3. Appen Limited (APX) is an Australian company that operates inthe machine learning and artificial intelligence space;specifically, they provide data used to train models. The shareprice (at the time of writing) is $28.85. You decide to construct aportfolio that is short in a European call for APX stock withstrike price $35, and long in a European call (for APX stock) withstrike price $30. Both calls have the same maturity date, and arefor the same number of shares.

(a) Evaluate your total payoff per share if the value of APX atmaturity is: (a) $40; (b) $32; and (c) $28.

(b) Consider, more generally, any portfolio which is short in aEuropean call with strike K1 and long in a European call withstrike K2, where these two calls have the same maturity andunderlying asset, but K1 > K2. Write an equation for the payoffof this portfolio at maturity, in terms of S(T), K1, and K2.

(c) Plot the payoff at maturity in terms of the asset price,S(T). Label the key points on your figure.

(d) The above portfolio is called a bull spread. Explain when,and why, a bull spread is superior to just holding a European callat strike K2.

(e) A bear spread is a portfolio consisting of a short put withstrike K1 and a long put with strike K2 with the same maturitiesand underlying asset but K1 < K2. Write an equation for itspayoff at maturity (depending on the asset price S(T)), and plotthis payoff, labelling all key points.