Pricing principle via Tsallis relative entropy in incomplete market

TL;DR

This paper introduces a new pricing principle for incomplete markets using Tsallis relative entropy, ensuring time consistency, arbitrage-freeness, and compatibility with existing pricing methods, while exploring its mathematical properties and asymptotic behavior.

Contribution

It develops a novel pricing framework based on Tsallis entropy for non-attainable claims in incomplete markets, linking it to backward SDEs and existing pricing measures.

Findings

Pricing principle is time consistent and arbitrage-free.

Functional lies between minimal martingale and certainty equivalent prices.

Asymptotic behavior analyzed for ambiguity aversion coefficient.

Abstract

A pricing principle is introduced for non-attainable $q$-exponential bounded contingent claims in an incomplete Brownian motion market setting. The buyer evaluates the contingent claim under the ``distorted Radon-Nikodym derivative'' and adjustment by Tsallis relative entropy over a family of equivalent martingale measures. The pricing principle is proved to be a time consistent and arbitrage-free pricing rule. More importantly, this pricing principle is found to be closely related to backward stochastic differential equations with generators $f(y)|z|^2$ type. The pricing functional is compatible with prices for attainable claims. Except translation invariance, the pricing principle processes lots of elegant properties such as monotonicity and concavity etc. The pricing functional is showed between minimal martingale measure pricing and conditional certainty equivalent pricing under $q$-exponential utility. The asymptotic behavior of the pricing principle for ambiguity aversion coefficient is also investigated.