{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:27:23Z","timestamp":1760243243985,"version":"build-2065373602"},"reference-count":46,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2014,5,14]],"date-time":"2014-05-14T00:00:00Z","timestamp":1400025600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We study the problem of finding probability densities that match given European call option prices. To allow prior information about such a density to be taken into account, we generalise the algorithm presented in Neri and Schneider (Appl. Math. Finance 2013) to find the maximum entropy density of an asset price to the relative entropy case. This is applied to study the impact of the choice of prior density in two market scenarios. In the first scenario, call option prices are prescribed at only a small number of strikes, and we see that the choice of prior, or indeed its omission, yields notably different densities. The second scenario is given by CBOE option price data for S&amp;P500 index options at a large number of strikes. Prior information is now considered to be given by calibrated Heston, Sch\u00f6bel\u2013Zhu or Variance Gamma models. We find that the resulting digital option prices are essentially the same as those given by the (non-relative) Buchen\u2013Kelly density itself. In other words, in a sufficiently liquid market, the influence of the prior density seems to vanish almost completely. Finally, we study variance swaps and derive a simple formula relating the fair variance swap rate to entropy. Then we show, again, that the prior loses its influence on the fair variance swap rate as the number of strikes increases.<\/jats:p>","DOI":"10.3390\/e16052642","type":"journal-article","created":{"date-parts":[[2014,5,14]],"date-time":"2014-05-14T11:22:27Z","timestamp":1400066547000},"page":"2642-2668","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["The Impact of the Prior Density on a Minimum Relative Entropy Density: A Case Study with SPX Option Data"],"prefix":"10.3390","volume":"16","author":[{"given":"Cassio","family":"Neri","sequence":"first","affiliation":[{"name":"Lloyds Banking Group, 10 Gresham Street, London EC2V 7AE, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lorenz","family":"Schneider","sequence":"additional","affiliation":[{"name":"Center for Financial Risks Analysis (CEFRA), EMLYON Business School,23 avenue Guy de Collongue, 69130 Ecully, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2014,5,14]]},"reference":[{"key":"ref_1","first-page":"373","article-title":"\u00dcber die Beziehung zwischen dem zweiten Hauptsatze der mechanischen W\u00e4rmetheorie und der Wahrscheinlichkeitsrechnung respektive den S\u00e4tzen \u00fcber das W\u00e4rmegleichgewicht","volume":"76","author":"Boltzmann","year":"1877","journal-title":"Wien. 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