基于埃尔米特基函数的期权价格非参数估计
Nonparametric estimates of option prices via Hermite basis functions
AI总结：
本文提出用重标度埃尔米特多项式线性组合近似对数收益率密度，构造Black-Scholes模型的非参数扰动以定价欧式期权，并通过样本外误差检验表明该方法仅在执行价格接近观测样本时表现较好。
摘要：
我们考虑欧式期权的近似定价公式，其方法是将标的资产对数收益率的密度用重标度埃尔米特多项式的线性组合来近似。由此得到的模型可视为经典Black-Scholes模型的扰动，且具有非参数性质，因为在固定到期时间下，对数收益率的分布仅被假设具有平方可积密度。我们广泛研究了这类近似模型的经验表现，以样本外相对定价误差来衡量，并考察其阶数（粗略地说，即多项式展开的次数）以及多种基于观测数据进行校准的方式。经验结果表明，与基于隐含波动率曲线插值和外推的简单非参数估计相比，这类近似定价公式仅在期权执行价格与观测样本中的执行价格相距不太远时表现较好。
英文摘要：
We consider approximate pricing formulas for European options based on approximating the logarithmic return's density of the underlying by a linear combination of rescaled Hermite polynomials. The resulting models, that can be seen as perturbations of the classical Black-Scholes one, are nonpararametric in the sense that the distribution of logarithmic returns at fixed times to maturity is only assumed to have a square-integrable density. We extensively investigate the empirical performance, defined in terms of out-of-sample relative pricing error, of this class of approximating models, depending on their order (that is, roughly speaking, the degree of the polynomial expansion) as well as on several ways to calibrate them to observed data. Empirical results suggest that such approximate pricing formulas, when compared with simple nonparametric estimates based on interpolation and extrapolation on the implied volatility curve, perform reasonably well only for options with strike price not too far apart from the strike prices of the observed sample.
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