Robust Option Pricing Under Change of Numéraire

  • Guyue HuEmail author
  • Weixia Xu
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 994)


In this paper, we consider the problem of option pricing from the perspect of minimax algorithm, an online learning framework. We introduce numéraire, which is a unit of account in economics, to the market dynamic as a multi-round game between two players: the investor and the nature. In this way, we are able to apply the online learning framework namely minimax algorithm in game theory. We model the repeated games between the investor and the nature as a price process under different numéraires, thus permit arbitrary choice of numéraire, and study this model under no arbitrage condition of a complete market. We also relax the constraint of convex payoff functions in previous works by characterizing the explicit mixed-strategy Nash equilibrium in a single-round game, and then generalize this result to multi-round games.


Online learning Minimax Numéraire 


  1. 1.
    Abernethy, J., Bartlett, P.L., Frongillo, R., Wibisono, A.: How to hedge an option against an adversary: Black-Scholes pricing is minimax optimal, pp. 2346–2354 (2013)Google Scholar
  2. 2.
    Abernethy, J., Frongillo, R.M., Wibisono, A.: Minimax option pricing meets Black-scholes in the limit. In: Proceedings of the Forty-fourth Annual ACM Symposium on Theory of Computing, pp. 1029–1040. ACM (2012)Google Scholar
  3. 3.
    Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–654 (1973)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brigo, D., Mercurio, F.: Interest Rate Models - Theory and Practice: With Smile. Inflation and Credit. Springer, Heidelberg (2007). Scholar
  5. 5.
    Cox, J.C., Ross, S.A., Rubinstein, M.: Option pricing: a simplified approach. J. Financ. Econ. 7(3), 229–263 (1979)CrossRefGoogle Scholar
  6. 6.
    DeMarzo, P., Kremer, I., Mansour, Y.: Online trading algorithms and robust option pricing. In: Proceedings of the Thirty-eighth Annual ACM Symposium on Theory of Computing, pp. 477–486. ACM (2006)Google Scholar
  7. 7.
    Eraker, B.: Do stock prices and volatility jump? Reconciling evidence from spot and option prices. J. Financ. 59(3), 1367–1403 (2004)CrossRefGoogle Scholar
  8. 8.
    Eraker, B., Johannes, M., Polson, N.: The impact of jumps in volatility and returns. J. Financ. 58(3), 1269–1300 (2003)CrossRefGoogle Scholar
  9. 9.
    Geman, H., El Karoui, N., Rochet, J.C.: Changes of numeraire, changes of probability measure and option pricing. J. Appl. Probab. 32, 443–458 (1995)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Van der Hoek, J., Elliott, R.J.: Binomial Models in Finance. Springer, New York (2006). Scholar
  11. 11.
    Hull, J.C.: Options, Futures & Other Derivatives. Prentice Hall, Upper Saddle River (2009)zbMATHGoogle Scholar
  12. 12.
    Jamshidian, F.: An exact bond option formula. J. Financ. 44(1), 205–209 (1989)CrossRefGoogle Scholar
  13. 13.
    Lam, H., Liu, Z.: Robust dynamic hedgingGoogle Scholar
  14. 14.
    Sander, M.: Bondesson’s representation of the variance gamma model and Monte Carlo option pricing (2008)Google Scholar
  15. 15.
    Mansour, Y.: Learning, regret minimization and option pricing. In: Proceedings of the 11th Conference on Theoretical Aspects of Rationality and Knowledge, pp. 2–3. ACM (2007)Google Scholar
  16. 16.
    Merton, R.C.: Theory of rational option pricing. Bell J. Econ. Manag. Sci., 141–183 (1973)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Pan, J.: The jump-risk premia implicit in options: evidence from an integrated time-series study. J. Financ. Econ. 63(1), 3–50 (2002)CrossRefGoogle Scholar
  18. 18.
    Ross, S.A.: The arbitrage theory of capital asset pricing. J. Econ. Theory 13(3), 341–360 (1976)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.State Key Laboratory of High Performance Computing (HPCL)College of Computer, National University of Defense TechnologyChangshaPeople’s Republic of China

Personalised recommendations