476 research outputs found
B-spline techniques for volatility modeling
This paper is devoted to the application of B-splines to volatility modeling,
specifically the calibration of the leverage function in stochastic local
volatility models and the parameterization of an arbitrage-free implied
volatility surface calibrated to sparse option data. We use an extension of
classical B-splines obtained by including basis functions with infinite
support. We first come back to the application of shape-constrained B-splines
to the estimation of conditional expectations, not merely from a scatter plot
but also from the given marginal distributions. An application is the Monte
Carlo calibration of stochastic local volatility models by Markov projection.
Then we present a new technique for the calibration of an implied volatility
surface to sparse option data. We use a B-spline parameterization of the
Radon-Nikodym derivative of the underlying's risk-neutral probability density
with respect to a roughly calibrated base model. We show that this method
provides smooth arbitrage-free implied volatility surfaces. Finally, we sketch
a Galerkin method with B-spline finite elements to the solution of the partial
differential equation satisfied by the Radon-Nikodym derivative.Comment: 25 page
A quantitative mirror on the Euribor market using implied probability density functions
This paper presents a set of probability density functions for Euribor outturns in three monthsâ time, estimated from the prices of options on Euribor futures. It is the first official and freely available dataset to span the complete history of Euribor futures options, thus comprising over ten years of daily data, from 13 January 1999 onwards. Time series of the statistical moments of these option-implied probability density functions are documented until April 2010. Particular attention is given to how these probability density functions, and their associated summary statistics, reacted to the unfolding financial crisis between 2007 and 2009. In doing so, it shows how option-implied probability density functions could be used to contribute to monetary policy and financial stability analysis. JEL Classification: C13, C14, G12, G13financial, financial market, options, probability density functions
Modelling the implied probability of stock market movements
In this paper we study risk-neutral densities (RNDs) for the German stock market. The use of option prices allows us to quantify the risk-neutral probabilities of various levels of the DAX index. For the period from December 1995 to November 2001, we implement the mixture of log-normals model and a volatility-smoothing method. We discuss the time series behaviour of the implied PDFs and we examine the relations between the moments and observable factors such as macroeconomic variables, the US stock markets and credit risk. We find that the risk-neutral densities exhibit pronounced negative skewness. Our second main observation is a significant spillover of volatility, as the implied volatility and kurtosis of the DAX RND are mostly driven by the volatility of US stock prices. JEL Classification: C22, C51, G13, G15Option prices, risk-neutral density, spillover, Volatility
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Arithmetic variance swaps
Biases in standard variance swap rates can induce substantial deviations below market rates. Defining realised variance as the sum of squared price (not log-price) changes yields an `arithmetic' variance swap with no such biases. Its fair value has advantages over the standard variance swap rate: no discrete-monitoring or jump biases; and the same value applies for any monitoring frequency, even irregular monitoring and to any underlying, including those taking zero or negative values. We derive the fair-value for the arithmetic variance swap and compare with the standard variance swap rate by: analysing errors introduced by interpolation and integration techniques; numerical experiments for approximation accuracy; and using 23 years of FTSE 100 options data to explore the empirical properties of arithmetic variance (and higher-moment) swaps. The FTSE 100 variance risk has a strong negative correlation with the implied third moment, which can be captured using a higher-moment arithmetic swap
Calibration of the Hobson&Rogers model: empirical tests
The path-dependent volatility model by Hobson and Rogers is considered. It is known that this model can potentially reproduce the observed smile and skew patterns of different directions, while preserving the completeness of the market. In order to quantitatively investigate the pricing performance of the model a calibration procedure is here derived. Numerical results based on S&P500 option prices give evidence of the effectiveness of the model.
Option Pricing in an Imperfect World
In a model with no given probability measure, we consider asset pricing in
the presence of frictions and other imperfections and characterize the property
of coherent pricing, a notion related to (but much weaker than) the no
arbitrage property. We show that prices are coherent if and only if the set of
pricing measures is non empty, i.e. if pricing by expectation is possible. We
then obtain a decomposition of coherent prices highlighting the role of
bubbles. eventually we show that under very weak conditions the coherent
pricing of options allows for a very clear representation from which it is
possible, as in the original work of Breeden and Litzenberger, to extract the
implied probability. Eventually we test this conclusion empirically via a new
non parametric approach.Comment: The paper has been withdrawn because in the newer version it was
split into two different papers, each of which have been uploaded into Arxi
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