Options' Prices Under Arithmetic Brownian Motion and Their Implication for Modern Derivatives Pricing

Abstract

The pricing formulas for European call and put options under arithmetic Brownian motion (ABM) are derived via risk-neutral valuation using the martingale measure, and checked against the corresponding Black-Scholes-like partial differential equation (PDE). In quite a few limiting cases, the formulas are found to have the correct properties. For perpetual calls and very high standard deviation of the change in stock price, however, these formulas seem to violate the principle of no arbitrage, which suggest that the risk-neutral valuation or the Black-Scholes approach does not work for the ABM model of stock price evolution. This conclusion may have implications for pricing other non-equity derivatives.

Keywords: arithmetic Brownian motion, ABM, options' pricing formulas, risk-neutral valuation, violations of no arbitrage

JEL Classification: G13, G12, A23