[论文解读] Arbitrage in Fractal Modulated Markets When the Volatility is Stochastic

Erhan Bayraktar, H. Vincent Poor|ArXiv.org|Jan 22, 2005

Stochastic processes and financial applications参考文献 27被引用 3

一句话总结

本文在随机波动率的分形调制Black-Scholes模型中构建了套利策略，其中基础资产价格过程由分数布朗运动（fBm）或时间改变的fBm驱动。通过利用此类过程的零二次变差特性，作者建立了一个支持连续、适应性积分器的随机积分框架，从而在无需估计Hurst参数H的情况下实现显式套利。

ABSTRACT

In this paper an arbitrage strategy is constructed for the modified Black-Scholes model driven by fractional Brownian motion or by a time changed fractional Brownian motion, when the volatility is stochastic. This latter property allows the heavy tailedness of the log returns of the stock prices to be also accounted for in addition to the long range dependence introduced by the fractional Brownian motion. Work has been done previously on this problem for the case with constant `volatility' and without a time change; here these results are extended to the case of stochastic volatility models when the modulator is fractional Brownian motion or a time change of it. (Volatility in fractional Black-Scholes models does not carry the same meaning as in the classic Black-Scholes framework, which is made clear in the text.) Since fractional Brownian motion is not a semi-martingale, the Black-Scholes differential equation is not well-defined sense for arbitrary predictable volatility processes. However, it is shown here that any almost surely continuous and adapted process having zero quadratic variation can act as an integrator over functions of the integrator and over the family of continuous adapted semi-martingales. Moreover it is shown that the integral also has zero quadratic variation, and therefore that the integral itself can be an integrator. This property of the integral is crucial in developing the arbitrage strategy. Since fractional Brownian motion and a time change of fractional Brownian motion have zero quadratic variation, these results are applicable to these cases in particular. The appropriateness of fractional Brownian motion as a means of modeling stock price returns is discussed as well.

研究动机与目标

• 将现有分数布朗运动模型中的套利结果扩展至具有随机波动率的情形。
• 开发一种与fBm和时间改变的fBm兼容的随机积分框架，这些过程并非半鞅。
• 证明连续、适应性过程若具有零二次变差，可作为积分函数和连续半鞅的积分器。
• 表明此类积分同样具有零二次变差，从而可在套利构建中递归用作积分器。
• 提供一种具有经济可解释性的套利策略，且无需估计Hurst参数H。

提出的方法

• 通过在划分不断缩小下Stieltjes和的依概率收敛来定义随机积分，而非使用Wick型积分。
• 证明任意几乎必然连续、适应性且具有零二次变差的过程，可作为连续适应半鞅及积分器函数的积分器。
• 建立结果积分同样具有零二次变差的性质，从而可在构建交易策略时实现递归使用。
• 将该框架应用于Hurst参数H ∈ (1/2, 1]的分数布朗运动（fBm）及时间改变的fBm。
• 基于随机波动率下股票价格的动态，构建显式套利策略。
• 证明该策略独立于Hurst参数H，因其仅通过价格过程进入，而不通过交易规则。

实验结果

研究问题

• RQ1 能否在由分数布朗运动驱动且具有随机波动率的修正Black-Scholes模型中构建套利？
• RQ2 fBm和时间改变的fBm的零二次变差特性是否能为非半鞅过程提供一致的随机积分框架？
• RQ3 此类积分器能否递归使用，以构建具有保证盈利的自融资交易策略？
• RQ4 所得到的套利策略对Hurst参数H的取值是否稳健，还是需要估计H？
• RQ5 与Wick型积分相比，所提出的Stieltjes型积分框架在经济可解释性和套利生成方面有何差异？

主要发现

• 在由分数布朗运动或其时间改变版本驱动、且波动率为随机的修正Black-Scholes模型中，存在显式套利策略。
• 通过Stieltjes和收敛定义的随机积分，对任意几乎必然连续、适应性且具有零二次变差的过程均具有良好的定义性。
• 此类积分保持零二次变差性质，从而可在构建交易策略时递归用作积分器。
• 套利策略无需估计Hurst参数H，因其依赖于股票价格过程，但不直接依赖于H。
• 该框架支持生成无风险利润的连续交易策略，证明了在具有随机波动率的分形调制市场中存在套利。
• 结果表明，尽管基于fBm的模型不具备半鞅性质，但在所选积分理论下，仍可支持具有经济可解释性的套利策略。