论文标题
复制的内核希尔伯特空间方法，用于奇特的本地随机波动麦基恩 - 维拉索夫模型
A Reproducing Kernel Hilbert Space approach to singular local stochastic volatility McKean-Vlasov models
论文作者
论文摘要
受到与财务模型校准有关的挑战的激励，我们考虑了数值求解单一的McKean-Vlasov方程$$ D x_t =σ（t，x_t）x_t \ frac {\ sqrt v_t v_t v_t v_t} {\ sqrt {\ sqrt { $ v $是一个改编的扩散过程。该方程可被视为奇异的局部随机波动率模型。尽管这种模型在从业人员中很受欢迎，但不幸的是，它的良好性尚未得到充分理解，总的来说，根本无法保证。我们基于繁殖核Hilbert Space（RKHS）技术开发了一种新颖的正则化方法，并表明正则化模型已得到充分。此外，我们证明了混乱的传播。我们从数字上证明，由于典型的本地波动率模型，因此，正规化模型能够完美复制期权价格。我们的结果也适用于更通用的McKean-Vlasov方程。
Motivated by the challenges related to the calibration of financial models, we consider the problem of numerically solving a singular McKean-Vlasov equation $$ d X_t= σ(t,X_t) X_t \frac{\sqrt v_t}{\sqrt {E[v_t|X_t]}}dW_t, $$ where $W$ is a Brownian motion and $v$ is an adapted diffusion process. This equation can be considered as a singular local stochastic volatility model. Whilst such models are quite popular among practitioners, unfortunately, its well-posedness has not been fully understood yet and, in general, is possibly not guaranteed at all. We develop a novel regularization approach based on the reproducing kernel Hilbert space (RKHS) technique and show that the regularized model is well-posed. Furthermore, we prove propagation of chaos. We demonstrate numerically that a thus regularized model is able to perfectly replicate option prices due to typical local volatility models. Our results are also applicable to more general McKean--Vlasov equations.
