EUROPEAN
CENTRAL
BANK
WO R K I N G PA P E R S E R I E S
WORKING PAPER NO. 198
EXTRACTING RISK NEUTRAL
PROBABILITY DENSITIES BY
FITTING IMPLIED VOLATILITY
SMILES: SOME METHODOLOGICAL
POINTS AND AN APPLICATION
TO THE 3M EURIBOR FUTURES
OPTION PRICES
BY ALLAN BODSKOV
ANDERSEN
/
AND TOM WAGENER
December 2002
EUROPEAN
CENTRAL
BANK
WO R K I N G PA P E R S E R I E S
WORKING PAPER NO. 198
EXTRACTING RISK NEUTRAL
PROBABILITY DENSITIES BY
FITTING IMPLIED VOLATILITY
SMILES: SOME METHODOLOGICAL
POINTS AND AN APPLICATION
TO THE 3M EURIBOR FUTURES
OPTION PRICES1
BY ALLAN BODSKOV
ANDERSEN2
/
AND TOM WAGENER3
December 2002
1 The authors wish to thank seminar participants at Danmarks Nationalbank and the Aarhus School of Business, and participants in the
workshop “The measures and determinants of financial market uncertainty” at the European Central Bank premises 27-28 May 2002, for
helpful discussions and suggestions. Niels Christian Beier, Peter E. Storgaard, Nicki Sorensen,
Christian Olgaard,
and an anonymous referee
/
/
deserve special thank for comments to earlier drafts of the paper.The usual disclaimer applies. The opinions expressed herein are those of
the authors and do not necessarily represent those of the Danmarks Nationalbank nor of the European Central Bank.
2 Danmarks Nationalbank, Havnegade 5, DK-1093 Copenhagen K, email: [email protected], corresponding author
3 Danmarks Nationalbank, Havengade 5, DK-1093 Copenhagen K, email: [email protected].
© European Central Bank, 2002
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All rights reserved.
Reproduction for educational and non-commercial purposes is permitted provided that the source is acknowledged.
The views expressed in this paper do not necessarily reflect those of the European Central Bank.
ISSN 1561-0810
Contents
Abstract
4
Non-technical summary
5
1
Introduction
7
2
The common foundation of the methods
9
3
Extracting RNDs - methodologies
10
4
Fitting the implied Black-Scholes volatility smile
4.1 Specifying the implied volatility function
4.2 Estimation of the spline
12
15
18
5
Empirical applications
5.1 Application 1: constant horizon RNDs
5.1.1 Data
5.1.2 Results
5.2 Application 2: the market reaction to the 11 September attacks
20
21
24
24
26
6
Concluding remarks
26
References
28
Appendix A
31
Figures
32
European Central Bank working paper series
43
ECB • Working Paper No 198 • December 2002
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