Ornstein-Uhlenbeck随机波动率下的期权定价：一个线性模型
Option pricing under Ornstein-Uhlenbeck stochastic volatility: a linear model
摘要：
本文研究了随机波动率模型下的期权定价问题，重点研究了指数Ornstein-Uhlenbeck过程和Stein-Stein过程的线性逼近。事实上，我们表明，他们承认相同的限制动态的波动率过程中的低波动的制度，在此情况下，我们得到的风险中性概率密度的特征函数的精确表达式。这个表达式允许我们利用刘易斯和利普顿推导的公式计算期权价格。我们详细分析了普通香草电话的情况下，可靠的隐含波动率表面是可用的液体工具。我们还计算了前四个累积量的解析表达式，这对于实现简单的两步校准过程至关重要。它已被测试对一组数据的选择权交易的米兰证券交易所。我们提出的数据分析揭示了一个很好的配合与市场隐含的表面，并证实了线性近似的准确性。
英文摘要：
We consider the problem of option pricing under stochastic volatility models, focusing on the linear approximation of the two processes known as exponential Ornstein-Uhlenbeck and Stein-Stein. Indeed, we show they admit the same limit dynamics in the regime of low fluctuations of the volatility process, under which we derive the exact expression of the characteristic function associated to the risk neutral probability density. This expression allows us to compute option prices exploiting a formula derived by Lewis and Lipton. We analyze in detail the case of Plain Vanilla calls, being liquid instruments for which reliable implied volatility surfaces are available. We also compute the analytical expressions of the first four cumulants, that are crucial to implement a simple two steps calibration procedure. It has been tested against a data set of options traded on the Milan Stock Exchange. The data analysis that we present reveals a good fit with the market implied surfaces and corroborates the accuracy of the linear approximation.
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