Consider a one-period model with only two possible end-of-period states.

Three assets are traded in an arbitrage-free market. Asset 1 is a risk-free asset with a price of 1 and an end-of-period dividend of Rf , the risk-free gross rate of return. Asset 2 has a price of S and offers a dividend of uS in state 1 and dS in state 2.

(a) Show that if the inequality d < Rf < u does not hold, there will be an arbitrage.

Asset 3 is a call-option on asset 2 with an exercise price of K. The dividend of asset 3 is therefore Cu ≡ max(uS − K, 0) in state 1 and Cd ≡ max(dS − K, 0) in state 2.

(b) Show that a portfolio consisting of θ1 units of asset 1 and θ2 units of asset 2, where

θ1 = (Rf )

−1 uCd − dCu u − d , θ2 = Cu − Cd

(u − d)S will generate the same dividend as the option.

(c) Show that the no-arbitrage price of the option is given by C = (Rf )

−1 qCu + (1 − q)Cd

, where q = (Rf − d)/(u − d).