3. Derivative Pricing (35 points). Consider the forex model: = dX = uXdt +oXdz dBd = raBadt dBf = rfBfdt. = rf is the X is the process of the exchange rate. rd is the domestic risk-less return, and foreign riskless return. The spot exchange rate X is quoted, by our definition units of domestic currency units of foreign currency For example, if X is quoted at 10 NOK/USD, it means 1 USD is worth 10 NOK. We assume all foreign currency holdings are invested in the foreign risk-free asset. Assume that a European style derivative has terminal payoff (XT). Denote the arbitrage-free price of this derivative f(t, x) when the exchange rate is Xt = x at date t. |(a) Derive the boundary value PDE (the Black-Scholes equation) which f(t, x) satisfies (30 points). (b) Derive the risk-neutral valuation formula. Point out which theorem gives you this formula from the Black-Scholes equation (5 points). 3. Derivative Pricing (35 points). Consider the forex model: = dX = uXdt +oXdz dBd = raBadt dBf = rfBfdt. = rf is the X is the process of the exchange rate. rd is the domestic risk-less return, and foreign riskless return. The spot exchange rate X is quoted, by our definition units of domestic currency units of foreign currency For example, if X is quoted at 10 NOK/USD, it means 1 USD is worth 10 NOK. We assume all foreign currency holdings are invested in the foreign risk-free asset. Assume that a European style derivative has terminal payoff (XT). Denote the arbitrage-free price of this derivative f(t, x) when the exchange rate is Xt = x at date t. |(a) Derive the boundary value PDE (the Black-Scholes equation) which f(t, x) satisfies (30 points). (b) Derive the risk-neutral valuation formula. Point out which theorem gives you this formula from the Black-Scholes equation (5 points)