Black Scholes

To price these options, we need the implied volatility which is obtained from market prices of 5 year options. In the absence of such data, we use the realised volatility over a five year period, (we would have preferred to calculate the five year historical volatility), but we only have the rolling 90 day volatility. At this stage, we debated whether or not to exclude the crisis period of 2008-2009 from our calculation of volatility. While there is an argument for excluding this period as an outlier, we question the validity of this assumption and use what visually appears to be the 5 year historical average volatility of 25%.

Plugging this into the Black Scholes formula (along with a strike of 24$, stock price of 21$, rate of 0.81%), the value of a call option is 3.89$. This implies that the value of the stock option grant (15000 options) 58,349$.

2a. While Black-Scholes is not the appropriate way to value employee stock options, financial accounting principles require companies to use it in financial statements. Below we list the factors that make Black-Scholes (BS) model less suitable for valuing ESOs than for regular options:

• The BS model is not applicable to American options and thus is not suitable for valuing ESOs, which can be exercised before maturity.

• Using BS requires forecasting volatility of the stock over the life of the option. For ESOs, this duration is typically 5-10 years and as such estimating volatility for such a long period is extremely difficult and clearly yields a very rough estimate. Moreover, the assumption of BS, that volatility of the stock returns is constant over the life of the option, is even more unrealistic in the case of ESOs (vs. regular options), which have unusually long lives.

• BS assumes ESOs are tradable. If they were, it would be possible to take advantage of arbitrage opportunities arising from discrepancies with the BS value (if the options is worth less than the BS value, buy the option, sell the BS...