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Entropy, Relative Entropy and Variance Swaps
in Oberseminar Finanz- und Versicherungsmathematik, Technische Universität München, October 31, 2013 Technische Universität München 2013
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 generalize a previously presented algorithm to find the maximum entropy density of an asset price to the relative entropy case. This is applied to study the impact the choice of prior density has in two market scenarios. In the first scenario, call option prices are prescribed at only a small number of strikes, and we
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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&P 500 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 ( entropy. Then we show, again, that the prior loses its influence on the fair variance swap rate as the number of strikes increases.
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