From Black-Scholes and Dupire formulae to last passage times of local martingales. Part A : The infinite time horizon

Approximating Functional of Local Martingale Under the Lack of Uniqueness of Black-Scholes PDE

When the underlying stock price is a strict local martingale process under an equivalent local martingale measure, Black-Scholes PDE associated with an European option may have multiple solutions. In this paper, we study an approximation for the smallest hedging price of such an European option. Our results show that a class of rebate barrier options can be used for this approximation. Among of them, a specific rebate option is also provided with a continuous rebate function, which corresponds to the unique classical solution of the associated parabolic PDE. Such a construction makes existing …

Exact solution of a generalized version of the Black-Scholes equation

We analyze a generalized version of the Black-Scholes equation depending on a parameter $a\!\in \!(-\infty,0)$. It satisfies the martingale condition and coincides with the Black-Scholes equation in the limit case $a\nearrow 0$. We show that the generalized equation is exactly solvable in terms of Hermite polynomials and numerically compare its solution with the solution of the Black-Scholes equation.

Black-Scholes方程在无穷远处的边界条件

Boundary conditions at infinity for Black-Scholes equations

针对标的资产为严格局部鞅的欧式期权定价，提出一种数值程序，通过在无穷远处施加边界条件选出最小解作为衍生品价格，该程序满足离散最大值原理且数值实验验证了其准确性。

随机Knothe-Rosenblatt：随机局部波动率模型的闪电校准

Stochastic Knothe-Rosenblatt: Light-speed Calibration of Stochastic Local Volatility Models

本文提出一种基于随机Knothe-Rosenblatt的模块化校准方法，通过递归构造鞅精确匹配边际分布并接近参考动力学，适用于随机和路径依赖波动率模型，支持高效Sinkhorn算法及多资产扩展。

迈向预测市场中的Black-Scholes：一种统一的核函数与交易商手册

Toward Black Scholes for Prediction Markets: A Unified Kernel and Market Maker's Handbook

本文提出一种统一的核函数，用于预测市场中的概率报价与对冲，通过引入对数it跳跃扩散模型，解决信念波动、跳跃和跨事件风险问题，并构建校准流程和衍生品层。