RESEARCH on PRICING DERIVATIVES BASED ON RISK MEASURES

本研究从资产负债管理的角度，运用风险度量方法，研究了在市场不完全或存在交易成本、卖空约束、税收等摩擦的情况下，欧美衍生品的定价问题。引入了法不变的概念，刻画了法不变的一致性风险度量，研究了一种新的有效的衍生产品定价的数值计算方法，并用最小二乘法对该方法的有效性进行了严格的证明。我们还介绍了一种新的基于Malliavin微积分的欧式衍生品定价的数值计算方法，该方法应用于随机微分方程和自由李代数，并发展了该方法。具体内容如下。引入了m-相似马氏算子和m-L相似随机变量的新概念，给出了逼近方法的灵活性，研究了迭代随机积分的代数结构。利用数值计算中常用的龙格-库塔方法，构造了确定的m-相似马尔可夫算子和m-L-相似随机变量，并研究了实用算法。

This research focused on the pricing of American or European derivatives in the case where the market is not complete or where there exist frictions such as transaction cost, short sale constraint, tax etc., from view point of Asset Liability Management by using risk measures.First, we did research on coherent risk measures proposed by Artzner, Delbaen, Eber and Heath, which is quite practical for banks. We introduced a new notion, law invariance, and characterized law invariant coherent risk measures.Then we did research on a new effective numerical computation method for pricing derivatives.We gave a rigorous proof of the effectiveness of the method proposed by Longstaff-Schwartz such that the value function is approximated with polynomials by the least square method. We also discussed the bound of this method.We also introduced a new numerical computation method for pricing of European derivatives based on Malliavin calculus applied to stochastic differential equations and on free Lie algebra, and we developed this method. The details are the following. We introduced new notions, m-similar Markov operators and m-L similar random variables, and then we gave flexibility of the approximation methods and studied the algebraic structure of iterated stochastic integrals. Also, we used the Runge-Kutta method which is effective in numerical computation in ordinary differential equations, to construct definite m-similar Markov operators and m-L-similar random variables, and we did research on practical algorithm.