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.