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+O,apter 33: Mathematical Considerations for Index Products 
+EUROPEAN EXERCISE 
+647 
+To account for European exercise, one basically ignores the fact that an in-the-money 
+put option's minimum value is its intrinsic value. European exercise puts can trade at 
+a discount to intrinsic value. Consider the situation from the viewpoint of a conver­
+sion arbitrage. If one buys stock, buys puts, and sells calls, he has a conversion arbi­
+trage. In the case of a European exercise option, he is forced to carry the position to 
+expiration in order to remove it: He cannot exercise early, nor can he be called early. 
+Therefore, his carrying costs will always be the maximum value to expiration. These 
+carrying costs are the amount of the discount of the put value. 
+For a deeply in-the-money put, the discount will be equal to the carrying 
+charges required to carry the striking price to expiration: 
+Carry = s Ji - 1 ] L (1+ r)t 
+Less deeply in-the-money puts, that is, those with deltas less than - 1.00, would 
+not require the full discounting factor. Rather, one could multiply the discounting 
+factor by the absolute value of the put' s delta to arrive at the appropriate discounting 
+factor. 
+FUTURES OPTIONS 
+A modified Black-Scholes model, called the Black Model, can be used to evaluate 
+futures options. See Chapter 29 on futures for a futures discussion. Essentially, the 
+adjustment is as follows: Use 0% as the risk-free rate in the Black-Scholes model and 
+obtain a theoretical call value; then discount that result. 
+Black model: 
+Call value= e-rt x Black-Scholes call value [using r = 0%] 
+where 
+r is the risk-free interest rate 
+and t is the time to expiration in years. 
+The relationship between a futures call theoretical value and that of a put can 
+also be discussed from the model: 
+Call = Put + e-rf(J - s) 
+where 
+f is the futures price 
+ands is the striking price.