Closed‐Form Approximation of Stock‐Based Awards With Moving‐Average Vesting Conditions
利用障碍期权和巴黎期权的估值结果，为具有移动平均行权条件的市场奖励和SPAC交易提供了闭式近似估值方法，作为蒙特卡洛模拟的快速替代方案，并考虑了不同行权策略。
ABSTRACT A market‐based compensation award with a moving‐average condition becomes vested when the ‐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.