Explicit form of approximate transition probability density functions of diffusion processes
扩散过程近似转移概率密度的显式形式AI 解读 ↓
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AI 根据所提供的论文材料归类：摘要明确将近似转移密度用于期权定价，并讨论一般扩散状态变量下的资产定价用途。
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扩散过程近似转移概率密度的显式形式
- 研究问题
- 研究真实转移概率密度不可得时，如何为多元时变扩散构造显式近似，并将其用于最大似然估计和期权定价。
- 方法与思路
- 作者分别处理可约与不可约扩散，构造闭式近似转移密度，并进一步利用密度展开给出多元跳跃扩散系数的递推公式。
- 材料陈述的贡献
- 摘要称方法可处理漂移与波动函数较光滑的一般多元时变扩散；模拟支持其似然估计用途，期权价格与所用基准差异较小。
- 阅读关注点
- 应核对光滑性与可约性条件、近似展开阶数及误差控制，并检查近似密度用于似然和定价时的数值稳定性。
- 阅读边界
- 摘要仅概括光滑性要求，未给出严格条件、误差界或价格差异数值；所述模拟与定价比较不能保证所有扩散模型均适用。
原文摘要
A continuous-time diffusion process is very popular in modeling and provides useful tools to analyze particularly, but not restricted to, a variety of economic and financial variables. The transition probability density function (TPDF) of a diffusion process plays an important role in understanding and explaining the dynamics of the process. A new way to find closed-form approximate TPDFs for multivariate diffusions is proposed in this paper. This method can be applied to general multivariate time-inhomogeneous diffusion processes, as long as, roughly speaking, they have smooth drift and volatility functions. A diffusion process is said to be reducible if it can be converted into a unit diffusion process where the volatility is the identity matrix. We have established how to obtain the approximate TPDF of a reducible diffusion explicitly. When a diffusion process is not reducible, an explicit form of approximate TPDF can be obtained by using the results in Aït-Sahalia (2008) and Choi (2013). The TPDF expansion suggested here enables us to obtain a recursive formula for the coefficient of the approximate TPDF for a multivariate jump diffusion. Monte Carlo simulation studies of conducting maximum likelihood estimation (MLE) using our approximations provide convincing evidence that our TPDF expansion can be used for the MLE when the true TPDF is unavailable. We also applied our approximate TPDFs to option pricing. The differences between our option prices and those from the Extended Black–Scholes formula are shown to be quite small. This implies that our methods can be employed to price assets whose underlying state variables follow general diffusion models.