A commercial bank wants to sell a new derivative security. If Sy is the stock price at expiration, this new derivative pays Sq, for some > 0. Suppose that stock returns follow a geometric Brownian motion with annual mean rate of return equal to 10% and annual volatility o = 10% a year. The risk-free rate of return is r = 5% per year, the contract expires in two years, and the initial stock price is 5. You have been hired as an analyst to price the power contract with y = 0.5. Proceed in steps: a. Explain, in words, how a derivative is priced in continuous-time finance using the risk-neutral method. b. Write the SDE for the stock price using B, the Brownian motion under the objective probability measure. C. Rewrite the SDE for the stock price using B2, the Brownian motion under the risk-neutral measure. State which theorem you are using here. Also explain in words how the change of measure works. d. Use Ito's lemma to find the stochastic differential equation for In St under the risk-neutral measure. Integrate the SDE for In S to find the risk-neutral distribution of In Sy as of time 0, where T is the time to expiration. e. Express the price of the derivative (at t = 0) as a risk-neutral expectation. Evaluate this expectation. [Hint: If x is a lognormal distribution, i.e., In X ~ N(m, s2), then E [ex] = cm+**).]