Opuscula Math. 40, no. 4 (2020), 451-473
https://doi.org/10.7494/OpMath.2020.40.4.451

 
Opuscula Mathematica

Option pricing formulas under a change of numèraire

Antonio Attalienti
Michele Bufalo

Abstract. We present some formulations of the Cox-Ross-Rubinstein and Black-Scholes formulas for European options obtained through a suitable change of measure, which corresponds to a change of numèraire for the underlying price process. Among other consequences, a closed formula for the price of an European call option at each node of the multi-period binomial tree is achieved, too. Some of the results contained herein, though comparable with analogous ones appearing elsewhere in the financial literature, provide however a supplementary widening and deepening in view of useful applications in the more challenging framework of incomplete markets. This last issue, having the present paper as a preparatory material, will be treated extensively in a forthcoming paper.

Keywords: Black-Scholes formula, binomial model, martingale measures, numèraire.

Mathematics Subject Classification: 91B25, 60G46.

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  1. BIS, Is the unthinkable becoming routine?, Technical Report, Bank for International Settlements (2015).
  2. F. Black, M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy 81 (1973), 637-654.
  3. J.C. Cox, S.A. Ross, M. Rubinstein, Option pricing: A simplified approach, Journal of Financial Economics 7 (1979), 229-263.
  4. K.C. Engelen, The unthinkable as the new normal, The International Economy 29 (2015) 3.
  5. H. Geman, N. El Karoui, J.C. Rochet, Changes of numéraire, change of probability measure and option pricing, Journal of Applied Probability 32 (1995), 443-458.
  6. R.C. Merton, Theory of rational option pricing, Journal of Economy and Management Sciences 4 (1973), 141-183.
  7. M. Musiela, M. Rutkowski, Martingale Methods in Financial Modelling, Springer-Verlag Berlin Heidelberg, 2005.
  8. L.T. Nielsen, Understanding \(N(d_1)\) and \(N(d_2)\): Risk-adjusted probabilities in the Black-Scholes model, Revue Finance 14 (1993), 95-106.
  9. L.T. Nielsen, Pricing and Hedging of Derivative Securities, Oxford University Press, 1999.
  10. S. Shreve, Stochastic Calculus for Finance I: the Binomial Asset Pricing Model, Springer-Verlag New York, 2004.
  11. S. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models, Springer-Verlag New York, 2004.
  12. D. Williams, Probability with Martingales, Cambridge University Press, Cambridge UK, 1991.
  • Communicated by Massimiliano Ferrara.
  • Received: 2020-02-07.
  • Accepted: 2020-05-09.
  • Published online: 2020-07-09.
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Cite this article as:
Antonio Attalienti, Michele Bufalo, Option pricing formulas under a change of numèraire, Opuscula Math. 40, no. 4 (2020), 451-473, https://doi.org/10.7494/OpMath.2020.40.4.451

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