[PDF] Inversion Of Option Prices For Implied Risk Neutral Probability Density Functions eBook

Author : Chen Wang
Publisher :
Page : 27 pages
File Size : 30,86 MB
Release : 2014
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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 : José Campa
Publisher :
Page : 64 pages
File Size : 20,29 MB
Release : 1997
Category : Foreign exchange options
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This paper uses a rich new data set of option prices on the dollar-mark, dollar-yen, and key EMS cross-rates to extract the entire risk-neutral probability density function (pdf) over horizons of one and three months. We compare three alternative smoothing methods--cubic splines, an implied binomial tree (trimmed and untrimmed), and a mixture of lognormals--for transforming option data into the pdf. Despite their methodological ifferences, the three approaches lead to a similar pdf distinct from the lognormal benchmark, and usually characterized by skewness and leptokurtosis. We then document a striking positive correlation between skewness in these pdfs and the spot rate. The stronger a currency the more expectations are skewed towards a further appreciation of that currency. We interpret this finding as a rejection that these exchange rates evolve as a martingale, or that they follow a credible target zone, explicit or implicit. Instead, this this positive correlation is consistent with target zones with endogenous realignment risk. We discuss two interpretations of our results on skewness: when a currency is stronger, the actual probability of further large appreciation is higher, or because of risk, such states are valued more highly.

Author : Fabio Fornari
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Page : 0 pages
File Size : 36,87 MB
Release : 2006
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The paper contributes to the stochastic volatility literature by developing simulation schemes for the conditional distributions of the price of long term bonds and their variability based on non-standard distributional assumptions and volatility concepts; itillustrates the potential value of the information contained in the prices of options on long and short term lira interest rate futures for the conduct of monetary policy in Italy, at times when significant regime shifts have occurred. Risk-neutral probability density functions (PDFs) of interest rates are estimated for several episodes since 1992, using an extended version of the model developed by Söderlind and Svensson (1997), in which the true PDF of an asset price is approximated by a parametric mixture of gaussian distributions with displaced means and mean-reverting deterministic volatilities. This approach generalizes the familiar Black and Scholes option pricing model by letting the returns of the underlying asset follow a mean-reverting stochastic process governed by the existence of two regimes. The parameters of this functional form are obtained by minimising the squared deviations between predicted and true option prices. The ability of the model to fit observed option prices seems encouraging; a significant degree of skewness (so-called risk-reversal), large changes over time and fatness of the tails (which gives rise to the volatility smile effect) are the elements which characterize the estimated PDFs). The analysis of the information conveyed by PDFs of future interest rates begins in the wake of the 1992 EMS crisis, when a large negative skewness emerged in the distribution of lira-denominated bond futures, suggesting that the market factored in the possibility of a sharp decline in prices, triggered by a monetary tightening aimed at defending the exchange rate peg and by the mounting inflationary risk induced by a large devaluation. We subsequently examine a recent sequence of monetary easings, which provide an interesting variety of market participants' reactions. The official rate reduction of July 1996, which followed a sequence of monetary tightenings in 1994-5, appears to have been discounted by the market, since the estimated PDFs of bond and 3-month T-bill futures prices did not change and an almost gaussian shape was maintained. The subsequent reductions - October 1996 and January 1997 - showed instead a different kind of market reaction, especially for long rate PDFs which returned from negative skewness and fairly high kurtosis, observed before the policy move, to normality, with narrower interquartile range and less kurtosis after the interest rate reduction.