Question 1 Let X be derivative on [0, T] which is a positive FT measurable random variable. Dene lX) be the value of X at time t, measured in discounted dollars. A portfolio has bounded risk if there exists N such that for all t, 214m.)- 3 N k . Here l/chXk)' is the negative part of V(Xk) and it represents the amount that the portfolio is short on Xk. This basically tells us there is a limit to what we can borrowed. There is no arbitrage with bounded risk if no selfnancing portfolio with bounded risk is an arbitrage opportunity. (a) Let X and Y be derivatives on [0, T] . If X g Y as, then show that l/(X) S VMY). Further show that if X = Y as, then VD(X) = VMY). (b) Let 1. a, 2 0, 2. 0 S t S T and 3. X and Y be derivatives on [0, T] (b) Let 1. a 20, 2. 0 X =0 a.e. (iii) Vo is linear : Vo(X + Y) = Vo(X) + Vo(Y) and Vo(ax) = aVo(X) (iv) If t 2 0, then Vo(X) = Vo( V.(X)). (v) If A > 0 is bounded and Ft measurable, then Vi(AX) = AVI(X)