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S, denotes the price of a security at time z. The discounted security process e"" S,, where / denotes the continuously-compounded risk-free interest rate, is a martingale under the risk-neutral measure . (i) Express mathematically the fact that the discounted security process is a O-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 R, (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 Q-martingale. [4] I', is a process defined by V, =ed-DE[X |F,], where A is a function of ST , T is a fixed time, and F, denotes the filtration representing the history of the security price up wo time / . (Hii) Show that the discounted process e""', is also a 0-martingale. [3] (iv) Explain the significance of these results in derivative pricing. [3] [Total 11]