<p>In this thesis the double exponential jump-diffusion model is considered and the Laplace transform is used as a method for pricing both plain vanilla and path-dependent options. The evolution of the underlying stock prices are assumed to follow a double exponential jump-diffusion model. To invert the Laplace transform, the Euler algorithm is used. The thesis includes the programme code for European options and the application to the real data. The results show how the Kou model performs on the NASDAQ OMX Stockholm Market in the case of the SEB stock.</p>

In this thesis the double exponential jump-diffusion model is considered and the Laplace transform is used as a method for pricing both plain vanilla and path-dependent options. The evolution of the underlying stock prices are assumed to follow a double exponential jump-diffusion model. To invert the Laplace transform, the Euler algorithm is used. The thesis includes the programme code for European options and the application to the real data. The results show how the Kou model performs on the NASDAQ OMX Stockholm Market in the case of the SEB stock.

Guaranteed Minimum Withdrawal Benefits(GMWB) have become popular riders on variable annuities. The pricing of a GMWB contract was originally formulated as a singular stochastic control problem which results in a Hamilton Jacobi Bellman (HJB) Variational Inequality (VI). A penalty method method can then be used to solve the HJB VI. We present a rigorous proof of convergence of the penalty method to the viscosity solution of the HJB VI assuming the underlying asset follows a Geometric Brownian Motion. A direct control method is an alternative formulation for the HJB VI. We also extend the HJB VI to the case of where the underlying asset follows a Poisson jump diffusion. The HJB VI is normally solved numerically by an implicit method, which gives rise to highly nonlinear discretized algebraic equations. The classic policy iteration approach works well for the Geometric Brownian Motion case. However it is not efficient in some circumstances such as when the underlying asset follows a Poisson jump diffusion process. We develop a combined fixed point policy iteration scheme which significantly increases the efficiency of solving the discretized equations. Sufficient conditions to ensure the convergence of the combined fixed point policy iteration scheme are derived both for the penalty method and direct control method. The GMWB formulated as a singular control problem has a special structure which results in a block matrix fixed point policy iteration converging about one order of magnitude faster than a full matrix fixed point policy iteration. Sufficient conditions for convergence of the block matrix fixed point policy iteration are derived. Estimates for bounds on the penalty parameter (penalty method) and scaling parameter (direct control method) are obtained so that convergence of the iteration can be expected in the presence of round-off error.

A robust computational framework is presented for the risk-neutral valuation of capital guarantees written on discretely-reallocated portfolios following the Constant Proportion Portfolio Insurance (CPPI) strategy. Aiming to address the (arguably more realistic) cases where analytical results are unavailable, this framework accommodates risky-asset jumps, volatility surfaces, borrowing restrictions, nonuniform reallocation schedules and autonomous CPPI floor trajectories. The two-asset state space representation developed herein facilitates visualising the CPPI strategy, which in turn provides insight into grid design and interpolation. It is demonstrated that given a deterministic process for the risk-free rate, the pricing problem can be cast as solving cascading systems of 1D partial integro-differential equations (PIDEs). This formulation’s stability and monotonicity are studied. In addition to making more sense financially, the limited borrowing variant of the CPPI strategy is found to be better suited than the classical (unlimited borrowing) counterpart for bounded-domain calculations. Consequently, it is demonstrated how the unlimited borrowing problem can be approximated by imposing an artificial borrowing limit. For implementation validation, analytical solutions to special cases are derived. Numerical tests are presented to demonstrate the versatility of this framework.

In life insurance contracts, actuaries generally value premiums using deterministic mortality rates and interest rates. They have ignored them stochastically in most of the studies. However it is known that neither interest rates nor mortality rates are constant. It is also known that companies may encounter insolvency problems such as ruin, so the ruin probability need to be added to the valuation of the life insurance contracts process. Insurance companies should model their surplus processes to price some types of life insurance contracts and to see risk position. In this study, mortality rates and surplus processes are modeled and financial strength of companies are utilized when pricing life insurance contracts.

近年來，能源商品的價格隨著國際政治情勢、國際金融環境以及景氣循環的影響產生劇烈波動，基於避險的需求，衍生性商品交易量也逐漸增加。然而，在評價能源衍生性商品的過程中，即期價格動態模型的選擇對於訂價與避險的結果有著顯著的影響，如何選擇一個適當的動態模型以評價能源商品便成為本文研究的目標。在指數與股價選擇權的評價模型中，大多以Black and Scholes (1973)提出的選擇權評價模型作為基礎，但Black-Scholes模型是否適用於評價能源市場的選擇權價格卻是有待商榷。Schwartz (1997)提出以均數回歸模型 (Mean Reversion Model)描述能源即期價格，發現比Black-Scholes模型中所假設的即期價格動態模型更能描述能源市場即期價格的波動。本研究也考慮能源市場遇到重大事件而造成即期價格產生劇烈波動的情況，因此在模型中加入跳躍項以捕捉價格跳躍的現象。另外，能源商品的需求與季節變化有高度相關性，因此本文亦考量即期價格的變動會受到季節性的變動影響，在模型中加入季節性函數，以補捉季節性的價格變化。基於前述模型考量，本研究在各種描述能源商品即期價格特性的動態模型之下，推導各個模型的期貨選擇權定價公式，進一步測試各模型在金融風暴與非金融風暴期間的訂價誤差與避險誤差，以提供投資人或避險需求者於原油期貨選擇權模型選用上之參考。 / In recent years, the price of energy commodities has fluctuated with the international political situation and the international financial environment. For the sake of hedging demands, the trading volume of derivatives has been gradually increasing. In the process of valuation of energy derivatives, choices of the spot price dynamics model have a significant impact on pricing and hedging. Therefore, how to choose an appropriate dynamic model to evaluate the energy commodities has been main purpose of this study. Two main models are tested in this paper. One is the option pricing model supposed by Black and Scholes (1973), and another is the mean reversion model supposed by Schwartz (1997). This study also considered the volatility of the spot price in the energy market in case of major events, so the researcher adds the jump to explore the mean reversion model. In addition, the demand for energy commodities is highly correlated with seasonal variations. The vibration of spot price often affected by the seasonal variations is considered in the research. Therefore, the researchers also take the seasonal function into the research to capture the seasonal price changes. Based on considerations described above, the pricing formula for each model of futures option is evaluated in the research. The researcher further tests the pricing errors and hedging errors of each model during the financial crises and non-financial crises in order to provide the investors and hedging demanders with some suggestions about selecting oil futures option models.

Guaranteed Minimum Withdrawal Benefits(GMWB) have become popular riders on variable annuities. The pricing of a GMWB contract was originally formulated as a singular stochastic control problem which results in a Hamilton Jacobi Bellman (HJB) Variational Inequality (VI). A penalty method method can then be used to solve the HJB VI. We present a rigorous proof of convergence of the penalty method to the viscosity solution of the HJB VI assuming the underlying asset follows a Geometric Brownian Motion. A direct control method is an alternative formulation for the HJB VI. We also extend the HJB VI to the case of where the underlying asset follows a Poisson jump diffusion. The HJB VI is normally solved numerically by an implicit method, which gives rise to highly nonlinear discretized algebraic equations. The classic policy iteration approach works well for the Geometric Brownian Motion case. However it is not efficient in some circumstances such as when the underlying asset follows a Poisson jump diffusion process. We develop a combined fixed point policy iteration scheme which significantly increases the efficiency of solving the discretized equations. Sufficient conditions to ensure the convergence of the combined fixed point policy iteration scheme are derived both for the penalty method and direct control method. The GMWB formulated as a singular control problem has a special structure which results in a block matrix fixed point policy iteration converging about one order of magnitude faster than a full matrix fixed point policy iteration. Sufficient conditions for convergence of the block matrix fixed point policy iteration are derived. Estimates for bounds on the penalty parameter (penalty method) and scaling parameter (direct control method) are obtained so that convergence of the iteration can be expected in the presence of round-off error.

A robust computational framework is presented for the risk-neutral valuation of capital guarantees written on discretely-reallocated portfolios following the Constant Proportion Portfolio Insurance (CPPI) strategy. Aiming to address the (arguably more realistic) cases where analytical results are unavailable, this framework accommodates risky-asset jumps, volatility surfaces, borrowing restrictions, nonuniform reallocation schedules and autonomous CPPI floor trajectories. The two-asset state space representation developed herein facilitates visualising the CPPI strategy, which in turn provides insight into grid design and interpolation. It is demonstrated that given a deterministic process for the risk-free rate, the pricing problem can be cast as solving cascading systems of 1D partial integro-differential equations (PIDEs). This formulation’s stability and monotonicity are studied. In addition to making more sense financially, the limited borrowing variant of the CPPI strategy is found to be better suited than the classical (unlimited borrowing) counterpart for bounded-domain calculations. Consequently, it is demonstrated how the unlimited borrowing problem can be approximated by imposing an artificial borrowing limit. For implementation validation, analytical solutions to special cases are derived. Numerical tests are presented to demonstrate the versatility of this framework.

受2008年金融海嘯影響，人壽保險業因資本及信用市場之系統性風險而導致帳列資產價值大幅減損，進一步影響壽險公司清償能力，而主管機關為兼顧審慎監理與市場穩定原則，而採行資本監理寬容措施，卻使得資本不足之保險公司缺口擴大。另外，保險安定基金以保費為基礎徵收單一費率，加劇保險公司間交叉補貼之情形。因此，如何透過以責任準備金為基礎，計算公平合理之風險差別費率，以避免產生影響其他保險公司正常經營之系統性風險，抑或引發保險公司道德風險，為本文研究之主要議題。 本文與過去文獻主要之差異為：(1) 資產模型依資產配置方式，使用蒙地卡羅模擬詳盡現金流路徑，著重於描述壽險業之情境；(2) 股票型風險性資產加入跳躍過程 (Jump) 與隨機波動兩種情境，以表達壽險業資產端承受資本市場變動加劇之風險；(3) 考慮政府監理寬容措施，以描述主管機關對於壽險業監理態度。 依蒙地卡羅模擬法試算保險安定基金公平費率，研究結果發現：(1)監理寬容期限增加時，安定基金公平費率增加；(2)監理標準提高，安定基金公平費率有先降後升之效果；(3)保險公司財務槓桿比例增加時，安定基金公平費率上升。 / Due to the global financial crisis in 2008 that resulted in systematic risks in the equity and credit market, it creates significant deprecation in the life insurers’ balance sheet which affect insurers’ solvency. In order to retain prudent supervision and market stability, the authority has announced capital temporal relief plan that may make insolvency insurer worse. Recent occurrences of financial distress to some insurers have raised questions about whether the current guaranty system that charge a flat levy rate in premium-based is adequate to protect policyholders. A risk-weighted levy rate in reserve-based has been proposed to establish reasonable contribution method which can avoid high risk insurers’ moral hazard and protect the other insurers from further systematic risks. A brief summary of the advantages of this paper is listed below：(1) By Monte Carol simulation method, detailed cash flow of insurer’s asset allocation can be used to describe the risk preference of life insurer. (2) Our stock model incorporates jump diffusion and stochastic volatility in order to reflect that life insurers face increasing volatility in capital market. (3) Consider regulatory forbearance to represent government’s attitude to life insurers. We calculate fair guaranty premium through Monte Carol simulation method. We find that： (1) Fair premium increases as extending the period of regulatory forbearance. (2) As regulatory criterion raises fair premium decreases at first, but increases if regulatory criterion reaches certain level. (3) Increasing leverage ratio of the insurer results in increasing fair premium.

Zugl.: Vallendar, WHU, Otto Beisheim School of Management, Diss., 2008.