英文标题：
《Calibration of Local Volatility Model with Stochastic Interest Rates by
  Efficient Numerical PDE Method》
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作者：
Julien Hok and Shih-Hau Tan
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最新提交年份：
2018
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英文摘要：
  Long maturity options or a wide class of hybrid products are evaluated using a local volatility type modelling for the asset price S(t) with a stochastic interest rate r(t). The calibration of the local volatility function is usually time-consuming because of the multi-dimensional nature of the problem. In this paper, we develop a calibration technique based on a partial differential equation (PDE) approach which allows an efficient implementation. The essential idea is based on solving the derived forward equation satisfied by P(t; S; r)Z(t; S; r), where P(t; S; r) represents the risk neutral probability density of (S(t); r(t)) and Z(t; S; r) the projection of the stochastic discounting factor in the state variables (S(t); r(t)). The solution provides effective and sufficient information for the calibration and pricing. The PDE solver is constructed by using ADI (Alternative Direction Implicit) method based on an extension of the Peaceman-Rachford scheme. Furthermore, an efficient algorithm to compute all the corrective terms in the local volatility function due to the stochastic interest rates is proposed by using the PDE solutions and grid points. Different numerical experiments are examined and compared to demonstrate the results of our theoretical analysis.
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中文摘要：
使用随机利率r（t）下资产价格S（t）的局部波动率类型模型，对长期期权或广泛类别的混合产品进行评估。由于问题的多维性，局部波动率函数的校准通常非常耗时。在本文中，我们开发了一种基于偏微分方程（PDE）方法的校准技术，该方法可以有效地实现。其基本思想是基于求解P（t；S；r）Z（t；S；r）满足的导出正演方程，其中P（t；S；r）表示（S（t）；r（t））和Z（t；S；r）状态变量中随机贴现因子的投影（S（t）；r（t））。该解决方案为校准和定价提供了有效和充分的信息。基于Peaceman-Rachford格式的扩展，采用交替方向隐式（ADI）方法构造PDE求解器。此外，利用偏微分方程解和网格点，提出了一种计算随机利率引起的局部波动函数中所有修正项的有效算法。对不同的数值实验进行了检验和比较，以验证我们的理论分析结果。
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Mathematical Finance        数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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