The Black-Scholes-Merton dual equation

Abstract

We derive the Black-Scholes-Merton dual equation, which has exactly the same form as the Black-Scholes-Merton equation. The new equation is general and works for European, American, Bermudan, Asian, barrier, lookback, etc. options and leads to new insights into pricing and hedging. Trivially, a put-call equality emerges - all the above-mentioned put (call) options can be priced as their corresponding calls (puts) by simply swapping stock price (dividend yield) for strike price (risk-free rate) simultaneously. More importantly, deltas (gammas) of such puts and calls are linked via analytic formulas. As one application in hedging, the dual equation is utilized to improve the accuracy of the recently proposed approach of hedging options statically with short-maturity contracts.