(a) Consider a European gap put option with a strike price of K₁ and a payment trigger of K₂. By analyzing its payoff at expiry date T and given a fixed strike price K₁, determine the optimal level of payment trigger K₂. Hence, establish a suitable upper bound for the value of a European gap option at time t T . [5 marks]

(b) Consider a non-dividend-paying asset with a current price S₀ of $20, a risk-free force of interest of 3%, a continuously compounded real-world drift of 7% and a volatility of 15%. Consider a 2-year European cash-or-nothing option written on the asset with the following payoff:

30 if S1>23 and S2<1.25S1

15 if S1<23 and S2<1.25S1

0 otherwise

Derive and calculate the current value of the option using the risk-neutral sequential pricing approach.