108156.pdf

7. Options: General Properties 171

Figure 7.8 Time valueCE(S)−(S−X)+of a European call option

For a European option we have to wait until the exercise timeTto realise the
payoff. The risk that the stock price will rise aboveXin the meantime may be
considerable, which reduces the value of the option.

Figure 7.9 Time valuePE(S)−(X−S)+of a European put option

The time value of an American call option is the same as that of a European
call (if there are no dividends) and Figure 7.8 applies. For an American put a
typical graph of the time value is shown in Figure 7.10.

Figure 7.10 Time valueCA(S)−(S−X)+of an American put option

Figures 7.8, 7.9 and 7.10 also illustrate the following assertion.

Proposition 7.19

For any European or American call or put option with strike priceX, the time
value attains its maximum atS=X.

Proof

We shall present an argument for European calls. ForS ≤X the intrinsic