108156.pdf

156 Mathematics for Finance

other hand, we have the lower bound

S(0)−Xe−rT≤CE,

which follows immediately by put-call parity, sincePE≥0. Moreover, put-call
parity implies that
PE<Xe−rT,

sinceCE<S(0), and
−S(0) +Xe−rT≤PE,

sinceCE≥0.
These results are summarised in the following proposition and illustrated
in Figure 7.4. The shaded areas correspond to option prices that satisfy the
bounds.

Figure 7.4 Bounds on European call and put prices

Proposition 7.3

The prices of European call and put options on a stock paying no dividends
satisfy the inequalities

max{ 0 ,S(0)−Xe−rT}≤CE<S(0),

max{ 0 ,−S(0) +Xe−rT}≤PE<Xe−rT.

For dividend-paying stock the bounds are

max{ 0 ,S(0)−div 0 −Xe−rT}≤CE<S(0)−div 0 ,

max{ 0 ,−S(0) + div 0 +Xe−rT}≤PE<Xe−rT.

Exercise 7.11

Prove the above bounds on option prices for dividend-paying stock.