Question

Consider a European call option on a non-dividend-paying stock. The strike price is K, the time to expiration is T, and the price of one unit of a zero-coupon bond (with face value one) maturing at T is B(T). Denote the price of the call by C. Show that C > max{0, So – KB(T)}, where So is the current stock price.

Answer 1

Payoff from the call option at t = T = C_T = max (0, S_T - K)

Hence, the price of the call option today

= C

= PV of all the future payoff from the instrument

= PV of [max (0, S_T - K)]

= max [0, PV (S_T - K)]

= max [0, PV (S_T) - PV (K)]

PV (S_T) = present value of the future stock price = Current stock price = S₀

PV (K) = K / (1 + discount rate)^T

B(T) = Price of a zero coupon bond paying face value of one at t = T will be

= FV / (1 + (1 + discount rate)^T = 1 / (1 + discount rate)^T

Hence, B(T) = 1 / (1 + discount rate)^T

Hence, PV (K) = K / (1 + discount rate)^T= K x B(T)

Hence, C =  max [0, PV (S_T) - PV (K)] = max {0, S₀ - KB(T)}