在不完全市場下，衍生性金融商品可利用上套利和下套利價格來訂出價格區間。我們運用效用無差異定價於此篇論文中，此定價方式為尋找一個初始交易價，會使在起始時交易商品和無交易商品於商品到期日之最大期望效用相等。利用主要的對偶結果，我們證明在指數效用函數下，效用無差異定價區間會比上套利和下套利定價區間小。 / In incomplete markets, prices of a contingent claim can be obtained between the upper and lower hedging prices. In this thesis, we will use utility indifference pricing to nd an initial payment for which the maximal expected utility of trading the claim is indierent to the maximal expected utility of no trading. From the central duality result, we show that the gap between the seller's and the buyer's utility indierence prices is always smaller than the gap between the upper and lower hedging prices under the exponential utility function.

投資型商品連動於特定資產，保險人除了面臨原有的核保風險，更需承擔部分的財務風險。傳統保險商品的純保費價格等於其預期損失，而投資型商品的保險給付依據投資標的波動，保險人的預期損失不易估算，傳統精算的評價方法不完全適用於投資型商品。保證最低給付的給付結構使得投資型商品其有選擇權的特質，Brennan與Schwartz(1976)首先利用選擇權定價理論探討附有保證最低給付投資型商品之價值與避險策略，爾後亦有許多文獻以此方向加以著墨，但選擇權定價理論是基於市場為完全市場的假設，保險市場為不完全市場，以完全市場假設之理論評定保險商品之價值實不合理。本為假設保險人面臨的風險為核保風險及財務風險，財務市場為完全市場，保險人可以藉由市場上的各種金融商品建構避險組合規避財務風險；而預期死亡人數與實際死亡人數所產生的核保誤差，保險人無法利用避險組合完全地規避，因此保險市場為不完全市場。 在不完全市場中請求權的價值牽涉投資者主觀的風險偏好，不存在唯一的平賭測度，請求權的價格也不唯一，最適避險策略依請求權的價格調整，所以投資型保險商品的價格不再等於其公平價值，真正的成交價格應落於買賣價差之中。本文引用Mercurio(1996)的結果，利用二次效用函數，以極大化保險人期末財富之效用為目標，建構生存險的合理價格範圍。以二元樹模型描述股票的波動，分別模擬五年、十年及十五年投資型生存險之價差範圍，保險人的風險規避程度、保單期限以及保證金額的高低將影響商品價差範圍的大小。 關鍵字：不完全市場、效用函數，買賣價差、最適避險策略 / Investment-linked life (LIL) insurance policies integrate the attributes from the mutual fund by introducing the investment options to the policyholders and life insurance through the benefit payments shielding the unexpected events of the insured. Since the execution of the implied options depends on the policyholder's health status. Actuarial equivalent principal and non-arbitrage pricing theory are used in evaluating the prices for LIL insurance policies. Brennan and Schwartz (1976) initially employ the option pricing theory in examining the pricing and hedging strategy for LIL insurance policies with minimum guarantees. Most published literatures are focusing on this issue adopting the B-S methodology. Since the values of the LIL policies cannot be replicated uniquely through the self-financing strategies due to underwriting risks of the insurance market. Insurance market does not satisfy the completeness assumptions, Due to lack of a unique martingale measure under market incompleteness, the utility assumption of the policyholder is involved in the pricing issue. Insurance pricing must consider the risk attitude of the investors in the market. Hence the cost the LIL insurance policies are not necessarily equal to the fair market prices. The market value should fall within the range of the bid and ask prices. In this study, we follow the approach in Mercurio (1996) by adopting the quadratic utility function and compute the reasonable range of the prices based on maximizing the terminal health utility function. Binary tree method is used in modeling the asset dynamics. Then the numerical computations are performed using endowment LIL insurance policies with 5, 10 and 15 years of duration. Based on the results, we find that the risk attitude of the policyholder, the policy duration and minimum amounts of the guarantees significantly affect the bid-ask price spread of LIL insurance policies. Keywords: market incompleteness; utility function; bid-ask spread; optimal hedging strategy.

以往財務理論在討論財務市場行為時，均是在完全市場( e market)的假 設上做分析。但是從現實的角度來看，由於時間 (time)與不確定(uncertainty) 等因素，導致市場往往是不完全的(incomplete)；亦即資產或證券的數目往往小 於未來各種可能發生的不確定狀態的數目。當各種不同狀態事件發生時,如果沒有足夠的 資產或證券工具來涵括(span)所有投資人所欲達成的跨時所得移轉，則有某些風 險因子不能以既有的財務工具來複製，此時投資人將無法完全地規避該項風險。 本文的分析方法屬於微分拓樸數學工具的應用。從基本可微分流行(differentiable manifold) 性質，我們可得到整體性的觀點，優於以往在歐氏空間，Hilbert 空 間等拓樸空間只考慮在局部的性質.首先由最基本平滑流形(smooth manifo-[念 切入，介紹平滑流形與歐氏空間的關係。再由市場均衡的定義找出均衡流形( equilibrium manifold)。結合Debreu(1970)與Balasko(1976,1988) 建立的正 規經濟，探討其性質.加入財務市場時，透過正規經濟的理論，證明不完全財物 市場均衡generic 存在性。最後，利用不完全財務市場均衡generic的存在性，證明單 一商品及多商品的資本資產定價(CAPM)；並且建立一以消費為基礎的資本資 產定價模式。

傳統人壽保險契約之分析利用等價原則(principal of equivalience) 來對商品評價。即保險人所收保費之現值等於保險人未來責任(保險金額給付)之現值。然而對於權益連結壽險商品而言，其結合傳統商品之風險(如利率風險、死亡率風險等)與財務風險，故更增加其評價困難性。過去研究中在假設預定利率為常數與死亡率為給定的情況下，利用Black-Scholes (1973)評價公式推導出公式解。然而Black-Scholes評價公式是建構在完全市場上，對於權益連結壽險商品而言其已不符合完全市場之假設，因此本文放寬完全市場之假設來對此商品重新評價與避險。 在財務市場上，對於不完全市場(incomplete markets)下請求權(contingent claims)之評價與避險，已發展出數個不同評價方法。本文利用均數變異避險(mean-variance hedging)方法(Follmer&Sondermann ，1986)所衍生之風險極小化(risk-minimization)觀念來對此保險衍生性金融商品評價與避險，並找到一風險衡量測度(Moller , 1996、1998a、2000)來評估發行此商品保險人需承受多少風險。 / In this study, actuarial equivalent principle and no-arbitrage pricing theory are used in pricing and valuation for unit-linked life insurance policies.　Since their market values cannot be replicated through the self-finance strategies due to market incompleteness, the theoretical setup in Black and Scholes (1973) and Follmer and Sondermann (1986) are adopted to develop the pricing and hedging strategies. Counting process is employed to characterize the transition pattern of the policyholder and the linked assets are modeled through the geometric Brownian motions. Equivalent martingale measures are adapted to derive the pricing formulas. Since the benefit payments depend on the performance of the underlying portfolios and the health status of the policyholder, mean-variance minimization criterion is employed to evaluate the financial risk. Finally pricing and hedging issues are examined through the numerical illustrations. Monte Carlo method is implemented to approximate the market premiums according to the payoff structures of the policies. In this paper, we show that the risk-minimization criterion can be used to determine the hedging strategies and access the minimal intrinsic risks for the insurers.