The Exact Value for European Options on a Stock Paying a Discrete   Dividend
  In the context of a Black-Scholes economy and with a no-arbitrage argument, we derive arbitrarily accurate lower and upper bounds for the value of European options on a stock paying a discrete dividend. Setting the option price error below the smallest monetary unity, both bounds coincide, and we obtain the exact value of the option. 
Jo\~ao Amaro de Matos, Rui Dil\~ao and Bruno Ferreira