Extracting Implied Stock Returns from Options Prices

Abstract

This paper proposes a new method for extracting the market’s expected return of a stock from options prices while also calculating option-specific risk discounts (calls) and premiums (puts). However, first, I revisit the variable μ (expected return of a stock) as it relates to stock prices in the Black-Scholes formula derivation. I postulate that μ is itself a function of time and therefore the partial derivative equation Black, Scholes, and Merton solved was incomplete. Importantly, this undermines the conclusion Black, Scholes, and Merton came to, that an option’s price is not a function of the expected return of the underlying stock. To extract the expected return from options prices, I begin by proposing formulas for call and put prices introducing variables for strike price specific discounts and premiums. Known qualities of options, required to satisfy the no arbitrage assumption, are then used to solve for these discounts and premiums as a function of the implied expected price of a stock and σ. Finally, implied expected price and σ are solved for using numerical analysis.