Opuscula Math. 40, no. 4 (2020), 451-473
Option pricing formulas under a change of numèraire
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.
- BIS, Is the unthinkable becoming routine?, Technical Report, Bank for International Settlements (2015).
- F. Black, M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy 81 (1973), 637-654.
- J.C. Cox, S.A. Ross, M. Rubinstein, Option pricing: A simplified approach, Journal of Financial Economics 7 (1979), 229-263.
- K.C. Engelen, The unthinkable as the new normal, The International Economy 29 (2015) 3.
- 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.
- R.C. Merton, Theory of rational option pricing, Journal of Economy and Management Sciences 4 (1973), 141-183.
- M. Musiela, M. Rutkowski, Martingale Methods in Financial Modelling, Springer-Verlag Berlin Heidelberg, 2005.
- 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.
- L.T. Nielsen, Pricing and Hedging of Derivative Securities, Oxford University Press, 1999.
- S. Shreve, Stochastic Calculus for Finance I: the Binomial Asset Pricing Model, Springer-Verlag New York, 2004.
- S. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models, Springer-Verlag New York, 2004.
- 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.