以動態規劃法評價員工股票選擇權

Evaluation of Employee Stock Options Using Dynamic Programming

現有的員工股票選擇權評價方法，大都是採用Black-Scholes（B-S）的無套利方法，但在實際上，經理人股票選擇權有許多限制，例如閉鎖期、resetting、reloading等特性，使得持有人無法避險，因此現有的ESO評價是在這些不完全市場特性下，評估執行股票選擇權價值。但在不完全市場假設下，ESO是無法用無套利模式來評價，因此股價報酬率的隨機過程並非單純遵循Brownian motion，故評價模式不能用一般的B-S模式架構下計算。Sircar and Xiong（2007）雖考慮了不完全市場等特性，但他是將每個特性，如閉鎖期等特性分開考慮，我們則在Grenadier（1999）的架構下，採用Dixit and Pindyck（1994）動態規劃的方法，並考慮風險偏好，將閉鎖期等三種特性限制整合在同一架構下，導出員工股票選擇權之價值。最後，並以數值分析方法比較用傳統B-S方法、考慮效用之動態規劃法、與修正動態規劃法所計算出之ESO價值的差異。 The current model for evaluating employee stock options (ESOs) is in a complete market. But in real world, there are many restrictions in using ESO: vesting, resetting, and reloading. Therefore, stock return doesn’t follow Brownian motion process and Black-Scholes model cannot be applied in option with these incomplete market characteristics. Sircar and Xiong (2007) consider these incomplete market characteristics in a separate way. Instead, our model considers them in a integrated way, using framework of Grenadier (1999) and dynamic programming method of Dixit and Pindyck (1994). Finally, our numerical analysis shows different ESO values of three methods: traditional Black-Scholes model, dynamic programming method with utility, and modified dynamic programming method.

The current model for evaluating employee stock options (ESOs) is in a complete market. But in real world, there are many restrictions in using ESO: vesting, resetting, and reloading. Therefore, stock return doesn’t follow Brownian motion process and Black-Scholes model cannot be applied in option with these incomplete market characteristics. Sircar and Xiong (2007) consider these incomplete market characteristics in a separate way. Instead, our model considers them in a integrated way, using framework of Grenadier (1999) and dynamic programming method of Dixit and Pindyck (1994). Finally, our numerical analysis shows different ESO values of three methods: traditional Black-Scholes model, dynamic programming method with utility, and modified dynamic programming method.