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by Torben G. Andersena, Tim Bollerslevb, Peter F. Christoffersenc and Francis X. Dieboldd Volatility has been one of the most active and successful areas of research in time series econometrics and economic forecasting in recent decades. This chapter provides a selective survey of the most important theoretical developments and empirical insights to emerge from this burgeoning literature, with a distinct focus on forecasting applications. Volatility is inherently latent, and Section 1 begins with a brief intuitive account of various key volatility concepts. Section 2 then discusses a series of different economic situations in which volatility plays a crucial role, ranging from the use of volatility forecasts in portfolio allocation to density forecasting in risk management. Sections 3, 4 and 5 present a variety of alternative procedures for univariate volatility modeling and forecasting based on the GARCH, stochastic volatility and realized volatility paradigms, respectively. Section 6 extends the discussion to the multivariate problem of forecasting conditional covariances and correlations, and Section 7 discusses volatility forecast evaluation methods in both univariate and multivariate cases. Section 8 concludes briefly.
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by Sarves Verma, Gunhan Mehmet Ertosun, Wei Wang, Benjamin Ambruster, Kay Giesecke
Abstract In this work, we extend the idea of [9] and classify options both on the basis of moneyness and maturity i.e. we form maturity & moneyness buckets and study the impact of different PCA factors on implied volatility. We believe that this will give a clear idea to a trader; which factor to look for when hedging an option of a specific moneyness and a specific maturity. In this context, we also come across a novel way of looking at gamma and vega (greeks) using principal components. Further, we also develop a comprehensive model to incorporate the effect of maturity on implied volatility. Section II describes the methodology while Section III deals with our results & interpretation of those results. Finally we conclude with Section IV. Discuss this paper
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by Martin le Roux In this paper we present an econometric model of implied volatilities of S&P500 index options. Firstly, we model the dynamics the CBOE VIX index as a proxy for the general level of implied volatilities. We then describe a parametric model of the implied volatility surface for options with a term of up to two years. We show that almost all of the differences between the VIX and the implied volatility surface (i.e. smile and term structure effects) can be explained by one or two uncorrelated random factors. Finally, we present a model of the dynamics of these factors
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by Francesco Audrino, Dominik Colangelo Abstract: This paper introduces a new semi-parametric methodology for the implied volatility surface, which incorporates machine learning algorithms. Given a starting model, a tree boosting algorithm sequentially minimizes the residuals of observed and estimated implied volatility. To overcome the poor predicting power of existing models, we include a grid in the region of interest, and implement a cross-validation strategy to find an optimal stopping value for the tree boosting. Back testing the out-of-sample performance on a large data set of implied volatilities on S&P 500 options, we provide empirical evidence of the strong predictive potential of our methodology.
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by Amadeo Alentorn The volatility surface implied by option prices presents a structure that changes over time. The aim of this study is to present a framework to model the implied volatility of the FTSE options in real time, and to present a prototype application that implements this framework. We adapt the parametric models presented in Dumas et al (1998) to estimate the surfaces across moneyness instead of across strikes. We discuss how this framework can be used in applications of option pricing and risk management
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by Dr. Reinhold Hafner, Dr. Bernd Schmid Abstract We present a factor-based model of the stochastic evolution of the implied volatility surface. The model allows for the integrated and consistent pricing and hedging, risk management, and trading of equity index derivatives as well as volatility derivatives. We develop a unifying theory for the analysis of contingent claims under both the realworld measure and the risk-neutral measure in an environment of stochastic implied volatility. On the basis of transaction data, we provide extensive statistical analyses on the dynamics of the implied volatility surface of German DAX options and propose a four-factor model to describe its evolution. The model is validated and tested on market data.
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by Rama Cont, Valdo Durrleman, Jose Da Fonseca Abstract: We propose a market-based approach to the modelling of implied volatility, in which the implied volatility surface is directly used as the state variable to describe the joint evolution of market prices of options and their underlying asset. We model the evolution of an implied volatility surface by representing it as a randomly fluctuating surface driven by a finite number of orthogonal random factors. Our approach is based on a Karhunen-Loeve decomposition of the daily variations of implied volatilities obtained from market data on SP500 and DAX options.
We illustrate how this approach extends and improves the accuracy of the well-known 'sticky moneyness' rule used by option traders for updating implied volatilities. Our approach gives a justification for the use of 'Vegas' for measuring volatility risk and provides a decomposition of volatility risk as a sum of independent contributions from empirically identifiable factors.
Keywords: Implied volatility, volatility risk, risk management, portfolios of options Discuss this paper
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by Rama CONT and Jose da FONSECA Abstract The prices of index options at a given date are usually represented via the corresponding implied volatility surface, presenting skew/smile features and term structure which several models have attempted to re-produce. However the implied volatility surface also changes dynamically over time in a way that is not taken into account by current modeling approaches, giving rise to 'Vega' risk in option portfolios. Using time series of option prices on the SP500 and FTSE indices, we study the deforma tion of this surface and show that it may be represented as a randomly surface driven by a small number of orthogonal random factors. We identify and interpret the shape of each of these factors, study their dynamics and their correlation with the underlying index. Our approach is based on a Karhunen-Loeve decomposition of the daily variations of implied volatilities obtained from market data. A simple factor model compatible with the empirical observations is proposed. We illustrate how this approach model and improves the the well-known sticky moneyness Discuss this paper
Lectures Lecture 1: The Principles of Valuation Lecture 2: Dynamic Replication: Myths and Realities of Options Pricing Lecture 3: The Smile: Constraints and Problems Lecture 4: Arbitrage Bounds, Problems with Valuation, Models Lecture 5: Static Hedging and Implied Distributions Lecture 6: Extending Black Scholes & Local Volatility Models Lecture 7: More about Local Volatility Models Lecture 8: Implications of Local Volatility Models Lecture 9: Regimes of Volatility Lecture 10: Stochastic Volatility Models of the Smile Lecture 11: More on Stochastic Volatility Models of the Smile Lecture 12: Jump Diffusion Models of the Smile Lecture 13: Conclusions
Investors in equity options experience two problems that compound each other. In contrast to fixed-income and currency markets, there are thousands of underlyers and tens of thousands of options, and each underlyer can have a potentially large volatility skew. How can an options investor gauge which option provides the best relative value? In this paper, we make use of a method for estimating the fair volatility smile of any equity underlyer from information embedded in the time series of that underlyer Discuss this paper
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by Valdo Durrleman Given the quote price of a call or put option, the Black-Scholes implied volatility is the unique volatility parameter to be put into Black-Scholes formula to give the same price as the option quote price. This dissertation is concerned with the link between the implied volatility and the actual volatility of the underlying stock. Such a link is of particular practical interest since it relates the fundamental quantity for pricing financial derivatives (the actual volatility of the underlying stock), which is not observable, to directly observable quantities such as implied volatilities. The link that we derive in chapter 2 is a link between the dynamics of the two quantities. So far these quantities were mostly studied at a given time whereas we work at the level of processes. This is the main result of the dissertation. In chapter 1, we shall first review current practical problems in option pricing. Our aim there is twofold. First, we want to show that from a practical point of view, studying dynamics is very natural. Second, we shall identify two practical issues to which we shall propose answers in chapter 3. Although the main motivation of this dissertation comes from contemporary issues in the study of financial markets, chapter 2 also gives a solution to an inverse problem in the mathematical sense. One wishes to recover the structure of a stochastic process from a family of conditional expectations over its distribution. Besides the main result, this dissertation makes the following contributions. It brings new insights about implied volatility dynamics. In particular, it was observed that its motion was extremely Discuss this paper
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by Klebert Kentia Tonleu Volatility derivatives are products where the volatility is the main underlying notion. These products are particularly important for market investors as they use them to have insight into the level of volatility which empirical evidence show that it is stochastic. In this essay, we provide a short introduction to volatility derivatives. We start by motivating the change from constant volatility as assumed by the standard Black-Scholes model, to stochastic mean-reverting volatility Discuss this paper
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by Jim Gatheral, Merrill Lynch In the course of the following lectures, we will study why equity options are priced as they are. In so doing, we will apply many of the techniques students will have learned in previous semesters and develop some intuition for the pricing of both vanilla and exotic equity options. By considering specific examples, we will see that in pricing options, it is often as important to take into account the dynamics of underlying variables as it is to match known market prices of other claims. My hope is that these lectures will prove particularly useful to those who end up specializing in the structuring, pricing, trading and risk management of equity derivatives
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prsentation by Jim Gatheral Outline of this talk n A compound Poisson model of stock trading n The relationship between volatility and volume n Clustering n Correlation between volatility changes and log returns n Stochastic volatility n Dynamics of the volatility skew n Similarities between stochastic volatility models n Do stochastic volatility models fit option prices? n Jumps n The impact of large option trades Discuss this paper
Summary: Volatility has a key role to play in the determination of risk and in the valuation of options and other derivative securities. The widespread Black-Scholes model for asset prices assumes constant volatility. The purpose of this document is to introduce implied, local and stochastic volatility, to review evidence of non-constant volatility, and to consider the implications for option pricing of alternative random or stochastic volatility models. We focus on continuous time diffusion models for the volatility, but we also briefly discuss certain classes of discrete time models, such as ARV or ARCH. Discuss this paper
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Roger W. Lee Abstract Given the price of a call or put option, the Black-Scholes implied volatility is the unique volatility parameter for which the Bulack-Scholes formula recovers the option price. This article surveys research activity relating to three theoretical questions: First, does implied volatility admit a probabilistic interpretation? Second, how does implied volatility behave as a function of strike and expiry? Here one seeks to characterize the shapes of the implied volatility skew (or smile) and term structure, which together constitute what can be termed the statics of the implied volatility surface. Third, how does implied volatility evolve as time rolls forward? Here one seeks to characterize the dynamics of implied volatility.
The pricing accuracy and pricing performance of local volatility models crucially depends on absence of arbitrage in the implied volatility surface: an input implied volatility surface that is not arbitrage-free invariably results in negative transition probabilities and/ or negative local volatilities, and ultimately, into mispricings. The common smoothing algorithms of the implied volatility surface cannot guarantee the absence arbitrage. Here, we propose an approach for smoothing the implied volatility smile in an arbitrage-free way. Our methodology is simple to implement, computationally cheap and builds on the well-founded theory of natural smoothing splines under suitable shape constraints. Unlike other methods, our approach also works when input data are scarce and not arbitrage-free. Thus, it can easily be integrated into standard local volatility pricers.
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Emanuel Derman Iraj Kani Neil Chriss SUMMARY In options markets where there is a significant or persistent volatility smile, implied tree models can ensure the consistency of exotic options prices with the market prices of liquid standard options. Implied trees can be constructed in a variety of ways. Implied binomial trees are minimal: they have just enough parameters Discuss this paper
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ABSCHLUSSARBEIT Abstract Dynamic Semiparametric Factor Model (DSFM) is a convenient tool for analysis of implied volatility surfice (IVS). It offers dimension reduction of the IVS and can be therefore applied in hedging, prediction or risk mangement. However the estimation of the DSFM parameters is a complex procedure since it requires huge number of observation. Therefore the efficient implementation is a key issue for application possibilites of this model. In this master thesis we discuss implementation issues of DSFM. We describe key features of the model and present its implementation in statistical computing enviroment XploRe. Keywords: Dynamic Semiparametric Factor Model, Implied Volatility, Option Pricing Discuss this paper
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