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Generalized option pricing models : numerical solutions and the pricing of a new life insurance contract Schwartz, Eduardo Saul

Abstract

The option pricing model developed by Black and Scholes and extended by Merton gives rise to partial differential equations governing the value of an option. When the underlying stock pays no dividends - and in some very restrictive cases when it does -, a closed form solution to the partial differential equation subject to the appropriate boundary condition, has been obtained. But, in more relevant cases such as the one in which the stock pays discrete dividends, no closed form solution has been found. This study shows how to solve those equations by numerical methods. The equations in question are linear parabolic partial differential equations. Although solution procedures are well known in Science and Mathematics, they have not been treated extensively in the finance literature. The numerical procedure used consists in approximating partial derivatives by finite differences to obtain expressions which can be handled by the computer. A general numerical solution to the partial differential equation governing the value of an option on a stock which pays discrete dividends, is developed in this dissertation. In addition, the optimal strategies for exercising American options are derived. For a sufficiently large value of the stock clearly larger than the exercise price, it may pay to exercise the American option at dividend payment dates. This study shows how to determine the "critical stock price" above which it will pay to exercise the option. Traditionally life insurance contracts have provided for benefits fixed in terms of the local unit of account. The prolonged period of inflation during the last three decades has served to diminish the usefulness of such usually long-term currency denominated contracts. As a result, life insurance companies have increasingly moved toward the study and issuance of equity based products the benefits of which depend upon the performance of some reference portfolio of common stock. This dissertation deals with the pricing of equity-linked life insurance contracts with asset value guarantee. In these contracts the benefits depend upon the performance of a portfolio of common stock. The insurance company, however, guarantees a minimum level of benefits (payable on death or maturity) regardless of the investment performance of the reference portfolio. In an equity based life insurance policy with this type of provision the insurance company assumes part of the investment risk as well as the mortality risk. When the contract becomes a claim (either because of death of maturity), the policyholder is entitled to receive an amount equal either to the value of the reference portfolio at that time or to the guaranteed sum, whichever is the greater. This study shows how the benefits of the contract can be viewed in terms of options and it obtains the partial differential equations governing the value of these options, which can also be solved by the above mentioned numerical methods. It is shown that the premium payments can be interpreted as negative dividend payments on the reference portfolio. Another important problem in relation to these policies is the determination of the appropriate investment policies for insurance companies to enable them to hedge against the major risks associated with the provision of the guarantee. It is shown how the equilibrium option pricing model determines the optimal investment strategy to be followed by insurance companies. The need for an appropriate model for pricing equity-linked life insurance policies with asset value guarantee, has long been apparent in the actuarial literature. The model developed in this study gives normative rules for pricing such contracts. Moreover, the prices determined by these rules are equilibrium prices, that is, they would prevail in a perfectly competitive market and would have the property that if the insurance company were to charge them and were to follow the investment policy determined by the model, it will bear no risk and will make neither profit nor loss. It is the hope of the author of this study that the model will find practical applications in the life insurance industry.

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