Time dependent variable in the Black-scholes: an application in life insurance contracts

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Valuations of actuarial liabilities (benefits of insurance contracts) have been of major concern to managers and policy holders in the financial sector. These valuations are done through the standard BS-equation which assumes a constant rate of return. However literature has it that rate of return is time dependent and valuation under constant rate of return may impair the conclusions made on life insurance contracts. This study seeks to solve the modified Black-Scholes partial differential equation with the incorporation of time dependent rate of return in the valuation of life insurance contracts. Further, solutions to both the standard and modified model under two iterative techniques, Hopscotch and Crank-Nicolson, were investigated and compared. In line with these objectives, data was simulated and estimates for speed and level were computed for using Vasicek model. These estimates were now adopted in modeling the interest rate as a Cox-Ingersoll-Ross process. Solutions to both the standard and modified BS indicates that the Hopscotch converges faster than the Crank-Nicolson but the Crank-Nicolson gives consistent values than the Hopscotch. Further it was observed that value of life insurance contract from the modified model are much lower than that of the existing model.
A thesis submitted to the Department of Mathematics, Kwame Nkrumah University of Science and Technology In partial fulfillment of the requirement for the Degree of Master of Philosophy in Actuarial Science,