Continuously Monitored Snowball Option Pricing via Partial Differential Equation

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.