4. (a) Define a complete market. The price process of a traded security satisfies the following stochastic dif- ferential equation (SDE) dSt = u Stdt + o SidZt where Zt is a Brownian motion under the real-world probability measure P. Let r > O be the continuously compounded risk-free rate of interest, with r = M. (b) Show that the discounted stock price e-rt St is not a martingale under the real-world probability measure P. (c) Demonstrate how the discounted asset price can be a martingale under an equivalent martingale measure Q. Let Vt be the value at t of a self-financing portfolio, consisting of $4 shares of stocks and Vt units of cash bonds. (d) Show that d(e-rtV:) = d(e-rt St) 4. (a) Define a complete market. The price process of a traded security satisfies the following stochastic dif- ferential equation (SDE) dSt = u Stdt + o SidZt where Zt is a Brownian motion under the real-world probability measure P. Let r > O be the continuously compounded risk-free rate of interest, with r = M. (b) Show that the discounted stock price e-rt St is not a martingale under the real-world probability measure P. (c) Demonstrate how the discounted asset price can be a martingale under an equivalent martingale measure Q. Let Vt be the value at t of a self-financing portfolio, consisting of $4 shares of stocks and Vt units of cash bonds. (d) Show that d(e-rtV:) = d(e-rt St)