c. State the Feyman-Kac Theorem, elearly stating the applications of the theorem in option pricing. [2 marks], Hence, let F(S,T)=KS, where T>0 is a fixed time, K>0 a fixed price. Solve the following boundary value problem of a financial derivative using the Feyman-Kac stochastic representation theorem. tF+rSSF+22S2S22FrF=0 d. Using portfolios comprised of different positions in different securities, show that it is not optimal to exercise an American call option. [4 marks] e. Define the following terms with respect to a portfolio consisting of rt units of the risky asset, St, and ft units of the risk-free asset, Bt : (i) Arbitrage. [2 Marks] (ii) A complete market. [2 Marks] . (iii) A self-finaneing trading strategy. [3 Marks] . (iv) Cameron-Martin-Girsanov theorem. [3 Marks] , Question Three a. An exotic option allows the holder, at time t0 to receive (with no further payment) either a vanilla call or a vanilla put, both of which have the same expiry T>t0 and strike price Q. (i) State one foundational assumption in the pricing dynamics of this exotic option. [1 Mark] (ii) Show that prior to to this exotic option is equivalent to a portfolio of European calls, specifying their strike prices and expiry times. [4 Marks] (iii) Hence, show that the value of this exotic option at time 0 is the sum of the value of a T-call with strike and the value of a T-put with strike er(Tt0)Q [5 Marks] b. Option trading strategies involve taking different positions in different types of options, and are commonly employed by investors to hedge against risks. Differentiate between spreads and combinations with respect to option trading strategies. [2 Marks] c. For each of the following strategies, construnt the expiry payoff table, draw the expiry payoff diagram and explain what change of the underlying price the holder prefers. (i) Long Butterfly spread [4 Marks] (ii) Short Straugle involviug dall optiouk (4) Marks)