Solve the questions in the attachment.

S, denotes the price of a security at time . The discounted security process e "S, , where r denotes the continuously-compounded risk-free interest rate, is a martingale under the risk-neutral measure O. (i) Express mathematically the fact that the discounted security process is a Q-martingale. [1] B, denotes the accumulated value at time : of an initial investment of 1 unit of cash. (ii) (a) Write down an expression for B, . (b) Show that the discounted cash process is also a O-martingale. (C) Deduce that the discounted value of any self-financing portfolio (where transactions are made only by switching funds between the security and cash, with no injections or withdrawals of funds from the portfolio) will also be a O-martingale. [4] I' is a process defined by V, =e""(-DEo[X | F,], where X is a function of Sy, T is a fixed time, and F, denotes the filtration representing the history of the security price up to time t. (iii) Show that the discounted process e ", is also a O-martingale. [3] (iv) Explain the significance of these results in derivative pricing. [3] [Total 11](i) Explain what is meant by a "complete market". (ii) Give two reasons why in practice financial markets may not be complete. (iii) State the relevance of the concept of complete markets in derivative pricing. [Tote