Closed-Form Approximation of Stock-Based Awards With Moving-Average Vesting Conditions

A market-based compensation award with a moving-average condition becomes vested when the $d$-days moving-average stock price exceeds a predetermined threshold. The same mechanism applies to the economic characteristics of founder shares issued in connection with special purpose acquisition company (SPAC) transactions. This paper employs results from the valuation of barrier and Parisian options to provide a closed-form approximation on the fair value of market-based awards and SPAC transactions as a robust and fast alternative to Monte Carlo simulation techniques. Our method accounts for different exercise strategies, including immediate exercise upon vesting, exercise at maturity, and exercise at the midpoint between vesting and option maturity. We perform a plethora of numerical tests and conclude that our approximation performs well across different levels of equity volatility, risk-free interest rate, moving-average period, option moneyness, and time-to-maturity. We highlight the significant implications of our approach for companies, valuation practitioners, and audit teams.