Please help me to get clear and reliable answers for the question below.
(1) S State the principle of correspondence as it relates to mortality investigations. [1] Two small countries conduct population censuses on an annual basis. Country A records its population on 1 February every year based on an age definition of age lost birthday. Country B records its population on every 1 August using a definition of age nearest birthday. Each country records deaths as they happen based on age next birthday. Below are some data from the last few years. Country A Age last Population Population Population birthday ! February 20]] ] February 2012 ] February 2013 44 382,000 394,000 401,000 45 374,000 381,000 385,000 46 354,000 372,000 375,000 Country B Age nearest Population Population Population birthday I August 201 1 I August 2012 I August 2013 44 382,000 394,000 401,000 45 374,000 381,000 385,000 46 354,000 372,000 375,000 In the combined lands of Countries A and B in the calendar year 2012 there were 4,800 deaths of those aged 46 next birthday and 4,500 deaths of those aged 45 next birthday. The two countries decide to form an economic union, after which it will be mandatory to offer the same rates for life insurance to residents of each country. (ii) Estimate the death rate at age 45 years last birthday for the two countries combined. [6] (iii) Explain the exact age to which your estimate relates. [1] [Total 8]One particular company over which an investment bank writes European call options has experienced a severe fall in its share price. However, analysts have not revised their expectation that the share price will grow to f4 in six months. The table below shows the share price together with the price of the options. Date Share price Option price 1 November f3.00 E0.90 2 November f2.00 10.60 You may assume that the basic Black-Scholes framework is used to price the options. (i) Explain why the option price has fallen even though the expected return has increased according to the analysts. [3] (ii) State any requirements for the option price to have fallen to its level on 2 November. 131 [Total 6] (i) Outline the approach adopted by Shiller to test for "excessive volatility" and state the criticisms of his work. [?] (ii) State one difficulty of testing the strong form of the efficient market hypothesis and state the general conclusion of studies carried out on it. [2] [Total 9] (i) State the martingale representation theorem, including conditions for its application. defining all terms used. [3] Let S, denote the price of an underlying security at time f; / denotes the risk free rate of return expressed in continuously compounded form, B, represents an accumulated "bank account" at time / that earns the risk free rate of return. Let X be any derivative payment contingent on Fy, payable at some fixed future time T, where Fy is the sigma algebra generated by S,, for O S & ST. You may assume that, under the equivalent measure @, the process D, =ell S, is a martingale and that dS, - BID di + dD,) (ii) Show that the value of this derivative at time f
