基于连续时间随机游走模型的期权定价

Based upon the continuous time random walk which is usually used to describe the anomalous diffusion, this project will establish a class of sophisticated models for the asset price and consider a series of problems related to the option pricing under these models. Comparing to the option pricing models listed in the literature, the newly established models are expected not only to capture tail characteristics of asset returns, but also the influence of waiting times between trades. Under these models, the option price will be governed by fractional partial differential equations (FPDE), which are more difficult to analyze and solve than the corresponding integer order partial differential equations due to the “globalness” of the fractional operators. However, we will have managed to analyze the properties and propose efficient methods for solving option prices under these models. By reducing the pricing bias, this project is aiming to reduce arbitrage opportunities in trading and stabilize the financial market. It is expected that the successful implementation of this project will not only improve the transparency of the market, but also help the regulatory authorities to detect market abuse and carry out effective regulatory policies. From a theoretical point of view, since the application of fractional calculus in the option pricing filed is rather in an initial stage, the implementation of this project will significantly promote the development of financial engineering and applied mathematics.

本项目将基于反常扩散运动下最常见的连续时间随机游走，建立更贴近实际金融市场的刻画原生资产走势的精细模型，并在此类模型下考虑与期权定价相关的一系列问题。与传统的模型不同，我们拟建立的模型不仅能够抓住资产收益的尾部特征，也能将两次交易中的等待时间的影响考虑在内。在此类模型下，期权的价格由分数阶偏微分方程（FPDE）控制。与整数阶导数不同，分数阶导数具有全局性，这给求解和分析FPDE带来了很大的困难。在本项目中，我们将克服此类困难，从理论上定量地研究期权价格的性质并且设计高效的期权定价算法。该项目将致力于减少定价偏差，从而达到减少套利机会和稳定金融市场的目的。该项目的成功实施不仅会提高市场的透明度，也会有助于监管部门检测市场滥用情况和实施有效的监管政策。从理论角度来说，由于分数阶导数在期权定价领域的应用尚处于初始阶段，该项目的实施会显著推进金融工程以及应用数学学科的发展。

本项目基于基于反常扩散运动下最常见的连续时间随机游走模型，从理论上定量地研究了此类模型下与期权定价相关的一系列问题，并将所得理论与算法推广到其他衍生品定价领域。该项目在计划时间内完成了所有指标，主要获得两部分重要成果。第一部分成果是在分数阶导数模型下期权定价问题方面有了实质性进展：得到了美式看跌期权价格在FMLS模型下的一些性质，研究了连续时间随机游走模型与分数阶布朗运动的异同点,推导出了KoBoL模型下欧式期权价格的解析表达式。第二部分则是成功地将第一部分的结果推广和运用到了信用衍生品领域：在快速均值回归随机波动率模型、多尺度随机波动率模型、在传统模型和体制转化机制相结合等模型下解决了CDS(信用违约互换)的近似定价问题。该项目的成功实施，在一定程度上减少了由模型层面和方法层面带来的定价偏差，对我国衍生产品的发展、防范金融风险和稳定金融市场有着积极的意义。它不仅提高了市场的透明度，也有助于监管部门检测市场滥用和实施有效的监管政策。从理论角度来说，由于分数阶导数模型在衍生品定价领域的应用尚处于初始阶段，该项目的实施已引起了相关方面研究人员的兴趣和跟进，这对于金融工程以及应用数学领域的发展具有一定的推动作用。

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