连续观察期雪球期权定价的偏微分方程方法

Continuously Monitored Snowball Option Pricing via Partial Differential Equation

摘要:

雪球期权是一种带障碍结构的复杂金融衍生品，其收益结构依赖于标的资产在整个存续期内的价格路径。在 Black-Scholes 模型框架下，建立了连续观察条件下的雪球期权定价偏微分方程模型，并结合金融产品复制技术和偏微分方程求解理论，推导出显式定价公式。进一步通过数值方法（有限差分法和蒙特卡洛Monte-Carlo模拟）与显式解进行对比，评估了该公式在特定市场假设下的计算效果。此外，基于实际市场参数，探讨了波动率、障碍水平等关键因素对期权价格的影响。研究结果为雪球期权的定价实践提供了可行的工具参考，补充了相关产品在连续观察条件下的定价分析。

Abstract:

Snowball options are path-dependent exotic derivatives with embedded barrier structures， whose payoff profiles hinge on the price trajectory of the underlying asset over the full lifespan of the contract. Within the classical Black-Scholes model framework， this paper constructs a partial differential equation （PDE） model for the valuation of snowball options under continuous monitoring conditions. By integrating financial replication techniques and PDE-solving theories， an explicit pricing formula is derived. The computational efficacy of this closed-form solution is further evaluated through systematic comparisons with numerical methodologies， namely the finite difference method and Monte-Carlo simulation. Additionally， drawing on real-world market parameters， the paper probes into the impacts of key determinants， such as volatility and barrier levels， on option prices. The findings provide viable methodological guidance for the practical pricing of snowball options， and supplement the pricing analytics of relevant structured products under continuous monitoring scenarios.