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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 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.

Suggested Citation

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

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    1. Orlin Grabbe, J., 1983. "The pricing of call and put options on foreign exchange," Journal of International Money and Finance, Elsevier, vol. 2(3), pages 239-253, December.
    2. Schroder, Mark, 1999. "Changes of Numeraire for Pricing Futures, Forwards, and Options," Review of Financial Studies, Society for Financial Studies, vol. 12(5), pages 1143-1163.
    3. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-651, October.
    4. Jérôme Detemple, 1999. "American Options: Symmetry Properties," CIRANO Working Papers 99s-45, CIRANO.
    5. Peter Carr & Liuren Wu, 2014. "Static Hedging of Standard Options," Journal of Financial Econometrics, Oxford University Press, vol. 12(1), pages 3-46.
    6. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    7. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    8. Garman, Mark B. & Kohlhagen, Steven W., 1983. "Foreign currency option values," Journal of International Money and Finance, Elsevier, vol. 2(3), pages 231-237, December.
    9. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    10. Orlin J. Grabbe, "undated". "The Pricing of Call and Put Options on Foreign Exchange," Rodney L. White Center for Financial Research Working Papers 06-83, Wharton School Rodney L. White Center for Financial Research.
    11. Qiang Liu, 2010. "Optimal approximations of nonlinear payoffs in static replication," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 30(11), pages 1082-1099, November.
    12. David S. Bates, "undated". "The Cash Premium: Option Pricing Under Asymmetric Processes, with Applications to Options on Deutschemark Futures," Rodney L. White Center for Financial Research Working Papers 36-88, Wharton School Rodney L. White Center for Financial Research.
    13. Orlin J. Grabbe, "undated". "The Pricing of Call and Put Options on Foreign Exchange," Rodney L. White Center for Financial Research Working Papers 6-83, Wharton School Rodney L. White Center for Financial Research.
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