Valuation of surrender option using Crank-Nicolson and Hopscotch methods

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Most of the insurance contracts in Ghana contains the right to early termination and are also path-depend, due to the presence of path-dependence derivatives and the right to early termination of the contract, can make valuation of Life insurance contract in Ghana come with complexities. These complexities are aggravated with introduction of the new parameter (S). Termination of life insurance contract in Ghana among other factors may come as a result of many factors that policyholders face. This study seeks to modify the Black-Scholes partial differential equation by incorporating risk of being multimorbid, and investigate the suitability of using some existing numerical methods (Crank- Nicolson and Hopscotch) to value life insurance contract. Further comparison between the two methods were done to select an efficient method for the modified model. In line with these objectives, simulations for time of an individual to be multimorbid were performed and the survival for risk of multimorbidity computed. This study revealed that, the modified model is stable, consistent and hence suitable to solve. In the numerical analysis of the option valuation using the original Black-Scholes model, Crank-Nicolson method converges faster than Hopscotch method. On the other hand, numerical analysis of the option valuation using the Black-Scholes model with the incorporated multi-morbid survival rate, Hopscotch method converges faster than Crank-Nicolson method. Further, it is observed that, the Hopscotch method converges much faster and give higher values as the step sizes are increased for Black-Scholes partial differential equation of the life insurance contract in Ghana embedded with surrender option. Hence, making the Hopscotch method favour policyholders who might want to surrender in order to receive the surrender value (payoffs).
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 Acturial Science.