摘要(英) |
This paper utilizes the contingent claim approach to value option features in a retirement benefit. Extending from Michael Sherris (1997), the contributions of this paper are two-fold. First, we assume the valuation model contains three state variables: salary growth rate, crediting rate, and interest rate, where Sherris assumed the interest rate to be constant. Since retirement benefits involve a long-time membership, the assumption made by Sherris is impractical. Our model is still able to take some decrements such as retirement, resignation, death, and disability into consideration. Second, because business cycles are often observed in the real world, we consider that the crediting rate has some random jumps in the life of a pension fund, and the number of jumps follows a Poisson random variable.
A discrete lattice commonly used for valuing a financial option is unsatisfactory for multi-variances. A simulation is more efficient. In this paper we use the Longstaff and Schwartz’s simulation approach to calculate benefit values for a range of ages.
The results show that, after considering the stochastic character of interest rates, the costs of benefit valued by Sherris are overstated by about an average of 34 percent. When we examine the sensitivity of benefit values to parameters, such asρfr, ρsr, σr, and r0, they have little effect to the benefit costs except for r0. As for changing r0 from 0.1 to 0.5, the costs of benefit rise about 4%. Considering random jumps in crediting rate will thus not affect the cost of retirement benefits too much. |
參考文獻 |
Barraquand, J., and D. Martineau. “Numerical Valuation of High Dimensional Multivariate American Securities”, Journal of Financial and Quantitative Analysis, 30, 3 (1995), 383-405
Bell, I. F. and M. Sherris. “Greater of Benefits in Superannuation Funds” Quarterly Journal of Institute of Actuaries of Australia, (1991 June), 47-64
Bolye, P. P., “A Monte Carlo Approach” Journal of Financial Economics, 4 (1977), 323-338.
Bolye, P. P., M. Broadie, and P. Glasserman. “ Monte Carlo Method for Security Pricing.” Journal of Economic Dynamic and Control, 21, 8-9 (1977), 1627-1321.
Bolye, P.P. and E.S. Schwartz. “Equilibrium Prices of Guarantees Under Equity Linked Contracts.” Journal of Risk and Insurance, 43 (1976), 639-660.
Britt, S., “Greater of Benefits: Member Options in Defined Benefit Superannuation Plans.” Transactions of the Institute of Actuaries of Australia, (1991), 77-118.
Chan, K. C., A. K. Karolyi, F. Longstaff, and A. Sanders, “An Empirical Comparison of Alternative Models of the Short-Term Rates.” Journal of Finance, (1992), 1209-1227.
Chen, Z. L., “Guaranty Cost and Normal Contribution Costs for Pension Funds.” (2000).
Cox, J. C., J. E. Ingersoll, and S. A. Ross, “ An Intertemporal General Equilibrium Model of Asset Prices.” Econometrica, (1985).
Darlington, Richard B., “Regression and Linear Models” (1990), (New York: McGraw-Hill).
Dwight Grant, Gautam Vora, and David E. Weeks. “Simulation and the Early-Exercise Option Problem.” The Journal of Financial Engineering, 5 (1996).
Francis A. Longstaff, and Eduardo S. Schwartz “ Valuing American Option by Simulation: A simple Least-Squares Approach” Review of Financial Studies, 14 (2001), 113-148
Hull, J., “Options, Futures and Other Derivative Securities.” Fourth Edition.(2000), (Englewood Cliffs, N.J.: Prentice-Hall).
Hull, J. and A. White. “Valuing Derivative Securities Using the Explicit Finite Difference Method.” Journal of Financial and Quantitative Analysis, 25(1) (1990)
Johnson, H., “Options on the Maximum or the Minimum of Several Assets” Journal of Financial and Quantitative Analysis, 25 (1987), 277-283.
Sheldon M. Ross “Simulation.” Second Edition. (1997), 63-85. (San Diego: Academic Press)
Sherris, M. “The Valuation of Option Features in Retirement Benefits” Journal of Risk and Insurance, 62 (1995), 509-534.
Shimko, D. C., “The Equilibrium Valuation of Risky Discrete Cash Flows in Continuous Time.” Journal of Finance, 44 (1989), 1373-1383.
Shimko, D. C., “ The Valuation of Multiple Claim Insurance Contracts” Journal of Financial and Quantitative Analysis, 27(2) (1992a) 229-246.
Steven B. Raymar, and Michael J. Zwecher “Monte Carlo Estimation of American Call Options on the Maximum of Several Stocks” The Journal of Derivatives, (1997)
Stulz, R. M. “Options on the Minimum or the Maximum of Two Risky Assets”, Journal of Financial Economics, 10, (1982), 161-185.
Wilkie, A. D., “The Use of Option Pricing Theory for Valuing Benefits with Cap and Collar Guarantees”, Transactions of the 23rd International Congress of Actuaries, (1989), 227-286. |