Fengler Semiparametric Modeling of Implied Volatility

2005
ISBN: 978-3-540-30591-0
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 224 Seiten

Reihe: Springer Finance

Yet that weakness is also its greatest strength. People like the model because they can easily understand its assumptions. The model is often good as a ?rst approximation, and if you can see the holes in the assumptions you can use the model in more sophisticated ways. Black (1992) Expected volatility as a measure of risk involved in economic decision making isakeyingredientinmodern?nancialtheory:therational,risk-averseinvestor will seek to balance the tradeo? between the risk he bears and the return he expects. The more volatile the asset is, i.e. the more it is prone to exc- sive price ?uctuations, the higher will be the expected premium he demands. Markowitz (1959), followed by Sharpe (1964) and Lintner (1965), were among the ?rst to quantify the idea of the simple equation ‘more risk means higher return’ in terms of equilibrium models. Since then, the analysis of volatility and price ?uctuations has sparked a vast literature in theoretical and quan- tative ?nance that re?nes and extends these early models. As the most recent climax of this story, one may see the Nobel prize in Economics granted to Robert Engle in 2003 for his path-breaking work on modeling time-dependent volatility.

Autoren/Hrsg.

Fengler, Matthias R.

Inhaltsverzeichnis

1;Acknowledgements;8
2;Frequently Used Notation;9
3;Contents;13
4;1 Introduction;16
5;2 The Implied Volatility Surface;24
5.1;2.1 The Black-Scholes Model;24
5.2;2.2 The Self-Financing Replication Strategy;26
5.3;2.3 Risk Neutral Pricing;27
5.4;2.4 The BS Formula and the Greeks;29
5.5;2.5 The IV Smile;34
5.6;2.6 Static Properties of the Smile Function;42
5.6.1;2.6.1 Bounds on the Slope;42
5.6.2;2.6.2 Large and Small Strike Behavior;43
5.7;2.7 General Regularities of the IVS;45
5.7.1;2.7.1 Static Stylized Facts;45
5.7.2;2.7.2 DAX Index IV between 1995 and 2001;48
5.8;2.8 Relaxing the Constant Volatility Case;49
5.8.1;2.8.1 Deterministic Volatility;50
5.8.2;2.8.2 Stochastic Volatility;51
5.9;2.9 Challenges Arising from the Smile;55
5.9.1;2.9.1 Hedging and Risk Management;55
5.9.2;2.9.2 Pricing;57
5.10;2.10 IV as Predictor of Realized Volatility;57
5.11;2.11 Why Do We Smile?;58
5.12;2.12 Summary;61
6;3 Smile Consistent Volatility Models;62
6.1;3.1 Introduction;62
6.2;3.2 The Theory of Local Volatility;64
6.3;3.3 Backing the LVS Out of Observed Option Prices;66
6.4;3.4 The dual PDE Approach to Local Volatility;69
6.5;3.5 From the IVS to the LVS;70
6.6;3.6 Asymptotic Relations Between Implied and Local Volatility;75
6.7;3.7 The Two-Times-IV-Slope Rule for Local Volatility;77
6.8;3.8 The K-Strike and T -Maturity Forward Risk-Adjusted Measure;79
6.9;3.9 Model-Free (Implied) Volatility Forecasts;81
6.10;3.10 Local Volatility Models;82
6.10.1;3.10.1 Deterministic Implied Trees;82
6.10.2;3.10.2 Stochastic Implied Trees;95
6.10.3;3.10.3 Reconstructing the LVS;99
6.11;3.11 Excellent Fit, but...: the Delta Problem;103
6.12;3.12 Stochastic IV Models;106
6.13;3.13 Summary;109
7;4 Smoothing Techniques;112
7.1;4.1 Introduction;112
7.2;4.2 Nadaraya-Watson Smoothing;114
7.2.1;4.2.1 Kernel Functions;114
7.2.2;4.2.2 The Nadaraya-Watson Estimator;115
7.3;4.3 Local Polynomial Smoothing;117
7.4;4.4 Bandwidth Selection;119
7.4.1;4.4.1 Theoretical Framework;119
7.4.2;4.4.2 Bandwidth Choice in Practice;121
7.5;4.5 Least Squares Kernel Smoothing;130
7.5.1;4.5.1 The LSK Estimator of the IVS;130
7.5.2;4.5.2 Application of the LSK Estimator;132
7.6;4.6 Summary;138
8;5 Dimension-Reduced Modeling;140
8.1;5.1 Introduction;140
8.2;5.2 Common Principal Component Analysis;143
8.2.1;5.2.1 The Family of CPC Models;143
8.2.2;5.2.2 Estimating Common Eigenstructures;146
8.2.3;5.2.3 Stability Tests for Eigenvalues and Eigenvectors;149
8.2.4;5.2.4 CPC Model Selection;153
8.2.5;5.2.5 Empirical Results;154
8.3;5.3 Functional Data Analysis;170
8.3.1;5.3.1 Basic Set-Up of FPCA;171
8.3.2;5.3.2 Computing FPCs;172
8.4;5.4 Semiparametric Factor Models;175
8.4.1;5.4.1 The Model;177
8.4.2;5.4.2 Norming of the Estimates;181
8.4.3;5.4.3 Choice of Model Parameters;182
8.4.4;5.4.4 Empirical Analysis;186
8.4.5;5.4.5 Assessing Prediction Performance;197
8.5;5.5 Summary;199
9;6 Conclusion and Outlook;202
10;A Description and Preparation of the IV Data;204
10.1;A.1 Preliminaries;204
10.2;A.2 Data Correction Scheme;205
11;B Some Results from Stochastic Calculus;210
12;C Proofs of the Results on the LSK IV Estimator;216
12.1;C.1 Proof of Consistency;216
12.2;C.2 Proof of Asymptotic Normality;218
13;References;222
14;Index;236

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5.1 Introduction

The IVS is a complex, high-dimensional random object. In building a model, it is thus desirable to have a low-dimensional representation of the IVS. This aim can be achieved by employing dimension reduction techniques. Generally it is found that two or three factors with appealing .nancial interpretations are su.cient to capture more than 90% of the IVS dynamics. This implies for instance for a scenario analysis in risk-management that only a parsimonious model needs to be implemented to study the vega-sensitivity of an option portfolio, Fengler et al. (2002b). This section will give a general overview on dimension reduction techniques in the context of IVS modeling. We will consider techniques from multivariate statistics and methods from functional data analysis. Sections 5.2 and 5.3 will provide an in-depth treatment of the CPC and the semiparametric factor model of the IVS together with an extensive empirical analysis of the German DAX index data.

In multivariate analysis, the most prominent technique for dimension reduction is principal component analysis (PCA). The idea is to seek linear combinations of the original observations, so called principal components (PCs) that inherit as much information as possible from the original data. In PCA, this means to look for standardized linear combinations with maximum variance. The approach appears to be sensible in an analysis of the IVS dynamics, since a large variance separates out systematic from idiosyncratic shocks that drive the surface. As a nice byproduct, the structure of the linear combinations reveals relationships among the variables that are not apparent in the original data. This helps understand the nature of the interdependence between di.erent regions in the IVS.

In .nance, PCA is a well-established tool in the analysis of the term structure of interest rates, see Gouri´eroux et al. (1997) or Rebonato (1998) for textbook treatments: PCA is applied to a multiple time series of interest rates (or forward rates) of various maturities that is recovered from the term structure of interest rates. Typically, a small number of factors is found to represent the dynamic variations of the term structure of interest rates. The studies of Bliss (1997), Golub and Tilman (1997), Ni.keer et al. (2000), and Molgedey and Galic (2001) are examples of this kind of literature.

This approach does not immediately carry over to the analysis of IVs due to the surface structure. Consequently, in analogy to the interest rate case, empirical work .rst analyzes the term structure of IVs of ATM options, only, Zhu and Avellaneda (1997) and Fengler et al. (2002b). Alternatively, one smile at one given maturity can be analyzed within the PCA framework, Alexander (2001b). Skiadopoulos et al. (1999) group IVs into maturity buckets, average the IVs of the options, whose maturities fall into them, and apply a PCA to each bucket covariance matrix separately. A good overview of these methods can be found in Alexander (2001a).