[PDF] Implied Risk Neutral Probability Density Functions From Option Prices eBook

Author : Chen Wang
Publisher :
Page : 27 pages
File Size : 27,62 MB
Release : 2014
Category :
ISBN :

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This paper applies classic linear inverse theory to the estimation of the implied risk neutral probability density function (PDF) from option prices. To overcome non-uniqueness and instability inherent in the option inverse problem, smoothness requirement for the shape of a PDF and an initial model are introduced by a penalty function. Positivity constraints are included as a hard bond on the PDF values. Then the option inverse problem becomes a non-negative least-squares problem which can be solved by the classic methods such as the non-negative least squares program of Lawson and Hanson (1974). The best solution is not the solution that gives best fit, but the solution that gives the optimal trade-off between the goodness of fit and smoothness of the estimated risk natural PDF. The proposed inversion technique is compared to the models of Black-Scholes (BS), a mixture of two lognormals (MLN), Jarrow and Rudd's Edgeworth expansion (JR), and jump diffusion (JD) for the estimation of the PDF from the option prices associated with the September 2007 NYMEX natural gas futures. It is found that the inversion technique not only gives best goodness of fit, but also the significantly better model resolution. BS, JD and MLN models basically cannot resolve the densities far away from the strikes where option prices are observed and can resolve long wavelength features of the densities inside the strikes where option prices are observed. On the other hand, the inversion model can resolve not only the significant details of the densities inside the strikes where option prices are observed, but also the long wavelength features of the densities away from the strikes where option prices are observed. The empirical study for the last three months of the September 2007 futures contract shows that the shapes of the estimated PDFs become more symmetric as the futures contract is closer to the expiration date. The dispersion of the estimated PDFs decreases with decreasing the time to expire, indicating the resolution of uncertainty with time.

Author : Mr.Kevin C. Cheng
Publisher : International Monetary Fund
Page : 33 pages
File Size : 39,76 MB
Release : 2010-08-01
Category : Business & Economics
ISBN : 1455202150

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Building on the widely-used double-lognormal approach by Bahra (1997), this paper presents a multi-lognormal approach with restrictions to extract risk-neutral probability density functions (RNPs) for various asset classes. The contributions are twofold: first, on the technical side, the paper proposes useful transformation/restrictions to Bahra’s original formulation for achieving economically sensible outcomes. In addition, the paper compares the statistical properties of the estimated RNPs among major asset classes, including commodities, the S&P 500, the dollar/euro exchange rate, and the US 10-year Treasury Note. Finally, a Monte Carlo study suggests that the multi-lognormal approach outperforms the double-lognormal approach.

Author : Martin Mandler
Publisher :
Page : 24 pages
File Size : 14,45 MB
Release : 2002
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ISBN :

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In recent years different techniques to uncover the information on market expectations implicit in option prices have been developed. This paper proposes an approach to highlight statistically significant changes in risk-neutral probability density functions by comparing the distributional characteristics of statistics derived from risk-neutral densities to those of a benchmark sample. In an application we extract risk-neutral probability density functions from LIFFE-Euribor futures options and look for characteristic differences in market expectations related to meetings of the Governing Council of the ECB.

Author : Rupert De Vincent-Humphreys
Publisher :
Page : 39 pages
File Size : 39,46 MB
Release : 2012
Category : Options (Finance)
ISBN :

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"The prices of derivatives contracts can be used to estimate 'risk-neutral' probability density functions that give an indication of the weight investors place on different future prices of their underlying assets, were they risk-neutral. In the likely case that investors are risk-averse, this leads to differences between the risk-neutral probability density and the actual distribution of prices. But if this difference displays a systematic pattern over time, it may be exploited to transform the risk-neutral density into a 'real-world' density that better reflect agents' actual expectations. This work offers a methodology for performing this transformation. The resulting real-world densities may better represent market participants' views of future prices, and so offer an enhanced means of quantifying the uncertainty around financial variables. Comparison with their risk-neutral equivalents may also reveal new and useful information as to how attitudes towards risk are affecting pricing."--Abstract.

Author : Guillaume Leduc
Publisher :
Page : 17 pages
File Size : 18,23 MB
Release : 2017
Category :
ISBN :

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Risk neutral densities recovered from option prices can be used to infer market participantsņ expectations of future stock returns and are a vital tool for pricing illiquid exotic options. Although there is a broad literature on the subject, most studies do not address the likelihood of default. To fill this gap, in this paper we develop a novel method to retrieve the risk neutral probability density function from call options written on a defaultable asset. The primary advantage of the method is that default probabilities inferred by the model can be analytically expressed and, if available, can be incorporated as an input in a ፟lexible, robust and easily implementable manner.