State the general risk-neutral pricing formula for the price of a derivative at time t in terms of the derivative payoff TX at the maturity date T and the constant risk-free force of interest r .

Assume that the price of a share, which pays a constant force of dividend yield q , follows geometric Brownian motion. 

1. (a) Derive the formula for the price at time t of Derivative 1, which pays one at time T provided the share price at that time is less than K . 
2. (b) Derive the formula for the price at time t of Derivative 2, which pays the share price at time T provided the share price at that time is less than K . 
3. (c) Hence derive the formula for the price of a European put option with strike price K.