قیمت گزینه شرودینگر سیاه - شولز : "بیت" در مقابل "کیوبیت"

The celebrated Black-Scholes differential equation provides for the price of a financial derivative. The uncertainty environment of such option price can be described by the classical ‘bit’: a system with two possible states. This paper argues for the introduction of a different uncertainty environment characterized by the so called ‘qubit’. We obtain an information-based option price and discuss the differences between this option price and the classical option price.

One of the most celebrated equations in finance is the Black-Scholes differential equation, which draws quite heavily from the physics based notion of Brownian motion. In the finance literature there exists a close connection between the binomial option pricing model and the Black-Scholes model [1]. The Black-Scholes model can be derived from the binomial model. In the latter the stock price can take on two different positions at each time step. In analogy with information theory, the binomial model represents a bit: a system with two possible states (at each time step). Other examples where the bit notion is also implicitly used is in the so called ‘Arrow-Debreu’ paradigm, where future payments are a function of both time and the states of the world [2] and [3]. In the words of W.F. Sharpe, a well recognized finance academic, this paradigm is part of what he calls ‘nuclear financial economics’ [4]. This paper has as aim to show how the Black-Scholes option price does change when we extend the notion of bit into that of qubit. If the states ‘0’ and ‘1’ are the basis states of the bit then a qubit is a linear combination of those basis states. The paper is organized as follows. In the next section we rationalize, from a finance perspective, the use of the Schrödinger equation. We also consider the value of the option price within the qubit environment and provide for an example. Finally, we consider the existence of the Black-Scholes portfolio in the qubit environment and in the bit environment but for a different time scale.

We believe that this short paper can indicate that if we take the Black-Scholes price to be a state function directly, the parameter ℏ can be modified to indicate differing levels of uncertainty. Those varying levels of ℏ influence the qubit option price and make it to diverge from the Black-Scholes price. From a decision makers point of view the three models covered in this paper can be ranked in terms of increasing levels of uncertainty in the following way. The lowest uncertainty model would be the one where View the MathML source, since View the MathML source. The bit model or binomial model follows, since in that model the hedging delta and a Black-Scholes portfolio are two necessary constraints. The qubit model is the most uncertain, since the information carried per price step in the next period of time is quite less than in the bit model. The Black-Scholes portfolio is not even risk free in that case.