I am reviewing an old Math test to get ready for my next test and I am having trouble with this problem. Can anyone please answer the question and break down the steps to solve it along the way?

(25 points) Consider a stock process that follows an exponential Brownian motion under the measure ( mathbb{P} ) [ S_{t}=S_{0} mathrm{e}^{sigma W_{t}+mu t} ] (a) (10 points) Using the Girsanov theorem, construct a risk-free measure ( mathbb{Q} ), using a ( mathbb{Q} )-Brownian motion ( ilde{W}_{t} ) such that ( S_{t} ) is a ( mathbb{Q} )-martingale. Write ( S_{t} ) in terms of ( ilde{W}_{t} ). (b) (15 points) Assume now that there are no interest rates and that ( sigma=20 % ). Consider a claim that pays ( $ 85 ) if, after three years, the stock price ( S_{T} ) is smaller than ( 7 S_{0} / 3 ), if ( S_{0} ) is the initial value of the stock. What is the price of such a claim?

( 25 points) Consider a stock process that follows an exponential Brownian motion under the measure P St=S0eWt+t (a) (10 points) Using the Girsanov theorem, construct a risk-free measure Q, using a Q-Brownian motion W~t such that St is a Q-martingale. Write St in terms of W~t. (b) (15 points) Assume now that there are no interest rates and that =20%. Consider a claim that pays $85 if, after three years, the stock price ST is smaller than 7S0/3, if S0 is the initial value of the stock. What is the price of such a claim