A Family of Maximum Entropy Densities Matching Call Option Prices

We investigate the position of the Buchen-Kelly density in a family of entropy maximising densities which all match European call option prices for a given maturity observed in the market. Using the Legendre transform which links the entropy function and the cumulant generating function, we show that it is both the unique continuous density in this family and the one with the greatest entropy. We present a fast root-finding algorithm that can be used to calculate the Buchen-Kelly density, and give upper boundaries for three different discrepancies that can be used as convergence criteria. Given the call prices, arbitrage-free digital prices at the same strikes can only move within upper and lower boundaries given by left and right call spreads. As the number of call prices increases, these bounds become tighter, and we give two examples where the densities converge to the Buchen-Kelly density in the sense of relative entropy when we use centered call spreads as proxies for digital prices. As pointed out by Breeden and Litzenberger, in the limit a continuous set of call prices completely determines the density.Comment: 22 pages, 6 figure

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

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 (2011) 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 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, Schoebel-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.Comment: 24 pages, 2 figure

Statistical mechanics of the NN-point vortex system with random intensities on R2R^2

The system of N -point vortices on $mathbb{R}^2$ is considered under the hypothesis that vortex intensities are independent and identically distributed random variables with respect to a law $P$ supported on $(0,1]$. It is shown that, in the limit as $N$ approaches $infty$, the 1-vortex distribution is a minimizer of the free energy functional and is associated to (some) solutions of the following non-linear Poisson Equation:$$-Delta u(x) = C^{-1}int_{(0,1]} rhbox{e}^{-eta ru(x)- gamma r|x|^2}P(hbox{d}r), quadforall xin mathbb{R}^2,$$where $displaystyle C = int_{(0,1]}int_{mathbb{R}^2}hbox{e}^{-eta ru(y) - gamma r|y|^2}hbox{d} yP(hbox{d}r)

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

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

Mécanique statistique des systèmes de Vortex

PARIS-DAUPHINE-BU (751162101) / SudocSudocFranceF

A Family of Maximum Entropy Densities Matching Call Option Prices

We investigate the position of the Buchen-Kelly density in a family of entropy maximising densities which all match European call option prices for a given maturity observed in the market. Using the Legendre transform which links the entropy function and the cumulant generating function, we show that it is both the unique continuous density in this family and the one with the greatest entropy. We present a fast root-finding algorithm that can be used to calculate the Buchen-Kelly density, and give upper boundaries for three different discrepancies that can be used as convergence criteria. Given the call prices, arbitrage-free digital prices at the same strikes can only move within upper and lower boundaries given by left and right call spreads. As the number of call prices increases, these bounds become tighter, and we give two examples where the densities converge to the Buchen-Kelly density in the sense of relative entropy when we use centered call spreads as proxies for digital prices. As pointed out by Breeden and Litzenberger, in the limit a continuous set of call prices completely determines the density.