Numerical valuation of surrender options

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Embedded in Life insurance contracts are surrender options and also path dependency. Surrender option stems from many reasons. Multi morbidity increases the rate of mortality and a variety of adverse health outcomes which may lead to surrendering. In Ghana, poverty levels coupled with social burdens can inform a multi-morbid person to surrender a life policy contract. The study seeks to incorporate the multi-morbid survival rate of a policy holder in the Black-Scholes model for option pricing. The solution to this model come along with its own complexities. Therefore the need to resort to numerical solutions for the option valuation. Further, a comparison is made of two finite difference algorithms in solving the proposed Black-Scholes equation ;the Crank-Nicolson method and the Implicit method. In line with these objectives, simulations of survival times were performed to compute the survival rate and the stability, consistency and convergence of these algorithms were investigated. It was observed that the algorithms were stable, consistent and converges to the exact solution. However the Explicit method of the finite difference approximation is found to be conditionally stable. Numerical solution to the Black -Scholes model and the proposed model indicates that the Crank-Nicolson method converges faster than the Implicit method for the Black-Scholes while the Implicit method converges faster than the Crank- Nicolson method. Finally it is observed that the Implicit method converges faster as the multi-morbid survival rate decreases below the short rate of the Black-Scholes 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, 2015