Probabilistic closed-form formulas for pricing nonlinear payoff variance and volatility derivatives under Schwartz model with time-varying log-return volatility
时变对数收益波动率的 Schwartz 模型下非线性方差与波动率衍生品定价AI 解读 ↓
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时变对数收益波动率的 Schwartz 模型下非线性方差与波动率衍生品定价
- 研究问题
- 在离散观测且对数收益波动率随时间变化的 Schwartz 单因子模型下，如何解析定价具有非线性收益的方差与波动率衍生品？
- 方法与思路
- 作者利用实现方差的加权非中心卡方分布表示计算相关期望，推导互换与看涨期权公式，并研究常波动率情形和近似公式。
- 材料陈述的贡献
- 摘要称给出方差及波动率互换、看涨期权的解析价格，分析常波动率情形的 vega，并以蒙特卡洛及数值实验考察准确性和效率。
- 阅读关注点
- 应核对定价测度、时变波动率设定和离散采样规则，以及闭式公式的计算要求、近似误差与敏感度推导的适用条件。
- 阅读边界
- 这是 arXiv 预印本，同行评审及期刊接收未核实。摘要未给出定价测度、公式或误差数值，模型内模拟表现不能证明市场定价准确性。
原文摘要
This paper presents closed-form analytical formulas for pricing volatility and variance derivatives with nonlinear payoffs under discrete-time observations. The analysis is based on a probabilistic approach assuming that the underlying asset price follows the Schwartz one-factor model, where the volatility of log-returns is time-varying. A difficult challenge in this pricing problem is to solve an analytical formula under the assumption of time-varying log-return volatility, resulting in the realized variance being distributed according to a linear combination of independent noncentral chi-square random variables with weighted parameters. By utilizing the probability density function, we analytically compute the expectation of the square root of the realized variance and derive pricing formulas for volatility swaps. Additionally, we derive analytical pricing formulas for volatility call options. For the payoff function without the square root, we also derive corresponding formulas for variance swaps and variance call options. Additionally, we study the case of constant log-return volatility; simplified pricing formulas are derived and sensitivity with respect to volatility (vega) is analytically studied. Furthermore,we propose simple closed-form approximations for pricing volatility swaps under the Schwartz one-factor model. The accuracy and efficiency of the proposed methods are demonstrated through Monte Carlo simulations, and the impact of price volatility and the number of trading days on fair strike prices of volatility and variance swaps is investigated across various numerical experiments.