Question

1 In this problem c(K,T) denotes the price of a European call option with strike price K and strike time T, p(K,T) is the price of the identical put option, r is the risk-free rate and So is the current price of the underlying security. Which of the following are correct? i 0 <c(50,T) - c(55,T) <5e-rT ii 50e-rT <p(45, T) - c(50,T) + So < 55e-rT iii 45e-T <p(45, T) - c(50,T) + So < 50e-rT

Answer 1

Solution:

Since the maximum possible value of the difference between the two calls is:

Max (C (K1) – C (K2)) =e−rT(K2−K1)

using K1= 50 and K2= 55, we have:

                                  C (50, T) – C (55,T)≤5e−rT

Since by ”direction property”: if K1 ≤ K2 then

                                   C (K1) ≥ C (K2)

Again usingK1= 50 andK2= 55, we have

                                 C (50, T) − C(55,T)≥0

Therefore, I is true.

By the Put Call Parity for a nondividend paying stock:

C(S,K,T)−P(S,K,T) =S−Ke−rT⇔P(S,K,T)−C(S,K,T) +S=Ke−rT

For K= 50, the equality becomes:

P(S,50,T)−C(S,50,T) +S= 50e−rT

Since by ”direction property”: if K1≤K2then

P (K1)≤P(K2)

usingK1= 45 andK2= 50, we have:

P(45)≤P(50)⇒P(S,45,T)−C(S,50,T) +S≤50e−rT

On the other hand, forK= 45, the Put Call Parity becomes:

          P(S,45,T)−C(S,45,T) +S= 45e−rT

Since by ”direction property”: if K1≤K2then

C (K1) ≥ C (K2)

usingK1= 45 andK2= 50, we have:

C (50) ≤C(45)⇒P(S,45,T)−C(S,50,T) +S≥45e−rT

Therefore, III is true. Thus, II is not true.