Multi-step Reflection Principle and Barrier Options

TL;DR

This paper introduces multi-step barrier options with explicit pricing formulas under Black-Scholes, enabling flexible barrier levels and approximations, and develops a generalized reflection principle for Brownian motion.

Contribution

It presents a novel multi-step reflection principle and explicit pricing formulas for multi-step barrier options, expanding the tools for complex barrier option modeling.

Findings

Explicit pricing formulas for multi-step barrier options.

Generalized reflection principle for Brownian motion.

Potential applications in complex financial products.

Abstract

This paper examines a class of barrier options-multi-step barrier options, which can have any finite number of barriers of any level. We obtain a general, explicit expression of option prices of this type under the Black-Scholes model. Multi-step barrier options are not only useful in that they can handle barriers of different levels and time steps, but can also approximate options with arbitrary barriers. Moreover, they can be embedded in financial products such as deposit insurances based on jump models with simple barriers. Along the way, we derive multi-step reflection principle, which generalizes the reflection principle of Brownian motion.