金融保险中的若干模型与分析
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摘要
随着人类社会的不断进步与发展,金融保险产品逐渐成为人们日常生活的必须品.金融保险中的数学建模、定量分析尤显重要了.本文研究了金融保险中的若干随机模型,主要研究工作包括:
1.考虑到保险公司经营规模的不断扩大,建立了完全离散的两险种风险模型;给出了破产概率满足的瑕疵离散更新方程;讨论了该模型的调节系数,在调节系数存在的条件下得到了破产概率的估计;利用离散更新不等式估计了破产概率,这种方法适用于调节系数不存在的情形.
2.建立了两险种马氏风险模型,得到了破产概率满足的积分方程;利用推广后的更新技巧估计了破产概率收敛速度的界,在此基础上进一步将模型拓广到一般情形的两险种马氏风险模型并给出其破产概率的估计.
3.通过分析带有随机收益的风险模型,建立了带有马氏随机收益的马氏风险模型,给出其破产概率满足的积分方程,并讨论其在随机收益和索赔额满足指数分布、混合指数分布情形下的破产概率.
4.基于金融市场存在摩擦的现状,研究了借款利率大于存款利率且投资者拥有或借入某种股票需交纳比例费用的摩擦金融市场中的欧式未定权益的套期保值问题,得到了欧式未定权益的上、下套期保值价格,并证明了最优上、下套期保值策略的存在性,进而得到了欧式未定权益无套利价格区间.
5.研究了标的资产价格由几何Levy过程定义的几何平均亚式期权的定价问题,通过选择股票作为标准单位资产,使用鞅方法,得到了一个简单有效的定价方法,同时得到了价格过程所满足的一个积分微分方程.
The finance and insurance become more and more important with the society forward, therefore their mathematical models and quantitative analysis are especially important. In this dissertation, some stochastic models are studied and the following questions are studied mainly.
1. In view of the expand of the insurance company,the discrete risk model with two-type-insurance is established. The defective discrete renewal equation is given. The adjustment coefficient is studied and the ruin probabilities are estimated when the adjustment coefficient exists. The ruin probabilities are estimated by the discrete renewal inequality and this method is applicable to the situation which the adjustment coefficient doesn't exist.
2. The Markov risk model with two-type-insurance is constructed, the integral equation of the ruin probability is given;the ruin probabilities are estimated by the general renewal technique;the model is extended to a general case.
3. Prom the analysis of the risk model with random incomes, the Markov risk model with Markov random incomes is studied, the integral equation of the ruin probability is given and the probabilities which the incomes and the claims are submitted to exponential distribution and mixed exponential distribution respectively.
4. On the basis of the existence of frictions in the financial market, the problem of hedging European Contingent Claims in the market that has frictions in the form of a higher interest rate for borrowing than for lending and percentage management costs for holding or borrowing risk assets is studied. The upper-hedging price and the lower-hedging price of a European Contingent Claim are given, the existence of the optimal upper-hedging portfolio for hedging and the optimal lower-hedging portfolio for hedging are showed and the arbitrage-free interval is obtained.
5. The pricing problem of geometric average Asian option with fixed strike price the price process of underlying asset follows a geometric Levy model, the pricing formula of the payoff of geometric average Asian option is derived and the simpler integro-differential equation which the price process satisfies is showed.
1.考虑到保险公司经营规模的不断扩大,建立了完全离散的两险种风险模型;给出了破产概率满足的瑕疵离散更新方程;讨论了该模型的调节系数,在调节系数存在的条件下得到了破产概率的估计;利用离散更新不等式估计了破产概率,这种方法适用于调节系数不存在的情形.
2.建立了两险种马氏风险模型,得到了破产概率满足的积分方程;利用推广后的更新技巧估计了破产概率收敛速度的界,在此基础上进一步将模型拓广到一般情形的两险种马氏风险模型并给出其破产概率的估计.
3.通过分析带有随机收益的风险模型,建立了带有马氏随机收益的马氏风险模型,给出其破产概率满足的积分方程,并讨论其在随机收益和索赔额满足指数分布、混合指数分布情形下的破产概率.
4.基于金融市场存在摩擦的现状,研究了借款利率大于存款利率且投资者拥有或借入某种股票需交纳比例费用的摩擦金融市场中的欧式未定权益的套期保值问题,得到了欧式未定权益的上、下套期保值价格,并证明了最优上、下套期保值策略的存在性,进而得到了欧式未定权益无套利价格区间.
5.研究了标的资产价格由几何Levy过程定义的几何平均亚式期权的定价问题,通过选择股票作为标准单位资产,使用鞅方法,得到了一个简单有效的定价方法,同时得到了价格过程所满足的一个积分微分方程.
The finance and insurance become more and more important with the society forward, therefore their mathematical models and quantitative analysis are especially important. In this dissertation, some stochastic models are studied and the following questions are studied mainly.
1. In view of the expand of the insurance company,the discrete risk model with two-type-insurance is established. The defective discrete renewal equation is given. The adjustment coefficient is studied and the ruin probabilities are estimated when the adjustment coefficient exists. The ruin probabilities are estimated by the discrete renewal inequality and this method is applicable to the situation which the adjustment coefficient doesn't exist.
2. The Markov risk model with two-type-insurance is constructed, the integral equation of the ruin probability is given;the ruin probabilities are estimated by the general renewal technique;the model is extended to a general case.
3. Prom the analysis of the risk model with random incomes, the Markov risk model with Markov random incomes is studied, the integral equation of the ruin probability is given and the probabilities which the incomes and the claims are submitted to exponential distribution and mixed exponential distribution respectively.
4. On the basis of the existence of frictions in the financial market, the problem of hedging European Contingent Claims in the market that has frictions in the form of a higher interest rate for borrowing than for lending and percentage management costs for holding or borrowing risk assets is studied. The upper-hedging price and the lower-hedging price of a European Contingent Claim are given, the existence of the optimal upper-hedging portfolio for hedging and the optimal lower-hedging portfolio for hedging are showed and the arbitrage-free interval is obtained.
5. The pricing problem of geometric average Asian option with fixed strike price the price process of underlying asset follows a geometric Levy model, the pricing formula of the payoff of geometric average Asian option is derived and the simpler integro-differential equation which the price process satisfies is showed.
引文
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