QUANTUM THEORY FOR THE BINOMIAL MODEL IN FINANCE THEORY

In this paper, a quantum model for the binomial market in nance is pro- posed. We show that its risk-neutral world exhibits an intriguing structure as a disk in the unit ball of R 3 , whose radius is a function of the risk-free interest rate with two thresh- olds which prevent arbitrage opportunities from this quantum market. Furthermore, from the quantum mechanical point of view we re-deduce the Cox-Ross-Rubinstein binomial option pricing formula by considering Maxwell-Boltzmann statistics of the system of N distinguishable particles. In nance theory the binomial market is a useful and very popular technique for pricing a stock option, in which only one risky asset is binomial. Although the binomial market is a very ideal model, a realistic model may be assumed to be composed of a much large number of binomial markets. This is the assumption that underlies a widely used numerical procedure rst proposed by Cox, Ross, and Rubinstein (2) , in which Bernoulli's random variables are used to describe the only one risky asset. There seems to be a prior no reason why the binomial market must be modelled by using the Bernoulli's random variables, even though the binomial market is a hypothesis and an ideal model. In this paper, a quantum model for the binomial market is proposed. We show that its risk- neutral world exhibits an intriguing structure as a disk in the unit ball of R 3 , whose radius is a function of the risk-free interest rate with two thresholds which prevent arbitrage opportunities from this quantum market. Moreover, from the quantum point of view we re-deduce the Cox- Ross-Rubinstein binomial option pricing formula by considering Maxwell-Boltzmann statistics of the system of N distinguishable particles. Therefore, it seems that it is of some interest that we use quantum nancial models in nance theory. Indeed, some mathematical methods on applications of quantum mechanics to nance have been presented in (3,4), including quantum trading strategies, quantum hedging, and quantum version of no-arbitrage.