Inside Volatility Filtering E-Book

Alireza Javaheri

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Wiley Finance Editions
• Sprache: Englisch

Seitenzahl: 313

Veröffentlichungsjahr: 2015

Beschreibung

A new, more accurate take on the classical approach to volatility evaluation Inside Volatility Filtering presents a new approach to volatility estimation, using financial econometrics based on a more accurate estimation of the hidden state. Based on the idea of "filtering", this book lays out a two-step framework involving a Chapman-Kolmogorov prior distribution followed by Bayesian posterior distribution to develop a robust estimation based on all available information. This new second edition includes guidance toward basing estimations on historic option prices instead of stocks, as well as Wiener Chaos Expansions and other spectral approaches. The author's statistical trading strategy has been expanded with more in-depth discussion, and the companion website offers new topical insight, additional models, and extra charts that delve into the profitability of applied model calibration. You'll find a more precise approach to the classical time series and financial econometrics evaluation, with expert advice on turning data into profit. Financial markets do not always behave according to a normal bell curve. Skewness creates uncertainty and surprises, and tarnishes trading performance, but it's not going away. This book shows traders how to work with skewness: how to predict it, estimate its impact, and determine whether the data is presenting a warning to stay away or an opportunity for profit. * Base volatility estimations on more accurate data * Integrate past observation with Bayesian probability * Exploit posterior distribution of the hidden state for optimal estimation * Boost trade profitability by utilizing "skewness" opportunities Wall Street is constantly searching for volatility assessment methods that will make their models more accurate, but precise handling of skewness is the key to true accuracy. Inside Volatility Filtering shows you a better way to approach non-normal distributions for more accurate volatility estimation.

Leseprobe

Table of Contents

Title Page

Copyright

Foreword

Acknowledgments (Second Edition)

Acknowledgments (First Edition)

Introduction (Second Edition)

Introduction (First Edition)

Summary

Contributions and Further Research

Data and Programs

Chapter 1: The Volatility Problem

Introduction

The Stock Market

The Derivatives Market

Jump Diffusion and Level-Dependent Volatility

Local Volatility

Stochastic Volatility

The Pricing PDE under Stochastic Volatility

The Generalized Fourier Transform

The Mixing Solution

The Long-Term Asymptotic Case

Local Volatility Stochastic Volatility Models

Stochastic Implied Volatility

Joint SPX and VIX Dynamics

Pure-Jump Models

Chapter 2: The Inference Problem

Introduction

Using Option Prices

Using Stock Prices

Recapitulation

Chapter 3: The Consistency Problem

Introduction

The Consistency Test

The “Peso” Theory

Trading Strategies

Non-Gaussian Case

A Word of Caution

Foreign Exchange, Fixed Income, and Other Markets

Chapter 4: The Quality Problem

Introduction

An Exact Solution?

Quality of Observations

Conclusion

Bibliography

Index

End User License Agreement

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Guide

Cover

Table of Contents

Foreword

Introduction (Second Edition)

Begin Reading

List of Illustrations

Chapter 1: The Volatility Problem

Figure 1.1 Comparing the volatility-of-volatility series expansion with the Monte-Carlo mixing model. The graph shows the price difference . We are taking and . This example was used in the original Heston paper.

Figure 1.2 Comparing the volatility-of-volatility series expansion with the Monte-Carlo mixing model. The graph shows the price difference . We are taking and . This example was used in the original Heston paper.

Figure 1.3 Comparing the volatility-of-volatility series expansion with the Monte-Carlo mixing model. The graph shows the price difference . We are taking and . This example was used in the original Heston paper.

Figure 1.4 The Gamma Cumulative Distribution Function for various values of the parameter . The implementation is based on code available in “Numerical Recipes in C.”

Chapter 2: The Inference Problem

Figure 2.1 A simple example for the joint filter. The convergence toward the constant parameter happens after a few iterations.

Figure 2.2 Joint EKF estimation for the parameter Prices were simulated with . The convergence remains mediocre. We shall explain this in the following section.

Figure 2.3 Joint EKF estimation for the parameter Prices were simulated with . The convergence remains mediocre. We shall explain this in the following section.

Figure 2.4 Joint EKF estimation for the parameter Prices were simulated with . The convergence remains mediocre. We shall explain this in the following section.

Figure 2.5 Joint EKF estimation for the parameter Prices were simulated with . The convergence remains mediocre. We shall explain this in the following section.

Figure 2.6 Joint EKF estimation for the parameter applied to the Heston model as well as to a modified model where the noise is reduced by a factor . As we can see, the convergence for the modified model is improved dramatically. This justifies our comments on large observation error.

Figure 2.7 SPX historic data (1996–2001) are filtered via EKF and UKF. The results are very close; however, the estimated parameters differ. Indeed, we find for the EKF and for the UKF. This might be due to the relative insensitivity of the filters to the parameter-set or the non-uniqueness of the optimal parameter-set. We shall explain this low sensitivity in more detail.

Figure 2.8 EKF and UKF absolute filtering-errors for the same time-series. As we can see, there is no clear superiority of one algorithm over the other.

Figure 2.9 Histogram for filtered data via EKF versus the normal distribution. The residuals are fairly normal.

Figure 2.10 Variograms for filtered data via EKF and UKF. The input corresponds to a sequence of Independent Gaussian Random-Variables. As we can see, the variograms are close to one.

Figure 2.11 Variograms for filtered data via EKF and UKF. The input corresponds to a Brownian Motion. As we can see, the variograms are close to .

Figure 2.12 Filtering errors: Extended Kalman filter and extended particle filter are applied to the one-dimensional Heston model. The PF has better performance.

Figure 2.13 Filtering errors: All filters are applied to the one-dimensional Heston model. The PFs have better performance.

Figure 2.14 Filters are applied to the one-dimensional Heston model. The time-series has a larger time-step . Naturally, the errors are larger than the case where .

Figure 2.15 EKF and GHF are applied to the one-dimensional Heston model. The time-series has a larger time-step . Naturally, the errors are larger than the case where .

Figure 2.16 The EPF without and with the Metropolis-Hastings step is applied to the one-dimensional Heston model. The time-series has a time-step . The improvement is hardly visible.

Figure 2.17 Simulated stock-price path via Heston using . This is an artificial time series following the Heston model.

Figure 2.18 has a good slope around =0.10.

Figure 2.21 is flat and irregular around .

Figure 2.22 via EKF for points. The true value is .

Figure 2.25 via EKF for points. The true value is .

Figure 2.26 Density for estimated from paths of length via EKF. The true value is . The sampling distribution is fairly unbiased, but is inefficient.

Figure 2.29 Density for estimated from paths of length via EKF. The true value is . The sampling distribution is inefficient and even has a bias.

Figure 2.30 Gibbs sampler for in . The true value is .

Figure 2.31 Gibbs sampler for in . The true value is .

Figure 2.32 Metropolis-Hastings algorithm for in . The true value is .

Figure 2.33 Metropolis-Hastings algorithm for in . The true value is .

Figure 2.34 Plots of the Incomplete Beta Function. Implementation is based on code from “Numerical Recipes in C.”

Figure 2.35 Comparison of EPF results for simulated and estimated jump-diffusion time-series. The filtered data match the real data fairly well.

Figure 2.36 The simulated arrival rates via and are quite different.

Figure 2.37 However, the simulated log stock prices are close.

Figure 2.38 he VGSA residuals histogram. The residuals are fairly normal.

Figure 2.39 The VGSA residuals variogram. The variogram is close to 1 as expected.

Figure 2.40 Simulation of VGG-based log stock prices with two different parameter-sets and . The observed time-series remain close.

Chapter 3: The Consistency Problem

Figure 3.1 Implied volatilities of close to ATM puts and calls as of January 2, 2002. Maturity is December 21, 2002, and Index at 1154.67 USDs. The bid-ask spread can clearly be observed.

Figure 3.2 The observations have little sensitivity to the volatility parameters. One-year simulation with , , . Cross-sectional uses and while time-series uses and . This is consistent with what we had seen previously.

Figure 3.3 The state has a great deal of sensitivity to the volatility parameters. One-year simulation with , , . Cross-sectional uses and while time-series uses and

Figure 3.4 The observations have a great deal of sensitivity to the drift parameters. One-year simulation with , , . Cross-sectional uses and while time-series uses and

Figure 3.5 The state has a great deal of sensitivity to the drift parameters. One-year simulation with , , . Cross-sectional uses and while time-series uses and .

Figure 3.6 Comparing SPX cross-sectional and time-series volatility smiles (with historic and ) as of January 2, 2002. The spot is at 1154.67 USD.

Figure 3.7 A generic example of a skewness strategy to take advantage of the undervaluation of the skew by options. This strategy could be improved by trading additional OTM puts and calls.

Figure 3.8 A generic example of a kurtosis strategy to take advantage of the overvaluation of the kurtosis by options. This strategy could be improved by trading additional puts and calls.

Figure 3.9 Historic spot level movements during the trade period.

Figure 3.10 Hedging PnL generated during the trade period. As we can see, losses occur on jumps.

Figure 3.11 GW (Grey Wolf Inc.) Historic prices (March 31, 2002, to March 31, 2003) show a high volatility-of-volatility but a weak stock-volatility correlation. The resulting negative skew is low.

Figure 3.12 The Historic GW (Grey Wolf Inc.) skew is low and not in agreement with the options prices. There is a skew trading opportunity here.

Figure 3.13 MSFT (Microsoft) historic prices (March 31, 2002, to March 31, 2003) show a high volatility-of-volatility and a strong negative stock-volatility correlation. The resulting negative skew is high.

Figure 3.14 The Historic MSFT (Microsoft) skew is high and in agreement with the options prices. There is no skew trading opportunity here.

Figure 3.15 NDX (Nasdaq) historic prices (March 31, 2002, to March 31, 2003) show a high volatility-of-volatility and a strong negative stock-volatility correlation. The resulting negative skew is high.

Figure 3.16 Historic NDX (Nasdaq) skew is high and in agreement with the options prices. There is no skew trading opportunity here.

Figure 3.17 Arrival rates for simulated SPX prices using and . We can see that they are quite different.

Figure 3.18 Gamma times for simulated SPX prices using and .

Figure 3.19 Log stock prices for simulated SPX prices using and . Unlike arrival rates, the spot prices are hard to distinguish. This is consistent with our previous observations.

List of Tables

Chapter 2: The Inference Problem

Table 2.1 The estimation is performed for SPX on 21, 2002, with Index 1079.88 USD for different maturities

Table 2.2 The true parameter-set used for data simulation

Table 2.3 The initial parameter-set used for the optimization process

Table 2.4 The optimal parameter-set . The estimation is performed individually for each parameter on the artificially generated time-series. Particle filters use simulations

Table 2.5 The Optimal EKF parameters and given a sample size . The true parameters are and . The initial values were and

Table 2.6 The true parameter-set used for data generation

Table 2.7 The initial parameter-set used for the optimization process

Table 2.8 Optimal EKF parameter-set for given a sample size . The four parameters are estimated jointly

Table 2.9 Optimal EKF parameter-set via the HRS approximation for given a sample size . The four parameters are estimated jointly

Table 2.10 The Optimal PF parameter-set for given a sample size . The four parameters are estimated jointly

Table 2.11 Real and optimal parameter-sets obtained via NGARCH MLE. The points were generated via the one-factor NGARCH with daily parameters

Table 2.12 Real and optimal parameter-sets obtained via NGARCH MLE as well as EKF. The points were generated via the two-factor GARCH Diffusion limit with annual parameters

Table 2.13 Optimal parameter-set for data points. The sampling is performed times a day and therefore the data-set corresponds to business days. The four parameters are estimated jointly

Table 2.14 Mean and (standard deviation) for the estimation of each parameter via EKF over paths of lengths and . The true values are

Table 2.15 MPE and RMSE for the VGSA model under a generic PF as well as the EPF

Chapter 3: The Consistency Problem

Table 3.1 Average Optimal Heston parameter-set (under the risk-neutral distribution) obtained via LSE applied to one-year SPX options on January 2002. Various strike-price sets were used

Table 3.2 Average Optimal Heston parameter-set (under the statistical distribution) obtained via filtered MLE applied to SPX between January 1992 and January 2004. Various filters were used in the MLE

Table 3.3 VGSA statistical parameters estimated via PF. The stock-drifts are and , respectively

Table 3.4 VGSA risk-neutral arrival-rate parameters estimated from Carr et al. [52]

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Inside Volatility Filtering

Secrets of the Skew

Second Edition

ALIREZA JAVAHERI

Copyright © 2015 by Alireza Javaheri. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.

The First Edition of this book was published by John Wiley and Sons in 2005 under the title Inside Volatility Arbitrage: The Secrets of Skewness.

Published simultaneously in Canada.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, [phone], fax [phone], or on the Web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, [phone], fax [phone], or online at http://www.wiley.com/go/permissions.

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Library of Congress Cataloging-in-Publication Data:

ISBN 978-1-118-94397-7 (cloth)

ISBN 978-1-118-94399-1 (epdf)

ISBN 978-1-118-94398-4 (epub)

Cover Design: Wiley

Cover Image: Alireza Javaheri

Foreword

The views represented herein are the author's own views and do not necessarily represent the views of Morgan Stanley or its affliates, and are not a product of Morgan Stanley research.

A revolution has been quietly brewing in quantitative finance for the last quarter century. In the 1980s and 1990s, quant groups were dominated by physicists and mathematicians well versed in sophisticated mathematical techniques related to physical phenomena such as heat flow and diffusion. Rivers of data flowed through the canyons of Wall Street, but little thought was given to capturing the data for the purpose of extracting the signal from the noise. The dominant worldview at the time viewed finance as primarily a social enterprise, rendering it fundamentally distinct from the physical world that the quants had left behind. I think that most quants of the time would have agreed with George Bernard Shaw, who said that:

Hegel was right when he said that the only thing we learn from history is that man never learns anything from history.

When an econometrician named Robert Litterman interviewed for a position at Goldman Sachs in 1986, the legendary Fischer Black began the interview by asking, “What makes you think that an econometrician has anything to contribute to Wall Street?”

I suspect we will never know how the candidate answered that question, but had the candidate been Mark Twain, the answer would surely have been that:

History doesn't repeat itself, but it does rhyme.

Six years after Black interviewed Litterman, they co-authored the Black–Litterman model that blends theoretical notions such as economic equilibrium with investor views guided by econometric time series models. Since then, econometricians and statisticians have enjoyed a steadily growing role in both practitioner and academic circles. In the May 2003 issue of Wilmott magazine, an article profiled past winners of the Nobel Memorial Prize in Economic Sciences. In a side note titled “A Winner You'd like to See,” the author of this book provided the following quote about an NYU professor named Robert Engle:

His work (with Bollerslev) on GARCH has made a real difference. Now, econometrics and quantitative finance are fully integrated thanks to him.… I think the magazine could boldly suggest his nomination to Bank of Sweden!

Half a year later, the Nobel committee appeared to have acted on Dr. Javaheri's advice, awarding the Nobel Memorial Prize in Economic Sciences jointly to Professor Engle and another econometrician. The stochastic volatility models that represent the major focus of this book were specifically cited by the Nobel Committee as the basis for Professor Engle's award. I believe that this watershed event changed the world of quantitative finance forever. Nowadays, few quants dispute the valuable insight that time series models bring to the difficult task of forecasting the future. At the Courant Institute of NYU where Dr. Javaheri and I co-teach a class, the master's program in Mathematical Finance has one class each semester devoted to financial econometrics. In particular, classes on machine learning and knowledge representation are routinely oversubscribed, alleviated only by the steady flow of graduates to industry. The success of the statistical approach to artificial intelligence has had a profound impact on financial activity, leading to steadily increasing automation of both knowledge accumulation and securities trading. While e-trading typically starts with cash instruments and vanilla securities, it is inevitable that it will eventually encompass trading activities that lean heavily on quantitative elements such as volatility trading. As a result, the second edition of this book serves its intended audience well, providing an up-to-date, comprehensive review of the application of filtering techniques to volatility forecasting. While the title of each chapter is framed as a problem, the contents of each chapter represent our best guess at the answer. Employing the advances that econometricians have made in the past quarter century, the fraction of variance explained is a truly impressive accomplishment.

Peter CarrGlobal Head of Market Modeling, Morgan StanleyExecutive Director, Masters in Math Finance Program, Courant Institute, New York University

Acknowledgments (Second Edition)

I'd like to thank the participants at the Global Derivatives 2012 and 2013 conferences for their valuable feedback. Many thanks go the students of the Baruch MFE program, where the first edition of the book was used for a Volatility Filtering course. Special thanks go to Peter Carr for his unique insights, as usual. Finally, I'd like to once again thank my wife, Firoozeh, and my children, Neda, Ariana, Alexander, and Kouros, for their patience and support.

Acknowledgments (First Edition)

This book is based on my PhD dissertation at Ecole des Mines de Paris. I would like to thank my advisor Alain Galli for his guidance and help. Many thanks to Margaret Armstrong and Delphine Lautier and the entire CERNA team for their support.

Special thanks to Yves Rouchaleau for helping make all this possible in the first place.

I would like to sincerely thank other committee members Marco Avellaneda, Lane Hughston, Piotr Karasinski, and Bernard Lapeyre for their comments and time.

I am grateful to Farshid Asl, Peter Carr, Raphael Douady, Robert Engle, Stephen Figlewski, Espen Haug, Ali Hirsa, Michael Johannes, Simon Julier, Alan Lewis, Dilip Madan, Vlad Piterbarg, David Wong, and the participants to ICBI 2003 and 2004 for all the interesting discussions and idea exchanges.

I am particularly indebted to Paul Wilmott for encouraging me to speak to Wiley and convert my dissertation into this book.

Finally, I'd like to thank my wife, Firoozeh, and my daughters, Neda and Ariana, for their patience and support.

Introduction (Second Edition)

The second edition of this book was written ten years after the first one. Needless to say, a great number of things happened during this period. Many readers and students provided useful feedback leading to corrections and improvements. For one thing, various people correctly pointed out that the original title Inside Volatility Arbitrage was quite misleading since the term arbitrage was used in a loose statistical manner.1 Hence, the change in name to Inside Volatility Filtering.

More to the point, the second edition benefited from various conference presentations I made, as well as courses I taught at the Courant Institute of Mathematics as well as at Baruch College.

A few new topics are introduced and organized in the following way:

The first chapter continues to be a survey of literature on the topic of volatility. A few sections are added to go over innovations such as Local Vol Stochastic Vol models, models to represent SPX/VIX dynamics, and Stochastic Implied Vol.

The second chapter still focuses on filtering and optimization. Two short sections on optimization techniques are added with a view to completeness.

The third chapter tackles the consistency question. I tried to further stress that this chapter merely provides a methodology and does not contain a robust empirical work. As such, the examples in this chapter remain anecdotal.

A fourth chapter is added to discuss some new ideas on the application of Wiener chaos to the estimation problem, as well as using better quality observations from options markets.

A few new references are added. However, with the ever-increasing body of literature I am certain many have been missed.

1

 As opposed to risk-less arbitrage in derivatives theory.

Introduction (First Edition)

Summary

This book focuses on developing Methodologies for Estimating Stochastic Volatility (SV) parameters from the Stock-Price Time-Series under a Classical framework. The text contains three chapters and is structured as follows:

In the first chapter, we shall introduce and discuss the concept of various parametric SV models. This chapter represents a brief survey of the existing literature on the subject of nondeterministic volatility.

We start with the concept of log-normal distribution and historic volatility. We then will introduce the Black-Scholes [40] framework. We shall also mention alternative interpretations as suggested by Cox and Rubinstein [71]. We shall state how these models are unable to explain the negative-skewness and the leptokurticity commonly observed in the stock markets. Also, the famous implied-volatility smile would not exist under these assumptions.

At this point we consider the notion of level-dependent volatility as advanced by researchers such as Cox and Ross [69, 70] as well as Bensoussan, Crouhy, and Galai [34]. Either an artificial expression of the instantaneous variance will be used, as is the case for Constant Elasticity Variance (CEV) models, or an implicit expression will be deduced from a Firm model similar to Merton's [199], for instance.

We also will bring up the subject of Poisson Jumps [200] in the distributions providing a negative-skewness and larger kurtosis. These jump-diffusion models offer a link between the volatility smile and credit phenomena.

We then discuss the idea of Local Volatility [38] and its link to the instantaneous unobservable volatility. Work by researchers such as Dupire [94], Derman, and Kani [79] will be cited. We shall also describe the limitations of this idea due to an ill-poised inversion phenomenon, as revealed by Avellaneda [17] and others.

Unlike Non-Parametric Local Volatility models, Parametric Stochastic Volatility (SV) models [147] define a specific stochastic differential equation for the unobservable instantaneous variance. We therefore will introduce the notion of two-factor Stochastic Volatility and its link to one-factor Generalized Auto-Regressive Conditionally Heteroskedastic (GARCH) processes [42]. The SV model class is the one we shall focus on. Studies by scholars such as Engle [99], Nelson [204], and Heston [141] will be discussed at this juncture. We will briefly mention related works on Stochastic Implied Volatility by Schonbucher [224], as well as Uncertain Volatility by Avellaneda [18].

Having introduced SV, we then discuss the two-factor Partial Differential Equations (PDE) and the incompleteness of the markets when only cash and the underlying asset are used for hedging.

We then will examine Option Pricing techniques such as Inversion of the Fourier transform, Mixing Monte-Carlo, as well as a few asymptotic pricing techniques, as explained for instance by Lewis [185].

At this point we shall tackle the subject of pure-jump models such as Madan's Variance Gamma [192] or its variants VG with Stochastic Arrivals (VGSA) [52]. The latter adds to the traditional VG a way to introduce the volatility clustering (persistence) phenomenon. We will mention the distribution of the stock market as well as various option pricing techniques under these models. The inversion of the characteristic function is clearly the method of choice for option pricing in this context.

In the second chapter we will tackle the notion of Inference (or Parameter-Estimation) for Parametric SV models. We shall first briefly analyze the Cross-Sectional Inference and will then focus on the Time-Series Inference.

We start with a concise description of cross-sectional estimation of SV parameters in a risk-neutral framework. A Least Squares Estimation (LSE) algorithm will be discussed. The Direction-Set optimization algorithm [214] will be also introduced at this point. The fact that this optimization algorithm does not use the gradient of the input-function is important, since we shall later deal with functions that contain jumps and are not necessarily differentiable everywhere.

We then discuss the parameter inference from a Time-Series of the underlying asset in the real world. We shall do this in a Classical (Non-Bayesian) [252] framework and in particular we will estimate the parameters via a Maximization of Likelihood Estimation (MLE) [134] methodology. We shall explain the idea of MLE, its link to the Kullback-Leibler [105] distance, as well as the calculation of the Likelihood function for a two-factor SV model.

We will see that unlike GARCH models, SV models do not admit an analytic (integrated) likelihood function. This is why we will need to introduce the concept of Filtering [136].

The idea behind Filtering is to obtain the best possible estimation of a hidden state given all the available information up to that point. This estimation is done in an iterative manner in two stages: The first step is a Time Update where the prior distribution of the hidden state, at a given point in time, is determined from all the past information via a Chapman-Kolmogorov equation. The second step would then involve a Measurement Update where this prior distribution is used together with the conditional likelihood of the newest observation in order to compute the posterior distribution of the hidden state. The Bayes rule is used for this purpose. Once the posterior distribution is determined, it could be exploited for the optimal estimation of the hidden state.

We shall start with the Gaussian case where the first two moments characterize the entire distribution. For the Gaussian-Linear case, the optimal Kalman Filter (KF) [136] is introduced. Its nonlinear extension, the Extended KF (EKF), is described next. A more suitable version of KF for strongly nonlinear cases, the Unscented KF (UKF) [174], is also analyzed. In particular we will see how this filter is related to Kushner's Nonlinear Filter (NLF) [181, 182].

EKF uses a first-order Taylor approximation upon the nonlinear transition and observation functions, in order to bring us back into a simple KF framework. On the other hand, UKF uses the true nonlinear functions without any approximation. It, however, supposes that the Gaussianity of the distribution is preserved through these functions. UKF determines the first two moments via integrals that are computed on a few appropriately chosen “sigma points.” NLF does the same exact thing via a Gauss-Hermite quadrature. However NLF often introduces an extra centering step, which will avoid poor performance due to an insufficient intersection between the prior distribution and the conditional likelihood.

As we shall observe, in addition to their use in the MLE approach, the Filters above could be applied to a direct estimation of the parameters via a Joint Filter (JF) [140]. The JF would simply involve the estimation of the parameters together with the hidden state via a dimension augmentation. In other words, one would treat the parameters as hidden states. After choosing initial conditions and applying the filter to an observation data set, one would then disregard a number of initial points and take the average upon the remaining estimations. This initial rejected period is known as the “burn in” period.

We will test various representations or State Space Models of the Stochastic Volatility models such as Heston's [141]. The concept of Observability [215] will be introduced in this context. We will see that the parameter estimation is not always accurate given a limited amount of daily data.

Before a closer analysis of the performance of these estimation methods, we shall introduce simulation-based Particle Filters (PF) [84, 128], which can be applied to non-Gaussian distributions. In a PF algorithm, the Importance Sampling technique is applied to the distribution. Points are simulated via a chosen proposal distribution and the resulting weights proportional to the conditional likelihood are computed. Since the variance of these weights tends to increase over time and cause the algorithm to diverge, the simulated points go through a variance reduction technique commonly referred to as Resampling [15]. During this stage, points with too small a weight are disregarded, and points with large weights are reiterated. This technique could cause a Sample Impoverishment, which can be corrected via a Metropolis-Hastings Accept/Reject test. Work by researchers such as Doucet [84], Smith, and Gordon [128] are cited and used in this context.

Needless to say, the choice of the proposal distribution could be fundamental in the success of the PF algorithm. The most natural choice would be to take a proposal distribution equal to the prior distribution of the hidden state. Even if this makes the computations simpler, the danger would be a non-alignment between the prior and the conditional likelihood as we previously mentioned. To avoid this, other proposal distributions taking into account the observation should be considered. The Extended PF (EPF) and the Unscented PF (UPF) [240] precisely do this by adding an extra Gaussian Filtering step to the process. Other techniques such as Auxiliary PF (APF) have been developed by Pitt and Shephard [213].

Interestingly, we will see that PF brings only marginal improvement to the traditional KF's when applied to daily data. However, for a larger time-step where the nonlinearity is stronger, the PF does help more.

At this point we also compare the Heston model to other SV models such as the “3/2” model [185] using real market data, and we will see that the latter performs better than the former. This is in line with the findings of Engle and Ishida [100]. We can therefore apply our inference tools to perform Model Identification.

Various Diagnostics [136] are used to judge the performance of the estimation tools. Mean Price Errors (MPE) and Root Mean Square Errors (RMSE) are calculated from the residual errors. The same residuals could be submitted to a Box-Ljung test, which will allow us to see whether they still contain auto-correlation. Other tests such as the Chi-Square Normality test as well as plots of Histograms and Variograms [115] are performed.

Most importantly, for the Inference process, we backtest the tools upon artificially simulated data, and we observe that although they give the correct answer asymptotically, the results remain inaccurate for a smaller amount of data points. It is reassuring to know that these observations are in agreement with work by other researchers such as Bagchi [20].

Here, we shall attempt to find an explanation for this mediocre performance. One possible interpretation comes from the fact that in the SV problem, the parameters affect the noise of the observation and not its drift. This is doubly true of volatility-of-volatility and stock-volatility correlation, which affect the noise of the noise. We should, however, note that the product of these two parameters enters in the equations at the same level as the drift of the instantaneous variance, and it is precisely this product that appears in the skewness of the distribution.

Indeed the instantaneous volatility is observable only at the second order of a Taylor (or Ito) expansion of the logarithm of the asset-price. This also explains why one-factor GARCH models do not have this issue. In their context the instantaneous volatility is perfectly known as a function of previous data points.

The issue therefore seems to be a low Signal to Noise Ratio (SNR). We could improve our estimation by considering additional data points. Using a high frequency (several quotes a day) for the data does help in this context. However, one needs to obtain clean and reliable data first.