1. (30 marks) Please answer the following questions as precisely as possible and use mathematical notation where necessa

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1 30 Marks Please Answer The Following Questions As Precisely As Possible And Use Mathematical Notation Where Necessa 1 (90.05 KiB) Viewed 77 times

1. (30 marks) Please answer the following questions as precisely as possible and use mathematical notation where necessary. (a) Write the definition of the Brownian motion (W.). What is the distribution of the random variable X defined by X = -3W+2W ? Justify carefully your answer. (4 marks) (b) Show that the Brownian motion (W) is a martingale (with respect to its own filtration (FM) (4 marks) (c) Write the definition of an Ito process. Can a typical trajectory of an Ito pro- cess be discontinuous? Give three different) examples of an Ito process. In each of the examples, provide the Ito representation in integral form and in differential form. In each of the examples, specify also the volatility process and the drift process. (3 marks) (d) Consider the process (Y) defined by Y, = W-6 SW ds, for all t > 0. Compute the differential dy, Which formula did you apply? Using your computation, deduce that (Y) is a martingale (with respect to the filtration generated by the Brownian motion (W.)). (5 marks) (e) Consider the processes (Y/) and (Z.) of the following form: Y; = 5W.W., for allt > 0 Z = W3, for all t > 0. Are the processes (Y) and (2) adapted to the filtration generated by the Brownian motion (W.)? Give a brief justification of your answers. Then, provide your own example of a process which is adapted to the filtra- tion generated by the Brownian motion, and your own example of a process which is not adapted to this filtration. (4 marks) (1) What is a zero-coupon bond with maturity T? Give the zero-coupon bond price P(1,7) in terms of the LIBOR rate and provide in detail a proof of this relation. Then, give the zero coupon bond price in terms of the continuously compounded rate Rt.T) and provide in detail a proof of this relation. (5 marks) (9) Give an example of an option on the LIBOR rate? Provide the expression of the pay-off and explain the notation used. Hint: You can use the lectures. (3 marks) (h) What is the smallest possible o-algebra (on the space of outcomes 2)? Are the constants measurable with respect to this o-algebra? Are non constant random variables measurable with respect to this o-algebra? (2 marks)