State and explain the different types of business cycles.

3 A life insurance company has written a tranche of with profits bonds that will mature in five years' time and is thinking about hedging the investment risks within the business. An actuarial student has performed a large number of stochastic runs and has produced a scatter-plot profile of the cost of guarantees and smoothing in five years' time against possible levels of the FTSE 100 index at that time. The insurance company wishes to flatten this profile (i.e. remove both the upside and the downside) using a simple hedging strategy. Cost of Guarantees & Smoothing. Em 15 10 -S -10 O 2000 4000 8000 10000 FTSE The FTSE 100 index currently stands at 4,000. (i) Suggest a hedge that meets the firm's requirements, using only one plain vanilla FTSE option and one FTSE forward deal. You should describe how the hedge would be set up and perform any necessary calculations. [6] (ii) Sketch a chart showing the cost of guarantees and smoothing net of the hedge payoff at the end of year five, assuming the hedge proposed in (i) is implemented from the outset. [2]1 Consider a tree of possible values for a stochastic process at times ? = 1 and 2, and two different probability measures P and Q that could apply to the same tree, as shown below: Tree of values with probabilities P Tree of values with probabilities Q 1 =0 t = 1 t = 2 1 =0 t = 1 t = 2 180 4- 180 120 100 100 80 80 80 80 60 60 30 30 (i) Show, with reasons, that: (a) P and Q are equivalent measures. (b) The process is a martingale under only one of the two measures. [4] (mi) (a ) Under measure P, calculate the one-period drift up and standard deviation on (i.e. from f = 0 to 1). (b) Calculate the values of the Radon-Nikodym derivative of Q with respect to P for both time periods. [5] (mii) (a) State the Cameron-Martin-Girsanov (CMG) theorem in continuous time. (b) For the first time period only, evaluate as far as you can the continuous time Radon-Nikodym derivative using a (previsible) drift-adjusting process y = H