172 Mathematics for Finance
value of the option is zero. SinceCE(S) is an increasing function ofSby
Proposition 7.9, this means that the time value is an increasing function ofS
forS≤X. On the other hand,CE(S′′)−CE(S′)≤S′′−S′for anyS′<S′′by
Proposition 7.10. It follows thatCE(S′′)−(S′′−X)+≤CE(S′)−(S′−X)+
ifX≤S′<S′′, which means that the time value is a decreasing function ofS
forS≥X. As a result, the time value has a maximum atS=X.
The proof for other options is similar.
Exercise 7.18
Prove Proposition 7.19 for put options.