Employee stock options （ESOs） have become an integral component of compensation in the US. In view of their significant cost to firms, the Financial Accounting Standards Board （FASB） has mandated expensing ESOs si...Employee stock options （ESOs） have become an integral component of compensation in the US. In view of their significant cost to firms, the Financial Accounting Standards Board （FASB） has mandated expensing ESOs since 2004. The main difficulty of ESO valuation lies in the uncertain timing of exercises, and a number of contractual restrictions of ESOs further complicate the problem. We present a valuation framework that captures the main characteristics of ESOs. Specifically, we incorporate the holder＇s risk aversion, and hedging strategies that include both dynamic trading of a correlated asset and static positions in market-traded options. Their combined effect on ESO exercises and costs are evaluated along with common features like vesting periods, job termination risk and multiple exercises. This leads to the study of a joint stochastic control and optimal stopping problem. We find that ESO values are much less than the corresponding Black-Scholes prices due to early exercises, which arise from risk aversion and job termination risk; whereas static hedges induce holders to delay exercises and increase ESO costs.展开更多
The pricing and hedging problem of foreign currency option with higher borrowing rate is discussed.The method to obtain the price and hedging portfolio of currency option is based on backward stochastic differential e...The pricing and hedging problem of foreign currency option with higher borrowing rate is discussed.The method to obtain the price and hedging portfolio of currency option is based on backward stochastic differential equations(BSDE for short) theory and Malliavin calculus technique.The sensitivity of the model parameters is also considered and some numerical simulations are given to illustrate our conclusion.展开更多
文摘Employee stock options （ESOs） have become an integral component of compensation in the US. In view of their significant cost to firms, the Financial Accounting Standards Board （FASB） has mandated expensing ESOs since 2004. The main difficulty of ESO valuation lies in the uncertain timing of exercises, and a number of contractual restrictions of ESOs further complicate the problem. We present a valuation framework that captures the main characteristics of ESOs. Specifically, we incorporate the holder＇s risk aversion, and hedging strategies that include both dynamic trading of a correlated asset and static positions in market-traded options. Their combined effect on ESO exercises and costs are evaluated along with common features like vesting periods, job termination risk and multiple exercises. This leads to the study of a joint stochastic control and optimal stopping problem. We find that ESO values are much less than the corresponding Black-Scholes prices due to early exercises, which arise from risk aversion and job termination risk; whereas static hedges induce holders to delay exercises and increase ESO costs.
基金supported by the National Nature Science Foundation of China(11221061,61174092,11126214,11126208)the National Science Fund for Distinguished Young Scholars of China(11125102)the Fundamental Research Funds for the Central Universities(2010QS05)
文摘The pricing and hedging problem of foreign currency option with higher borrowing rate is discussed.The method to obtain the price and hedging portfolio of currency option is based on backward stochastic differential equations(BSDE for short) theory and Malliavin calculus technique.The sensitivity of the model parameters is also considered and some numerical simulations are given to illustrate our conclusion.
