英文标题：
《Option pricing with Legendre polynomials》
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作者：
Julien Hok and Tat Lung Chan
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最新提交年份：
2017
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英文摘要：
Here we develop an option pricing method based on Legendre series expansion of the density function. The key insight, relying on the close relation of the characteristic function with the series coefficients, allows to recover the density function rapidly and accurately. Based on this representation for the density function, approximations formulas for pricing European type options are derived. To obtain highly accurate result for European call option, the implementation involves integrating high degree Legendre polynomials against exponential function. Some numerical instabilities arise because of serious subtractive cancellations in its formulation (96) in proposition 7.1. To overcome this difficulty, we rewrite this quantity as solution of a second-order linear difference equation and solve it using a robust and stable algorithm from Olver. Derivation of the pricing method has been accompanied by an error analysis. Errors bounds have been derived and the study relies more on smoothness properties which are not provided by the payoff? functions, but rather by the density function of the underlying stochastic models. This is particularly relevant for options pricing where the payoff of the contract are generally not smooth functions. The numerical experiments on a class of models widely used in quantitative finance show exponential convergence.
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中文摘要：
本文提出了一种基于密度函数勒让德级数展开的期权定价方法。关键的见解是，依靠特征函数与系列系数的密切关系，可以快速准确地恢复密度函数。基于密度函数的这种表示，导出了欧式期权定价的近似公式。为了获得欧式看涨期权的高精度结果，实现过程包括将高次勒让德多项式与指数函数进行积分。由于命题7.1中公式（96）中存在严重的减法对消，因此出现了一些数值不稳定性。为了克服这个困难，我们将这个量重写为二阶线性差分方程的解，并使用Olver提供的鲁棒稳定算法进行求解。定价方法的推导伴随着误差分析。已经推导出了误差界，并且研究更多地依赖于平滑特性，而这些特性不是由payoff提供的？函数，而是由基础随机模型的密度函数决定。这对于期权定价尤其相关，因为合同的回报通常不是平稳的函数。对一类广泛应用于定量金融的模型进行的数值实验表明，该模型具有指数收敛性。
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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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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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