An Executive Stock Option Model

Because of the feature of the freeze-out period, (also known as vesting) period, employee options are often modeled as forward stating options. A forward start option with time to maturity T starts at-the-money or proportionally in or out-of-the money after a known elapsed time t in the future. The strike price is set equal to a positive constant a times the asset price S after the known time t. Rubinstein (1990) uses the following formula for the call option:

c = Se^(b-r)t (e^(b-r)(T-t)N(d₁) - ae^-r(T-t)N(d₂))

where

d₁=(In(1/a)+(b+(Vol²/2)(T-t)) / (Vol x SQRT(T-t)),

d₂=d₁-Vol x SQRT(T-t)

Consider an employee who receives a call option with forward start three months from today. The options start 10% out-of-the-money, time to maturity is one year from today, the stock price is 60, the risk-free interest rate is 8%, the continuous dividend yield is 4%, and the expected volatility of the stock is 30%. In other words, S = 60, a = 1.1, t = 0.25, T = 1, r = 0.08, b = 0.08-0.04 = 0.04, and Vol = 0.30.

Build a spreadsheet model to calculate the call price with panels for inputs and panels for the model, similar to the Black-Scholes type analytical models. Submit your spreadsheet model. Use interim calculation steps, such as d1, d2, N(d1) and N(d2), before you reach your final valuation of the option. Round up your final result in $0.01.