ERIK AURELL
Matematiska Institutionen, Stockholms Universitet, S-106 91 Stockholm, Sweden
ROBERTO BAVIERA
Dipartimento di Fisica, Università dell'Aquila and Istituto Nazionale Fisica della Materia, Via Vetoio, I-67010 Coppito, L'Aquila, Italy
OLA HAMMARLID
Institutionen för Matematisk Statistik och Försäkringsmatematik, Stockholms Universitet, S-106 91 Stockholm, Sweden
MAURIZIO SERVA
Dipartimento di Matematica, Università dell'Aquila and Istituto Nazionale Fisica della Materia, Via Vetoio, I-67010 Coppito, L'Aquila, Italy
ANGELO VULPIANI
Dipartimento di Fisica, Università di Roma "La Sapienza" and Istituto Nazionale Fisica della Materia, P.le A. Moro 2, I-00185 Roma, Italy

Received 13 April 1999

We introduce and discuss a general criterion for the derivative pricing in the general situation of incomplete markets, we refer to it as the No Almost Sure Arbitrage Principle. This approach is based on the theory of optimal strategy in repeated multiplicative games originally introduced by Kelly. As particular cases we obtain the Cox–Ross–Rubinstein and Black–Scholes in the complete markets case and the Schweizer and Bouchaud–Sornette as a quadratic approximation of our prescription. Technical and numerical aspects for the practical option pricing, as large deviation theory approximation and Monte Carlo computation are discussed in detail.