Extracting expectations from currency option prices: a comparison of methods

This paper compares the goodness-of-fit and the stability of six methods used to extract risk-neutral probability density functions from currency option prices. We first compare five existing methods commonly employed to recover risk-neutral density functions from option prices. Specifically, we compare the methods introduced by Shimko (1993), Madan and Milne (1994), Malz (1996), Melick and Thomas (1997) and Bliss and Panigirtzoglou (2002). In addition, we propose a new method based on the piecewise cubic Hermite interpolation of the implied volatility function. We use data on 12 emerging market currencies against the US dollar and find that the piecewise cubic Hermite interpolation method is by far the method with the best accuracy in fitting observed option prices. We also find that there is a relative tradeoff between the goodness-of-fit and the stability of the methods. Thus, methods which have a better accuracy in fitting observed option prices appear to be more sensitive to option pricing errors, while the most stable methods have a fairly disappointing fitting. However, for the first two PDF moments as well as the quartiles of the risk-neutral distributions we find that the estimates do not differ significantly across methods. This suggests that there is a large scope for selection between these methods without essentially sacrificing the accuracy of the analysis. Nonetheless, depending on the particular use of these PDFs, some methods may be more suitable than othersRisk-neutral probability density functions, option pricing, exchange rate expectations

The forecasting performance of German stock option densities

In this paper the authors estimate risk-neutral densities (RND) for the largest euro-area stock market (the index of which is the German DAX), reporting their statistical properties, and evaluating their forecasting performance. The authors have applied an innovative test procedure to a new, rich, and accurate data set. They have two main results. First, They have recorded strong negative skewness in the densities. Second, they find evidence for a significant difference between the actual density and the risk-neutral density, leading to the conclusion that market participants were surprised by the extent of both the rise and the fall of the DAX.Stock market - Germany ; Stock options

Non-Parametric Extraction of Implied Asset Price Distributions

Extracting the risk neutral density (RND) function from option prices is well defined in principle, but is very sensitive to errors in practice. For risk management, knowledge of the entire RND provides more information for Value-at-Risk (VaR) calculations than implied volatility alone [1]. Typically, RNDs are deduced from option prices by making a distributional assumption, or relying on implied volatility [2]. We present a fully non-parametric method for extracting RNDs from observed option prices. The aim is to obtain a continuous, smooth, monotonic, and convex pricing function that is twice differentiable. Thus, irregularities such as negative probabilities that afflict many existing RND estimation techniques are reduced. Our method employs neural networks to obtain a smoothed pricing function, and a central finite difference approximation to the second derivative to extract the required gradients. This novel technique was successfully applied to a large set of FTSE 100 daily European exercise (ESX) put options data and as an Ansatz to the corresponding set of American exercise (SEI) put options. The results of paired t-tests showed significant differences between RNDs extracted from ESX and SEI option data, reflecting the distorting impact of early exercise possibility for the latter. In particular, the results for skewness and kurtosis suggested different shapes for the RNDs implied by the two types of put options. However, both ESX and SEI data gave an unbiased estimate of the realised FTSE 100 closing prices on the options' expiration date. We confirmed that estimates of volatility from the RNDs of both types of option were biased estimates of the realised volatility at expiration, but less so than the LIFFE tabulated at-the-money implied volatility.Comment: Paper based on Application of Physics in Financial Analysis,APFA5, Conference Presentation, Torino, Italy. 11.5 Page

Testing the Forecasting Performance of Ibex 35 Option-implied Risk-neutral Densities

Published also as: Documento de Trabajo Banco de España 0504/2005.risk-neutral densities, forecasting performance

Removing Maturity Effects of Implied Risk Neutral Densities and Related Statistics

When studying a time series of implied Risk Neutral Densities (RNDs) or other implied statistics, one is faced with the problem of maturity dependence, given that option contracts have a fixed expiry date. Therefore, estimates from consecutive days are not directly comparable. Further, we can only obtain implied RNDs for a limited set of expiration dates. In this paper we introduce two new methods to overcome the time to maturity problem. First, we propose an alternative method for calculating constant time horizon Economic Value at Risk (EVaR), which is much simpler than the method currently being used at the Bank of England. Our method is based on an empirical scaling law for the quantiles in a log-log plot, and thus, we are able to interpolate and extrapolate the EVaR for any time horizon. The second method is based on an RND surface constructed across strikes and maturities, which enables us to obtain RNDs for any time horizon. Removing the maturity dependence of implied RNDs and related statistics is useful in many applications, such as in (i) the construction of implied volatility indices like the VIX, (ii) the assessment of market uncertainty by central banks (iii) time series analysis of EVaR, or (iv) event studies.

The Forecasting Performance of German Stock Option Densities

In this paper we will be estimating risk-neutral densities (RND) for the largest euro area stock market (the index of which is the German DAX), reporting their statistical properties, and evaluating their forecasting performance. We have applied an innovative test procedure to a new, rich, and accurate data set. We have two main results. First, we have recorded strong negative skewness in the densities. Second, we find evidence for significant differences between the actual density and the risk-neutral density, leading to the conclusion that market participants were surprised by the extent of both the rise and the fall of the DAX. -- In dieser Arbeit werden "risikoneutrale" Dichtefunktionen über künftige DAXIndexstände aus täglich beobachteten Preisen europäischer Kauf- und Verkaufsoptionen mit verschiedenen Restlaufzeiten abgeleitet. Das hierbei verwendete Berechnungsverfahren beruht auf der Mischung von zwei Log-Normalverteilung, bei dem fünf Parameter (der Mischungsparameter, zwei Mittelwerte und zwei Standardabweichungen) so bestimmt werden, dass der quadratische Abstand zwischen beobachteten und impliziten Optionspreisen minimal ist. Die Preisnotierungen für die Derivative werden der Eurex entnommen und der Untersuchungszeitraum erstreckt sich von Dezember 1995 bis Mai 2002, also sowohl über die Boom- als auch über die Niedergangsphase des DAX. Die Vorhersagehorizonte der Dichten sind auf Grund der Datenlage auf sechs bis acht Wochen begrenzt. Die Vorhersagegüte dieser Dichten wird über verschiedene neuartige statistische Evaluierungsverfahren abgeschätzt. Im Ergebnis stellt sich folgendes heraus: Erstens: Die Dichten weisen im Durchschnitt eine negative Schiefe (negatives drittes Moment) auf, so dass das linke Ende der Dichte "dicker" ist als das rechte und die Marktteilnehmer somit einen bestimmten prozentualen Kursverlust als wahrscheinlicher einschätzten als einen Kursgewinn. Zweitens: Die Evaluierungstests für die Dichten machen deutlich, dass die tatsächlichen Dichten im Mittel deutlich von den risikoneutralen Dichten abweichen. Dabei kann ausgeschlossen werden, dass es sich lediglich um einen "Mittelwert"fehler handelt. Vielmehr scheinen die Markteilnehmer sowohl in der Aufschwung- als auch in der Abschwungphase von den Kursbewegungen des DAX überrascht worden zu.option prices,risk-neutral density,density evaluation,overlapping data

Tracing the general structure of Galactic molecular clouds using Planck data: I. The Perseus region as a test case

We present an analysis of probability distribution functions (pdfs) of column density in different zones of the star-forming region Perseus and its diffuse environment based on the map of dust opacity at 353 GHz available from the Planck archive. The pdf shape can be fitted by a combination of a lognormal function and an extended power-law tail at high densities, in zones centred at the molecular cloud Perseus. A linear combination of several lognormals fits very well the pdf in rings surrounding the cloud or in zones of its diffuse neighbourhood. The slope of the mean density scaling law $\langle\rho\rangle_L \propto L^\alpha$ is steep ($\alpha=-1.93$) in the former case and rather shallow ($\alpha=-0.77\pm0.11$) in the rings delineated around the cloud. We interpret these findings as signatures of two distinct physical regimes: i) a gravoturbulent one which is characterized by nearly linear scaling of mass and practical lack of velocity scaling; and ii) a predominantly turbulent one which is best described by steep velocity scaling and by invariant for compressible turbulence $\langle\rho\rangle_L u_L^3/L$, describing a scale-independent flux of the kinetic energy per unit volume through turbulent cascade. The gravoturbulent spatial domain can be identified with the molecular cloud Perseus while a relatively sharp transition to predominantly turbulent regime occurs in its vicinity.Comment: Accepted for publication in MNRAS; 16 pages with Appendix, 15 figure