The Black-Scholes-Merton dual equation

TL;DR

This paper introduces a dual equation to the Black-Scholes-Merton model that applies to various option types, revealing a put-call symmetry and providing practical formulas for hedging parameters.

Contribution

It derives a novel dual equation for options with homogeneous payoffs, extending the Black-Scholes-Merton framework and enabling new insights into pricing and hedging strategies.

Findings

A dual equation identical in form to Black-Scholes-Merton is derived.

A put-call symmetry is established, allowing pricing of puts via calls and vice versa.

Analytic formulas for delta and gamma are provided for practical hedging.

Abstract

We derive the Black-Scholes-Merton dual equation, which has exactly the same form as the Black-Scholes-Merton equation. The novel and general equation works for options with a payoff of homogeneous of degree one, including European, American, Bermudan, Asian, barrier, lookback, etc., and leads to new insights into pricing and hedging. Perceptibly, a put-call equality emerges - all the put options can be priced as their corresponding calls by simultaneously swapping stock price (dividend yield) for strike price (risk-free rate), and vice versa. Equally important, we provide simple analytic formulas for hedging parameters delta and gamma, which elevate the put-call equality to true practical applicability. For futures options, the put-call equality leads to "symmetric" properties between puts and calls or among puts (calls).