The pricing of lookback options and binomial approximation

TL;DR

This paper develops a refined binomial model for pricing European lookback options, deriving a closed-form formula and showing convergence to the Black-Scholes price through asymptotic analysis.

Contribution

It provides a new closed-form formula for lookback option prices in a discrete model and proves their convergence to continuous Black-Scholes prices.

Findings

Closed-form pricing formula for lookback options in a discrete model

Asymptotic expansion demonstrating convergence to Black-Scholes prices

Improved approximation of binomial cumulative distribution function

Abstract

Refining a discrete model of Cheuk and Vorst we obtain a closed formula for the price of a European lookback option at any time between emission and maturity. We derive an asymptotic expansion of the price as the number of periods tends to infinity, thereby solving a problem posed by Lin and Palmer. We prove, in particular, that the price in the discrete model tends to the price in the continuous Black-Scholes model. Our results are based on an asymptotic expansion of the binomial cumulative distribution function that improves several recent results in the literature.