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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• Communicated by Massimiliano Ferrara.
• Accepted: 2020-05-09.
• Published online: 2020-07-09.