Solve clearly.

(i) By considering a suitably chosen notional portfolio or portfolios (which you should specify carefully), show that the price p, of a European put option exercisable at time 7 with a strike price & on an underlying non-dividend- paying share with price S, at time / satisfies the inequality: P, 2 Ke "(-) -S, [4] (ii) Explain how you would modify your inequality if you knew that holders of the share on the day before the option expires are entitled to receive a cash dividend of 0.025, payable at time 7. [2] [Total 6]A 3-state time-homogeneous Markov jump process is determined by the following matrix of transition rates: 2 A= 0 0 0 The distribution at time 0 is ( ). Determine the distribution at time 1. An investigator wishes to construct a multiple decrement model of the mortality of a population, subdivided by the following causes of death. Cause 1: Cancer Cause 2: Heart disease Cause 3: All other causes You are given the following definitions: ty is the force of mortality due to cause i (/ = 1, 2, 3 ) at exact age x U4x is the probability that a life at exact age x dies due to cause i (/ = 1, 2, 3 ) before reaching exact age x+ (u20) UPx is the probability that a life aged exactly x is still alive at exact age x tu (u > 0) You may assume that "x _> u asu- 0. (i) Derive an expression for ,p, (t 2 0 ), in terms of the forces of mortality, using only the above functions in your derivation. (ii) Write down an integral expression for q, in terms of p, and the appropriate force(s) of mortality. (Note that 4x =14x .) (iii) Assuming that the force of mortality from each cause / is a constant /' between integer ages x and x+1, show that: qx = -xqx where qx =1-Px IM i= 1 ( Note that Px = 1Px .)