Financial Markets with Memory I: Dynamic Models
Financial Markets with Memory I: Dynamic Models
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DOI：
10.1081/sap-200050096
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发表时间：
2005-01
影响因子：
1.3
通讯作者：
Vo Anh;A. Inoue
中科院分区：
文献类型：
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作者：
Vo Anh;A. Inoue
摘要 这是两篇论文中的第一篇，其中我们考虑了股票的价格过程，该过程由随机微分方程定义，该过程由不同于布朗运动的过程 Y(⋅) 驱动。采用这种有色噪声输入的动机是对真实市场数据的分析。过程 Y(⋅) 由连续时间 AR(∞) 型方程定义，并且可以具有短记忆或长记忆。我们证明过程 Y(⋅) 具有良好的 MA(∞) 型表示。这种同时良好的 AR(∞) 和 MA(∞) 表示的存在使我们能够应用一种新的方法来计算相关的条件期望，从而获得投资组合优化等问题的各种显式结果。由上述股票价格过程定义的金融市场是完整的，如果系数恒定，则欧式看涨期权和看跌期权的价格由布莱克-斯科尔斯模型中的布莱克-斯科尔斯公式给出。然而，与后者不同的是，该模型允许历史波动率和隐含波动率之间存在差异。该模型包含一个特殊情况，与 Black-Scholes 模型相比，仅引入两个附加参数来描述市场记忆。基于真实市场数据的分析表明，这个带有两个附加参数的简单模型在捕捉市场记忆效应方面更加真实，同时保留了Black-Scholes模型的简单性和实用性。
ABSTRACT This is the first of two papers in which we consider a stock with price process defined by a stochastic differential equation driven by a process Y(⋅) different from Brownian motion. The adoption of such a colored noise input is motivated by an analysis of real market data. The process Y(⋅) is defined by a continuous-time AR(∞)-type equation and may have either short or long memory. We show that the process Y(⋅) has a good MA(∞)-type representation. The existence of such simultaneous good AR(∞) and MA(∞) representations enables us to apply a new method for the calculation of relevant conditional expectations, whence to obtain various explicit results for problems such as portfolio optimization. The financial market defined by the above stock price process is complete, and if the coefficients are constant, then the prices of European calls and puts are given by the Black-Scholes formulas as in the Black-Scholes model. Unlike the latter, however, the model allows for differences between the historical and implied volatilities. The model includes a special case in which only two additional parameters are introduced to describe the memory of the market, compared with the Black-Scholes model. Analysis based on real market data shows that this simple model with two additional parameters is more realistic in capturing the memory effect of the market, while retaining the simplicity and usefulness of the Black-Scholes model.