Futures and Options

σStoc

•w

Options: The Black-Scholes-Merton

Model

307 erican call

Exixixi IB

Using Black's approximation for an cm

Suppose that the option in Example 13.6 is American rather than European. The present value of the first dividend is given by 0.5e-009x2/12= 06B 926 The value of the option on th]ó assumption that it expires just before the final ex-dividend date can be calculated using the Black-Scholes-Merton formula ÿ with So = 40 - 06B = 39.5074ÿ K = 40ÿ r = 0.09ÿ σ= 0.30ÿ and T = 06B 926 167: It is $3.52. Black's approximation involves taking the greater of this value and the value of the option when it can be exercised only at the end of six months. From the previous exampl]ôÿ we know that the latter is $3.67. Black's approximation therefore gives the value of the American call as $3.67. them different from the regular options that trade on an exchange or in the over-thecounter marke t6 1. There is a vesting period during which the options cannot be exercised. This vesting period can be as long as four years. 2. When employees leave their jobs (voluntarily or otherwise) during the vesting period ÿ they forfeit the unvested options. 3. When employees leave (voluntarily or otherwise) after the vesting period ÿ they forfeit options that are out of the money and have to exercise options that are in the money almost immediately. 4. Employees are not permitted to sell the options. 5. When an employee exercises options ÿ the company issues new shares and sells them to the employee for the strike price. The fourth feature has important implications. If the employee ÿ for whatever reason ÿ wants to realize a cash b]ô nefit from options that are heldÿ the employee must exercise the options and sell the underlying shares. He or she cannot sell the options to someone else. This leads to a tendency for an employee stock option to b]ô] xercised earlier than the corresponding regular call option. Consider ÿ for exampl]ôÿ an option on a stock paying no...