具有随机波动率和位移扩散的Libor市场模型的快速校正
Fast calibration of the Libor Market Model with Stochastic Volatility and Displaced Diffusion
摘要：
本文证明了Edgeworth展开式和Gram-Charlier展开式在校正带有随机波动率和位移扩散的Libor市场模型（DD-SV-LMM）中的有效性。我们的方法汇集了两个研究领域;第一，自Wu和Zhang（2006）以来关于SV-LMM的结果，特别是关于矩母函数，第二，基于Edgeworth或Gram-Charlier展开的密度分布近似。通过探索四阶矩的分析易处理性，我们能够对偏度和峰度的正常波动率的参考Bachelier模型进行调整，并作为副产品推导出一个微笑公式，该公式将波动率与货币性与可解释的参数联系起来。作为一个主要的结论，我们的数值结果表明，减少了98%的计算时间的DD-SV-LMM校准过程相比，经典的数值积分方法开发的Heston（1993）。
英文摘要：
This paper demonstrates the efficiency of using Edgeworth and Gram-Charlier expansions in the calibration of the Libor Market Model with Stochastic Volatility and Displaced Diffusion (DD-SV-LMM). Our approach brings together two research areas; first, the results regarding the SV-LMM since the work of Wu and Zhang (2006), especially on the moment generating function, and second the approximation of density distributions based on Edgeworth or Gram-Charlier expansions. By exploring the analytical tractability of moments up to fourth order, we are able to perform an adjustment of the reference Bachelier model with normal volatilities for skewness and kurtosis, and as a by-product to derive a smile formula relating the volatility to the moneyness with interpretable parameters. As a main conclusion, our numerical results show a 98% reduction in computational time for the DD-SV-LMM calibration process compared to the classical numerical integration method developed by Heston (1993).
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