英文标题：
《High-order compact finite difference scheme for option pricing in
  stochastic volatility jump models》
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作者：
Bertram D\\\"uring, Alexander Pitkin
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最新提交年份：
2019
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英文摘要：
  We derive a new high-order compact finite difference scheme for option pricing in stochastic volatility jump models, e.g. in Bates model. In such models the option price is determined as the solution of a partial integro-differential equation. The scheme is fourth order accurate in space and second order accurate in time. Numerical experiments for the European option pricing problem are presented. We validate the stability of the scheme numerically and compare its performance to standard finite difference and finite element methods. The new scheme outperforms a standard discretisation based on a second-order central finite difference approximation in all our experiments. At the same time, it is very efficient, requiring only one initial $LU$-factorisation of a sparse matrix to perform the option price valuation. Compared to finite element approaches, it is very parsimonious in terms of memory requirements and computational effort, since it achieves high-order convergence without requiring additional unknowns, unlike finite element methods with higher polynomial order basis functions. The new high-order compact scheme can also be useful to upgrade existing implementations based on standard finite differences in a straightforward manner to obtain a highly efficient option pricing code.
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中文摘要：
我们推导了一种新的高阶紧致有限差分格式，用于随机波动率跳跃模型（如Bates模型）中的期权定价。在这种模型中，期权价格被确定为偏积分微分方程的解。该格式在空间上具有四阶精度，在时间上具有二阶精度。给出了欧式期权定价问题的数值实验。我们用数值方法验证了该格式的稳定性，并将其性能与标准有限差分法和有限元法进行了比较。在我们的所有实验中，新格式优于基于二阶中心有限差分近似的标准离散化。同时，它非常有效，只需对稀疏矩阵进行一次初始的$LU$因子分解即可进行期权价格估值。与有限元方法相比，它在内存需求和计算工作量方面非常节省，因为它实现了高阶收敛，而不需要额外的未知量，这与具有更高多项式阶基函数的有限元方法不同。新的高阶紧致方案还可用于以直接方式升级基于标准有限差分的现有实现，以获得高效的期权定价代码。
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Computational Finance        计算金融学
分类描述：Computational methods, including Monte Carlo, PDE, lattice and other numerical methods with applications to financial modeling
计算方法，包括蒙特卡罗，偏微分方程，格子和其他数值方法，并应用于金融建模
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一级分类：Mathematics        数学
二级分类：Numerical Analysis        数值分析
分类描述：Numerical algorithms for problems in analysis and algebra, scientific computation
分析和代数问题的数值算法，科学计算
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