An approximation procedure for valuing a call option on a dividend-paying share of stock. It was developed by Fischer Black (co-developer of the Black-Scholes Model). Black’s approximation essentially involves using the Black-Scholes model. However, it makes adjustments to the stock price and expiration date in order to account for the early exercise feature of an American call.
For example, assume a dividend-paying 6-month American option where: dividends, at times t1 and t2, D1= D2= USD0.7, underlying’s current price S0 = USD50, strike price K= USD50, interest rate r= 8%, ex-dividend dates t1= 3/12, t2= 4/12. The holder must consider whether to exercise the option before the first ex-dividend date or before the second ex-dividend date, by comparing K[1- e-r (t2-t1)] with the dividend:
K[1- e-r (t2-t1)] = 50 [1- e-0.08×0.25]= 0.99, this value is greater than the dividend, and therefore it should not be exercised before the first ex-dividend date.
K[1- e-r (T-t2)] = 50 [1- e-0.08×0.1667]= 0.66, this value is less than the dividend, the option should be exercised before the second ex-div date.
The black’s approximation requires that the current underlying’s price is adjusted by the present value of the first dividend (the present value is 0.7 e-0.08×0.25= 0.68). By solving the Black-Scholes formula with:
S0 = 50- 0.68= 49.32, strike price K= 50, interest rate r= 8%, σ = 0.25, and T= 4/12 (or 0.3333), the American option’s value is USD 3.14
Black’s approximation is based on comparing this value with the value of an equivalent European call (where:
S0 = 50, strike price K= 50, interest rate r= 8%, σ = 0.25, and T= 6/12, and the option’s value is USD 4.52 ). The greater value among the two (USD 4.52) is that of the American call.
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