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The Black-Scholes-Merton dual equation

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  • Shuxin Guo
  • Qiang Liu

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).

Suggested Citation

  • Shuxin Guo & Qiang Liu, 2019. "The Black-Scholes-Merton dual equation," Papers 1912.10380, arXiv.org, revised May 2024.
    • Handle: RePEc:arx:papers:1912.10380
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    References listed on IDEAS

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