Abstract
The application of entropy in finance can be regarded as the extension of the information entropy and the probability entropy. It can be an important tool in various financial methods such as measure of risk, portfolio selection ,option pricing and asset pricing. A typical example for the field of option pricing, is the Entropy Pricing Theory (EPT) introduced by Les Gulko [1996]. The Black-Scholes model [1973] exhibits the idea of no arbitrage which implies the existence of universal risk-neutral probabilities but unfortunately it does not guarantees the uniqueness of the risk-neutral probabilities. In a second step the parameterization of these risk-neutral probabilities needs a frame of stochastic calculus and to be more specific for example the Black and Scholes frame is controlled by Geometric Brownian Motion (GBM). This implies the existence of risk-neutral probabilities in the field of option pricing and their uniqueness is vital. The Shannon entropy can be used in particular manners to evaluate entropy of probability density distribution around some points but in the case of specific events for example deviation from mean and any sudden news for stock returns up (down), needs additional information and this concept of entropy can be generalized. If we want to compare entropy of two distributions by considering the two events i.e. deviation from mean and sudden news then Shannon entropy [1964] assumes implicit certain exchange that occurs as a compromise between contributions from the tail and main mass of the distribution. This is important now to control this trade-off explicitly. In order to solve this problem the use of entropy measures that depend on powers of probability for example Tsallis [1988], Kaniadakis [2001], Ubriaco [2009], Shafee [2007] and Reyni [1961] provide such control. In this article we use entropy measures depend on the powers of the probability. We propose some entropy maximization problems in order to obtain the risk neutral densities. We present also the European call and put in this frame work.
Highlights
The application of entropy in Finance can be regarded as the extension of both information entropy and probability entropy
Ubriaco (2009) proposed a new entropy measure based on fractional calculus and showed this new entropy has the same properties as Shannon entropy except additivity
In this article we have presented some approaches to obtain risk-neutral densities using three different types of entropy measures
Summary
The application of entropy in Finance can be regarded as the extension of both information entropy and probability entropy. The famous Black-Scholes model [1] assumes the condition of no arbitrage which implies the universe of riskneutral probabilities The uniqueness of these risk-neutral probabilities is very crucial. The Entropy Pricing Theory (EPT) was introduced by Les Gulko as an alternative method for the construction of risk-neutral probabilities without relying on stochastic calculus [8,9,10]. For maximum entropy distribution of asset returns, application of entropy in finance, entropy maximization problems, and others can be found in [3,7,13,15,16,26]. In this article we use three different types of entropy measures to find risk-neutral densities using the framework of EPT for stock options [20,23,24].
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