Abstract: Under the assumptions that asset price is a geometric Brownian process and the price volatility is constant, Black-Scholes model is an ideal pricing model for European options. Despite the success and popularity of BS model, studies in empirical finance reveal that the implied volatility obtained from financial market data is not a constant but shows the implied volatility “smile” phenomena, thus the assumption of constant volatility is unrealistic. Based on fast mean-reverting stochastic volatility model, this paper investigates the pricing problem of collar option. The pricing formula of the collar option with stochastic volatility is also derived. Based on the implied volatility and historic data obtained from the SPDR S&P 500 ETF option market, two parameters of the fast mean-reverting stochastic volatility model are calibrated. With these calibrated parameters and the pricing formula, the numerical experiment on option price is carried out. The experiment result shows that collar option with stochastic volatility is underpriced compared to option with constant volatility.

Key words: stochastic volatility, mean-reverting, collar option, implied volatility

摘要： 基于快速均值回归随机波动率模型, 研究双限期权的定价问题, 同时推导了考虑均值回归随机波动率的双限期权的定价公式。 根据金融市场中SPDR S&P 500 ETF期权的隐含波动率数据和标的资产的历史收益数据, 对快速均值回归随机波动率模型中的两个重要参数进行估计。 利用估计得到的参数以及定价公式, 对双限期权价格做了数值模拟。 数值模拟结果发现, 考虑了随机波动率之后双限期权的价格在标的资产价格偏高的时候会小于基于常数波动率模型的期权价格。

关键词: 随机波动率, 均值回归, 双限期权, 隐含波动率