# Existence of Maximum Entropy Problem Solution in a General N-Dimensional Case

## DOI:

https://doi.org/10.51408/1963-0012## Keywords:

Entropy, Boundary, Distribution, Options## Abstract

In the following paper, we will define conditions, which need to be satisfied in order for the maximum entropy problem applied in European call options to have a solution in a general n-dimensional case. We will also find a minimum right boundary for the price range in order to have at least one risk neutral measure satisfying the option pricing formula. The results significantly reduce the computational time of optimization algorithms used in maximum entropy problem.

## References

Y. Alhassid, N. Agmon and R. D. Levine, “An upper bound for the entropy and its applications to the maximal entropy problem”, Chem. Phys. Lett., vol. 53, pp. 22, 1978.

Y.Alhassid, N. Agmon and R. D. Levine, “An algorithm for finding the distribution of maximal entropy”, Journal of Computational Physics, vol. 30, pp. 250-258, 1979.

R. D. Levine and M. Tribus, “The maximum entropy formalism”, Cambridge MA: MIT Press, pp. 207-209, 1978.

Y. Alhassid and R. D. Levine, “Experimental and inherent uncertainties in the information theoretic approach”, Chem. Phys. Lett., vol. 73, pp. 16-20, 1980.

C. E. Shannon, “A mathematical theory of communication”, Bell Systems Technical Journal, vol. 27, pp. 379-423, 1948.

P. W. Buchen and M. Kelly, “The maximum entropy distribution of an asset inferred from option prices”, Journal of Financial and Quantitative Analysis, vol. 31, pp. 143- 159, 1996.

C. Neri and L. Schneider, “A family of maximum entropy densities matching call options prices”, arXiv:1102.0224v1 [q-fin.PR], 2011.

J. Borwein, R. Choksi and P. Marechal, “Probability distributions of assets inferred from option prices via the principle of maximum entropy”, SIAM J. OPTIM., vol. 14, no 2, pp. 464-478, 2003.

L. S. Rompolis, “A new method of employing the principle of maximum entropy to retrieve the risk neutral density”, http://web.xrh.unipi.gr/attachments/Seminars/2008

[Yu. Xishen and Li Yang, “Pricing american options using a nonparametric entropy approach”, Hindawi Publishing Corporation, pp. 16, article ID 369795, 2014.

M.Rubinstein, “Implied binomial trees.” Finance Working Paper, vol. 49, no 3, pp. 771-818, 1994.

N. D. Margaryan “Assessment of asset price distributions using maximum entropy method”, Proc. of Engineering Academy of Armenia, vol. 14, no 1, pp. 57-61, 2017.

N. D. Margaryan “An algorithmic approach to solving the maximum entropy problem”, Proc. of Engineering Academy of Armenia, vol. 14, no 3, pp. 371-374, 2017.

N. D. Margaryan “A boundary for the existence of solution to the maximum entropy problem applied in european call options”, Proc. of the Yerevan State University, vol. 52, no 1, pp. 3-7, 2018.

## Downloads

## Published

## How to Cite

*Mathematical Problems of Computer Science*,

*49*, 97–102. https://doi.org/10.51408/1963-0012

## Issue

## Section

## License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.