亚式期权三叉树定价模型分析

【作者】 王志刚；

【导师】 钱晓松；

【作者基本信息】 扬州大学 ， 应用数学， 2013， 硕士

【摘要】 本文研究当市场中原生资产的价格过程服从扩散模型时亚式期权三叉树方法及其相关性质。在期权定价理论中,二叉树方法是一种经典的数值方法和离散模型,而三叉树定价方法是对二叉树方法的扩展,其在计算精度和计算速度上都优于二叉树方法。亚式期权是一种强路径有关期权,它在期权到期日的收益不仅与当天原生资产的价格有关,而且还依赖于在整个期权有效期内原生资产的价格平均值。利用Jiang等人[1]研究扩散模型中亚式期权二叉树模型的方法,我们建立了亚式期权三叉树方法与一类显式差分格式的等价性,并在粘性解理论框架下证明了美式亚式期权三叉树方法的收敛性。亚式期权普通二叉树方法由于路径变量的引入而成为一个双状态变量模型,使得该方法运算量很大,通常无法运用到实际金融领域中,亚式期权的三叉树方法存在同样问题。Cheuk等人[2]利用变量化简的方法得到扩散模型中另一种强路径依赖期权回望期权的单状态二叉树模型,Dai[3]利用类似思想得到了扩散模型中几何平均亚式期权的单状态变量的二叉树模型,使得实际计算成为可行。本文将Dai[3]的结果推至三叉树模型中,给出了连续取样和离散取样情形下,固定敲定价格和浮动敲定价格的欧式、美式几何平均亚式期权的单状态变量的三叉树方法,并证明了它们与相应的显示差分格式的等价性。更多还原

【Abstract】 In this paper, we study the trinomial tree method for pricing Asian options and its related properties when the underlying asset in a financial market which follows the diffusion model. In the theory of pricing options, the binomial tree method is a classical numerical way and discrete model, while the trinomial tree method is extended to the binomial tree method, its on the computing precision and computing speed are better than the binomial tree method.Asian options are strong path-dependent related options whose payoffs depend not only on the underlying asset prices but also on the average of the underlying asset prices for the throughout the period of validity. In the paper, employing Jiang[1] and others research methods about the binomial tree methods for Asian options in diffusion models, we get the trinomial tree method for pricing Asian options and equivalence of one-state explicit difference scheme, and under the theoretical framework on the notion of viscosity solution,we prove the consistency of the trinomial tree for American-style Asian options. Asian options ordinary binomial tree method due to the introduction of the path variable as a double state variable model, making the computation is lager, it usually cannot be applied to the real financial field, while the trinomial tree method for Asian options have the same problem. Cheuk[2] and others by using the method of variational simplification get another strong path-dependent in the diffusion model of one-state variable binomial models for look back options, and Dai [3] using the similar idea got the one-state variable binomial tree models for geometric Asian options, which makes real calculation is feasible. In this paper,we push the Dai’s[3] results to the trinomial tree method models, which gives the one-state variable trinomial tree method for European/American-style fixed/floating strike geometric Asian option with the continuously sampled observations and discretely sampled observations, and proves their equivalence of the explicit difference scheme.更多还原