发表机构
Delft Institute of Applied Mathematics, Delft University of Technology; FF Quant Advisory B.V.（代尔夫特应用数学研究所，代尔夫特理工大学; FF量化咨询有限公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对高维密度恢复和期望计算中的维度灾难，提出COS-TT-CHF方法，通过压缩特征函数样本张量并映射到COS系数张量，实现线性复杂度，在GBM和VG下分别支持150维和100维，并给出误差控制和参数选择规则。
AI 中文摘要
Fang和Oosterlee（2008）提出的傅里叶-余弦（COS）方法能够从特征函数（ch.f.）半解析地恢复密度并计算期望。在高维情况下，展开合成和COS系数张量项的计算均随维度呈指数增长。我们通过两种张量列（TT）方法来解决这一维度灾难。COS-TT将COS系数张量替换为TT表示，消除了在线合成的指数成本，但离线构建成本仍然指数级。本文的主要贡献COS-TT-CHF则压缩特征函数样本张量，并通过另一种COS表示将其映射到原始傅里叶-余弦系数张量。除了与模型相关的一次特征函数评估成本外，其离线和在线成本随维度线性增长，而其误差仅随维度代数增长。COS-TT-CHF引入了频域截断和离散化误差，而COS-TT没有这些误差；我们推导了显式界限和实用的参数选择规则来控制这两种误差。作为副产品，该构造直接从特征函数得到余弦基函数TT，在平方可积条件下收敛到密度。我们进一步将可加标量聚合（包括相关变量的加权和）的不可分离函数的期望简化为一维COS计算，而无需对目标函数进行张量化。对于篮子期权定价，闭式TT收缩积分产生了半解析公式。在几何布朗运动（GBM）和方差伽马（VG）下的实验证实了参数选择规则，并表明COS-TT-CHF在GBM的150维和VG的100维下仍保持准确。因此，COS-TT-CHF使基于傅里叶的方法重新适用于以前被认为在高维中无法处理的问题。
英文摘要
The Fourier-cosine (COS) method of Fang and Oosterlee (2008) recovers densities and computes expectations semi-analytically from characteristic functions (ch.f.s). In higher dimensions, both expansion synthesis and COS coefficient-tensor construction scale exponentially with dimension. We address this curse of dimensionality with two tensor-train (TT) methods. A straightforward TT decomposition of the COS coefficient tensor gives COS-TT, which removes the exponential cost of online synthesis but not of offline decomposition. Our main contribution, COS-TT-CHF, instead compresses a ch.f. sample tensor and maps it to the original Fourier--cosine coefficient tensor through an alternative COS representation. Apart from the model-dependent cost of one ch.f. evaluation, both offline and online costs scale linearly with dimension, while error grows only algebraically. COS-TT-CHF introduces two additional errors: frequency-domain truncation and discretization errors. Both are controlled by parameter-selection rules from theoretical error analysis. We further propose an approach for expectations of nonseparable functions of additive scalar aggregates by computing the ch.f.s of the aggregates using tensorized COS methods, reducing the original multidimensional problem to a one-dimensional COS calculation. Closed-form formulas are derived for the TT-contraction integrals arising when the aggregate is a weighted sum, as in European basket option pricing. Experiments under geometric Brownian motion (GBM) and variance gamma (VG) show that, with the parameter-selection rules, COS-TT-CHF remains accurate through 150 dimensions for GBM and 100 for VG within a 30-minute offline computation budget on a laptop. As a by-product, our Fourier-cosine TT constructions yield cosine-basis functional TTs (FTTs) under weaker assumptions than in spectral FTT literature.
Comments38 pages, 8 figures. COS-TT and COS-TT-CHF were first developed and documented by the authors in terms of a Msc thesis; this paper provides systematic derivations, error analysis, parameter-selection rules, and extensive tests in higher-dimensional settings
