted formula that I could use to convert the daily volatility into a 30 day volatility? My initial thought is that the 30% volatility is an annual volatility and often this is calculated by determining daily volatility and multiplying by the sq root of 252 (to reflect number of trading days). So to convert to a 30-trading day average, I would calculate it as: 30% x sqrt(30) / sqrt(252). But then, this is more like a monthly volatility...don't know if that is conceptually correct.Thanks in advance for your help!

Interesting problem. My thoughts are that there are two volatilities one for determining the grant size and one for the option value (as the payoff is not based on the average of 30days for the final day's price..). The option value vol should be 30%. For the grant size, have you considered the vol for the series of 30 days moving average prices - this seems to me to be the most intuitive.

hi tttcheni think you need to do monte-carlo simulation asuming .3*sqrt(252) daily vol, and calculate 30 day moving average for each path.then calculate option value based on the results.however, pricing becomes difficult if common shares are actually issued by the company to pay for options held by persons. in this case volatility has to be adjusted to reflect volatility of existing shares + your quazywarrants. the problem is though, that you don't know exactly what is the amount of common stocks to be issued under stock grant programme. again monte-carlo simulation may help addressing this issue.

Am I correct that the number of shares depends on the maximum 30-day average price over any month in the next five years? If so, you can simulate or calculate it exactly as you would a monthly series of prices (except that the first month has only half a month to vary). You can use the 30% volatility (of course, divided by the square root of 12 to make it monthly).You could model this as a strip of up-and-in calls and get an analytic solution. Or you could simulate.

The number of shares depends on the average 30-day stock price. Once a hurdle is reached, the corresponding number of shares vest immediately. This is tested daily, based on then 30-day average closing price. So if in 100 trading days, the 30-day average stock price (i.e., average of closing prices for days 71-100) hits $100, the person gets 250 shares vested (assuming the average did not reach $100 before then) and can't lose them. Assume that it takes another 100 trading days before the 30-day average (in this case, average for days 171-200) reaches $110, person then gets another 250 shares vested, and so on. The person may never vest into all of the shares.As I'm not adept at Monte Carlo or other simulations, I was going to do this via a binomial model. I'd build the stock price tree starting with the current 30-day average stock price, and then build out the nodes based on volatility, time, and number of nodes. Then I can look at each node and determine how many shares vest at each node. Then work backward through the tree to get to a expected number of vested shares. This expected number is multiplied by today's stock price to get to a value.I'm just trying to determine whether the volatility used in building the stock price tree should be the annual vol of 30% (which is the daily vol multiplied by sqrt of 252) or something close to a monthly vol. I thought about using up and in barriers, but there's a feature that I didn't describe (wasn't germaine to my question on volatility) that I thought would not enable the use of barriers (but there may be types of barriers that handle this feature that I don't know about). Namely, none of the vesting begins until after two years, so even if the average stock price exceeded $130 in the first two years, nothing would vest at year 2 due to prior stock price average unless the 30-day average at 2 years was "in-the-money".If I read prior responses correctly, judo and chiron seem to be suggesting use of the annual vol (30%) and aaron suggests a monthly vol (my apologies if I'm wrong about what each is saying). I don't believe the vol est should be different whether I'm using a binomial model or monte carlo.

The trouble with a binomial model is these options are path dependent, so your tree does not recombine. Thus at each step you have 2^n nodes instead of n+1. This usually becomes intractable quickly. You could get around this somewhat by only forming a new node at a vesting point. That is you keep track of both share price and number of vested shares. You recombine nodes with the same number of vested shares, but not if that amount has changed. You end up with fewer than k*(n+1) terminal nodes, where k is the number of possible vesting states.What time step are you planning to use? You need daily time steps to capture the 30-day average, but this is more than you need for option valuation. One solution is to use monthly time steps, then do a separate calculation each month to compute the probability distribution of vesting states.Either way, you have to use the volatility of the stock, adjusted for the time step.