Please Solve the questions in the attachment clearly.
1 (0) A risky asset has value A at time t. A probability measure R is defined so that Av is an yle R-martingale, where v is calculated using the risk-free interest rate. Explain why R can be described as a risk-neutral probability measure. [2] Let X, =VER[C|F ], where C is a discrete random variable occurring at time T>t. Prove that X, is an R-martingale. [2] Hint: if X is a discrete random variable and Y is a vector of random variables, then E [ E [ x Y ]] =E [ x ] . (iii) Stating any results that you use, deduce that & is previsible, where dX, = QdD, and D. = AV . [2] [Total 6] 2 (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. 3 S, denotes the price of a security at time t . The discounted security process e "S, , where r denotes the continuously compounded risk-free interest rate, is a martingale under the risk-neutral measure Q. (i) Express mathematically the fact that the discounted security process is a Q-martingale. By denotes the accumulated value at time t 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 Q-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. Vt is a process defined by V, = e-DE[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 and including time t . (iii) Show that the discounted process e "V, is also a Q-martingale. (iv) Explain the significance of these results in derivative pricing.4 The diagram shows a two-step non-recombining binomial tree. The numerical values shown are X, , the possible values of a particular derivative at times / =0,1,2, based on a probability measure P that attributes equal probability to the two branches at each step. $ denotes the filtration of the derivative value process at time / . 25 20 15 13 12 6 0 1=0 i = 1 i = 2 (a) If X1 =20, what are the realised values of Fo and f ? (b) What is the value of Ep(X2 | 6) in this case? (c) What is the value of Ep(X2 | 6) in the case where X1 =6? (d) Hence calculate Ep[Ep(X2 | 6)| Fo]- (e) Calculate Ep(X2 | Fo) and comment on your answer. (ii) The risk-neutral probability measure Q attributes probabilities of 0.4 and 0.6 to the up-paths and down-paths at each branch of this tree. (a) Does your conclusion in (i)(e) still apply when the probability measure Q is used in place of P ? (b) State briefly why this type of result is useful. (c) Calculate the value of the derivative at time 0, presenting your calculations in the form of a tree. Ignore interest
