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Entropy 2014, 16(5), 2642-2668; doi:10.3390/e16052642
Article

The Impact of the Prior Density on a Minimum Relative Entropy Density: A Case Study with SPX Option Data

1,*  and 2,*
Received: 31 March 2014; in revised form: 7 May 2014 / Accepted: 9 May 2014 / Published: 14 May 2014
(This article belongs to the Special Issue Maximum Entropy and Its Application)
Abstract: 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&P500 index options at a large number of strikes. Prior information is now considered to be given by calibrated Heston, Schöbel–Zhu or Variance Gamma models. We find that the resulting digital option prices are essentially the same as those given by the (non-relative) Buchen–Kelly 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.
Keywords: entropy; relative entropy; Kullback–Leibler information number; asset distribution; option pricing; Fourier transform; variance swap entropy; relative entropy; Kullback–Leibler information number; asset distribution; option pricing; Fourier transform; variance swap
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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MDPI and ACS Style

Neri, C.; Schneider, L. The Impact of the Prior Density on a Minimum Relative Entropy Density: A Case Study with SPX Option Data. Entropy 2014, 16, 2642-2668.

AMA Style

Neri C, Schneider L. The Impact of the Prior Density on a Minimum Relative Entropy Density: A Case Study with SPX Option Data. Entropy. 2014; 16(5):2642-2668.

Chicago/Turabian Style

Neri, Cassio; Schneider, Lorenz. 2014. "The Impact of the Prior Density on a Minimum Relative Entropy Density: A Case Study with SPX Option Data." Entropy 16, no. 5: 2642-2668.


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