Homework 10. Consider the Black-Scholes model for a stock price (St), dSt = uS dt+oSdBt. So = 1, where (Bt) is a Brownian motion under P. In addition, let Bt = ent be the value of the savings account. Let ; = B+ + 4-20 t. (a) Explain why there exists a probability measure o equivalent to P such that (Bt) is a Brownian motion under O. (b) Show through calculation that (St/Bt) is not a martingale under . You may use that the solution to the Black-Scholes SDE is Se = e(302)t+oB Hint. The exponential martingale of Brownian motion may help you here! (c) The Black-Scholes model is complete and possesses the EMM Q which turns B4 = B+ + into a Brownian motion. By solely appealing to completeness of the model, explain why (St/Bt) could never have been a martingale under O. Tt o Homework 10. Consider the Black-Scholes model for a stock price (St), dSt = uS dt+oSdBt. So = 1, where (Bt) is a Brownian motion under P. In addition, let Bt = ent be the value of the savings account. Let ; = B+ + 4-20 t. (a) Explain why there exists a probability measure o equivalent to P such that (Bt) is a Brownian motion under O. (b) Show through calculation that (St/Bt) is not a martingale under . You may use that the solution to the Black-Scholes SDE is Se = e(302)t+oB Hint. The exponential martingale of Brownian motion may help you here! (c) The Black-Scholes model is complete and possesses the EMM Q which turns B4 = B+ + into a Brownian motion. By solely appealing to completeness of the model, explain why (St/Bt) could never have been a martingale under O. Tt o