Broeders認為，在或有退休金負債和最佳投資策略之間，存在著資產與負債的不對稱風險，因此，Broeders採用Klein(1996)模型，並藉著逆向工程，推導出最佳風險波動度，作為分析退休基金最佳投資策略之依據。但是，Klein(1996)卻忽略了選擇權可能產生潛在負債的影響，以及沒有考慮到，選擇權發行商在選擇權到期之前可能發生違約的狀況，以及受利率風險和信用風險所造成的影響。因此，本文為了改進上述的缺點，而加入隨機負債、提早違約與隨機利率的概念到本研究所提出之模型中。因而，本文發現，當加入提早違約與隨機利率到模型中，將有最佳的風險承擔能力，也就是說，他運用最少的風險承擔，就能夠使得退休基金之剩餘價值，有最大化的市場價值。而且使得退休基金之剩餘價值，擁有最多的盈餘。然而，當加入隨機負債到模型中，他必須承擔更多的風險，才能使得退休基金之剩餘價值，有最大化的市場價值。同時，他也讓退休基金之剩餘價值有更大的資金缺口。總之，這些模型均對退休基金的發行商和受益人之間，產生一個殘餘風險的不對稱分配。因此，須藉著最佳化波動度的分析，才能使得退休基金之剩餘價值，有最大化的市場價值。

Broeders (2010) claimed that there exists a risk of mismatch of assets and liabilities between the value of contingent pension liabilities and optimal investment policy. Therefore, he adopted Klein (1996) model, and the optimal investment policy for the pension fund in his context can be found by reverse-engineering option valuation formulas. However, Klein (1996) neglects the potential influence of debt that options may cause. Moreover, Klein’s model does not take into account the possible default of the option issuer prior to the maturity of the option, and is subjected to interest rate risk and credit risk. Therefore, the conception of stochastic liabilities, early-default and stochastic interest rates are incorporated to the proposed model further improve the weaknesses mentioned above. This article found that when added early-default and stochastic interest rates to Model II, it will have with the best risk-taking ability, that is, it uses minimal risk-taking, it can make that the market value of the pension surplus is maximized. Moreover, it had more profit for the pension surplus. Nevertheless, when this article added stochastic liabilities to Model I, it must bear more risks, to let that the market value of the pension surplus is maximized. Additionally, it had a greater pension shortage. However, all these models will generate an asymmetric distribution of residual risk between the sponsor and the beneficiaries of pension funds. Therefore, the market value of the pension surplus is maximized by the analysis of the optimal volatility.

一、中文文獻
[1] 陳絳珠著「連續時間模型下退休基金最適策略之研究」．政治大學風險管理與保險學系碩士論文．89年。
[2] 何嘉綺著「隨機控制理論應用於退休金之研究」．政治大學應用數學系碩士論文．89年。
[3] 陳勝源、杜化宇、周麗娟、黃凱群著「我國上市公司退休金保險費率、提撥率與勞退基金運用之探討－選擇權定價公式之應用」．管理評論第20卷第2期．pp.113-133．90年4月。
[4] 陳達新、周恆志合著「財務風險管理－工具衡量與未來發展」．雙葉書廊．95年。
[5] 謝珮芳著「負債驅動投資與雙重免疫策略於退撫基金之應用」．中山大學財務管理學系碩士班碩士論文．95年7月。
[6] 蘇恩德、洪偉屏、洪端禧、李勝榮著「考量盈餘風險與提撥風險下之最適退休基金管理」．管理科學研究．Vol.3, No.1．pp.37-60．95年。
[7] 毛建智著「台灣勞工退休基金最適資產配置研究」．長庚大學企業管理研究所碩士論文．96年6月。
[8] 陳松男著「金融工程學三版」．新陸書局．97年。
[9] 耿靖著「養老金全面風險管理」．文匯出版社．98年。
[10] 李翎竹、李志宏著「從風險觀點探討確定提撥與確定給付計畫之制度轉換選擇權」．中央研究院人文社會科學研究中心．人文及社會科學集刊．第22卷第1期．pp.77-107．99年3月。

二、英文文獻
[1] Black, F. and J. Cox,(1976), “Valuing Corporate Securities: Some Effects of Bond Indenture Provisions”, Journal of Finance, 31, 351-367.
[2] Blicksler, J. L. and A.H. Chen（1985）,”The Integration of Insurance and Taxes in Corporate Pension Strategy”, Journal of Finance, (July),pp.943-957.
[3] Blitzer, D. M. and S. Dash（2004）”Using Equity Duration in Pension Fund Asset Allocation.” www.standardandpoors.com, January 2004.
[4] Broeders, D.W.G.A.,(2010) “Valuation of Contingent Pension Liabilities and Guarantees under Sponsor Default Risk”, The Journal of Risk and Insurance, Vol. 77, No. 4, 911-934.
[5] Brown, J. R., and P. R. Orszag,(2006),”The Political Economy of Government- Issued Longevity Bonds”, Journal of Risk and Insurance, 73(4): 611-631.
[6] Chang, S. C.(1999）“Optimal Pension Funding Through Dynamic Simulations：the Case of Taiwan Public Employees Retirement System.” Insurance：Mathematics and Economics, 24：189-199.
[7] Crosbie, J. P.,（1999）“Modeling Default Risk,” KMV: San Francisco, California, U.S.A.
[8] Cui, J., F. de Jong, and E. H. M. Ponds,（2006）”Intergenerational Risk Sharing Within Funded Pension Schemes”, Netspar Discussion Paper No. 2006-11.
[9] Duan, J.C.,（1994）“Maximum Likelihood Estimation using Price Data of The Derivative Contract”. Mathematical Finance, 4, 155–167.
[10] Duan, J.C.,（2000）“Correction: Maximum Likelihood Estimation using Price Data of The Derivative Contract”. Mathematical Finance, 10, 461–462.
[11] Haberman, S. and J. H. Sung（1994）“Dynamic Approaches to Pension Funding.”, Insurance：Mathematics and Economics, 15,151-162.
[12] Hull, J. and A. White (1995),”The impact of default risk on the prices of options and other derivative securities.” Journal of Banking Finance,1995,19, 299-322.
[13] Jarrow, R., and S. Turnbull,(1995), “Pricing Derivatives on Financial Securities Subject to Credit Risk,” Journal of Finance, 50, 53–86.
[14] Johnson, H. and R. Stulz, (1987), “The Pricing of Options with Default Risk”. Journal of Finance, 42, 267–280.
[15] Kenneth B. and H. D. Skipper,（2000）”Life & Health Insurance, 13th edition”, Prentice Hall.
[16] Klein, P. (1996), “Pricing Black–Scholes Option with Correlated Credit Risk”. Journal of Banking Finance, 20, 1111–1129.
[17] Klein, P. and M. Inglis,（1999）, “Valuation of European options subject to financial distress and interest rate risk.” Journal of Derivatives 6, 44-56.
[18] Klein, P. and M. Inglis,（2001）, “Pricing Vulnerable European Option when The Option’s Payoff Can Increase The Risk of Financial Distress.” Journal of Banking Finance, 25,993–1012.
[19] Li, D. X. (2000). “On Default Correlation: A Copula Function Approach.” Working Paper 99-07, The Riskmetrics Group.
[20] Liao, S.-L. and H.-H. Huang,(2005). “Pricing Black-Scholes Options with Correlated Interest Rate Risk and Credit Risk: An Extension.” Quantitative Finance, 5, 443–457.
[21] Marcus, A. J.（1985）,”Spinoff / Terminations and the Value of Pension Insurance”, Journal of Finance, (July), 911-926.
[22] Merton, R. (1974). “On The Pricing of Corporate Debt: The Risk Structure of Interest Rates.” Journal of Finance,29, 449–70.
[23] Reddington, F. (1952),”Review of the Principles of Life Office Valuations”, Journal of the Institute of Actuaries, Vol. 78, 286 - 340
[24] Ronn, E.I. and A.K., Verma,（1986） “Pricing Risk-adjusted Deposit Insurance: An Option-based Model”. Journal of Finance,41, 871–895.
[25] Sharpe,W.F.(1976),”Corporate Pension Funding Policy”, Journal of Financial Economics, 3(3), 183-193.
[26] Sharpe,W.F.,(1992), ”Asset Allocation: Management Style and Performance Measurement,” The Journal of Portfolio Management, Winter, 7-19.
[27] Sklar, A. (1959), “Fonctions de Répartition à n Dimensions et Leurs Marges”, Publications de l'Institut de Statistique de l'Universite de Paris, 8, 229-231
[28] Waring, B. M.（2004） “Liability-Relative Investment I.” Journal of Portfolio Management, Summer 2004, 8-20.
[29] Waring, B. M. （2004）“Liability-Relative Investment II.” Journal of Portfolio Management, Fall 2004, 40-53.
[30] Williams, J. O.,（1997）,”Maximizing the Probability of Achieving Investment Goals”, Journal of Portfolio Management, 24, 77-81.