DOCSLIB.ORG

Volatility Markets Consistent Modeling, Hedging and Practical Implementation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Volatility Markets Consistent Modeling, Hedging and Practical Implementation Volatility Markets Consistent modeling, hedging and practical implementation vorgelegt von Diplom-Mathematiker Hans BÄuhler Berlin FakultÄatII - Mathematik und Naturwissenschaften der Technischen UniversitÄatBerlin zur Erlangung des akademischen Grades Doktor der Natuwissenschaften Dr. rer. nat. genehmigte Dissertation Promotionsausschuss: Vorsitzender: Prof. Dr. rer. nat. Stefan Felsner Berichter/Gutachter: Prof. Dr. rer. nat. Alexander Schied Berichter/Gutachter: Prof. Dr. rer. nat. Peter Imkeller Gutachter: Prof. Dr. rer. nat. Josef Teichmann Tag der wissenschaftlichen Aussprache: 26. Juni 2006 Berlin 2006, D83 1 To my parents, for all their love and patience. In Liebe, Hans. Contents 1 Introduction 5 1.1 Basic Assumptions . 10 1.1.1 Variance Swaps . 10 1.1.2 Options on Variance . 12 1.2 Mathematical Notation . 13 I Consistent Modelling 16 2 Consistent Variance Curve Models 17 2.1 Problem Statements and Overview . 17 2.1.1 Review of the Stochastic Volatility Case . 19 2.2 General Variance Curve Models . 21 2.2.1 The Martingale Property and Explosion of Variance . 24 2.2.2 Fixed Time-to-Maturity . 24 2.2.3 Fitting the Market with Exponential Variance Curve Models . 27 2.3 Consistent Variance Curve Functionals . 29 2.3.1 Markov Variance Curve Market Models . 30 2.3.2 HJM-Conditions for Consistent Parameter Processes . 31 2.3.3 Extensions to Manifolds: When does Z stay in Z ? . 32 2.4 Variance Curve Models in Hilbert Spaces . 35 3 Examples 37 3.1 Exponential-Polynomial Variance Curve Models . 37 3.2 Exponential Curves . 41 3.3 Variance Swap Volatility Curves . 41 3.4 Fitting Models . 43 II Hedging 47 4 Theory of Replication 48 4.1 Problem Statements and Overview . 48 4.2 Hedging in Complete Markets . 51 4.2.1 General Complete Markovian Markets . 51 4.2.2 Pricing with Local Martingales . 58 4.2.3 Hedging with Variance Swaps . 59 2 CONTENTS 3 4.2.4 Hedging in classic Stochastic Volatility Models . 65 5 Hedging in Practice 66 5.1 Model and Market . 66 5.1.1 Additional Market Instruments . 69 5.2 Parameter Hedging in Practise . 73 5.2.1 Constrained Parameter-Hedging in Practise . 74 5.3 Dynamic Arbitrage . 77 5.3.1 Entropy Swaps . 78 III Practical Implementation 82 6 A variance curve model 83 6.1 The Model . 83 6.1.1 Existence, Uniqueness and the Martingale Property . 87 6.2 Pricing . 92 6.2.1 Pricing General Payo®s using an unbiased Milstein Monte-Carlo Scheme . 93 6.2.2 Control Variates . 98 6.2.3 Pricing Options on Variance . 99 6.2.4 Pricing European Options on the Stock . 99 7 Calibration 102 7.1 Notation and Overview . 102 7.2 Numerical Calibration . 104 7.2.1 Intraday Use . 106 7.3 Phase 1: Market Data Adjustment . 107 7.3.1 The Balayage-Order . 107 7.3.2 Upper Pricing Measures . 108 7.3.3 Relative Upper Pricing Measures . 112 7.3.4 Test for Strict Absence of Arbitrage . 114 7.3.5 How to Produce a strictly Arbitrage-free Surface . 115 7.3.6 Summary . 117 7.4 Phase 2: State Calibration . 117 7.5 Phase 3: Parameter Calibration . 118 7.5.1 ATM calibration . 118 7.5.2 Calibration in Steps . 120 7.5.3 Examples . 120 7.5.4 Pricing Options on Variance . 124 8 Conclusions 129 A Variance Swaps, Entropy Swaps and Gamma Swaps 131 A.1 Basic Pricing and Hedging . 131 A.1.1 Variance Swaps . 132 A.1.2 Entropy Swaps . 134 A.1.3 Shadow Options . 135 CONTENTS 4 A.2 Deterministic Interest Rates and Proportional Dividends . 137 A.2.1 European Options . 138 A.2.2 Variance Swaps . 139 A.2.3 Gamma Swaps . 139 B Example Term Sheets 142 C Auxiliary Results 148 C.1 The Impact of incorrect Stochastic Volatility Dynamics . 148 D Transition Densities under Constraints 150 D.1 Expensive Martingales . 150 D.1.1 Construction of a Transition Kernel . 151 D.2 Incorporating Weak Information . 153 D.2.1 Mean-Variance Pricing . 153 D.2.2 Using Forward Started Call Prices . 154 Index 156 Bibliography 158 Chapter 1 Introduction Ever since Black, Scholes and Merton published their famous articles [BS73] and [M73], huge markets of ¯nancial derivatives on a wide range of underlying economic quantities have devel- oped. One of the most visible markets of underlyings is surely the equity market with index level and share price quotes being a common part of today's news programmes. Upon it rest deep exchange-based markets of \vanilla" derivatives on indices and single stocks. This development of course changes the way over-the-counter products (those which are agreed upon on a case-by-case basis between the counter-parties) are evaluated and risk-managed. While Black and Scholes (BS) used only the underlying stock price and the bond to hedge a derivative in their model,1 this cannot be justi¯ed anymore: their model is not able to capture what is today known as the \volatility skew", or \volatility smile", of the implied volatility of traded vanilla options. The root of the discrepancy is that volatility is not, as assumed in BS' model, a deterministic quantity. Rather, it is by itself stochastic. The stochastic nature of the instantaneous variance of the stock price process is particular important if we want to price and hedge heavily volatility-dependent exotic options such as options on realized variance or cliquet-type products.2 Such products cannot be priced correctly in the BS-model since their very risk lies in the movement of volatility (or variance, for that matter) itself. Beyond Black-Scholes There have been many approaches to remedy this problem: the most pragmatic idea is to infer an implied risk-neutral distribution from the observed market prices. To this end, Dupire [D96] has completely solved the problem of ¯nding a one-factor di®usion which reprices a continuum of market prices. His implied local volatility is today a standard tool for evaluating exotic derivatives. However, the resulting stock price dynamics are not overly realistic since the resulting dif- fusion is usually highly inhomogeneous in time. This implies that the model makes predictions about the future which are not matched by past market experience. Most notably, the implied volatility smile inside the model flattens out over time which is in contrast to the persistent presence of this phenomenon in reality. This in turn means that the dynamical behavior of the liquid options is not captured very well. 1In fact, the model is due to Samuelson [S65], but it is common to call it \Black&Scholes model". 2See section 1.1.2 for example payo®s. 5 CHAPTER 1. INTRODUCTION 6 Conceptually quite di®erent from this ¯tting approach are stochastic volatility models. In these models, a parsimonious description of the dynamics of both the stock price and its in- stantaneous variance is the starting point. Such a model is based on \structural" assumptions on the underlying stock price. For example, Heston's popular model [H93] assumes that the instantaneous variance of the stock price is a square-root di®usion whose increments are cor- related to the increments of the return of the stock price. Other popular stochastic volatility models are Hagan et al. [HKLW02], Schoebel/Zhu [SZ99] and Fouque et al. [FPS00] to name but a few. In addition, there are also models which incorporate jump processes (see for example Merton [M76], Carr et al. [CCM98]) or mixtures of both concepts such as Bates [B96]. A good reference on models based on L¶evyprocesses is Cont/Tankov [CT03]. Most of these structural models will lead per se to incomplete market models if only the stock and the cash bond are considered as tradable instruments. As a result, there is no unique fair price for most derivatives. To alleviate this problem in continuous models,3 we have to extend the range of tradable instruments. Broadly speaking, each additional source of randomness requires an additional traded instrument to be able to hedge the resulting risk. This is called \completion of the market" (see also Davis [Da04]). However, it not clear which traded instruments we have to choose to complete our market. Indeed, if we are to use the stock price together with a range of liquid reference options as hedging instruments, a more natural approach would be to model directly the evolution of the stock price and these reference options simultaneously. Such a framework has the advantage that the options are an integral part of the model and that the model yields hedging strategies directly in terms of the traded reference instruments. The most prominent approach has been to model call and put prices via their implied volatil- ities. This has been undertaken by Brace et al. [BGKW01], Cont et al. [CFD02], Fengler et al. [FHM03] and Ha®ner [H04], among others. However, to our knowledge, none of theses stochastic implied volatility models is able to ensure the absence of arbitrage situations (such as negative prices for butterfly trades) throughout the life of the model. For this reason, some authors have focused on the term-structure of implied volatilities for just one ¯xed cash strike. This approach has been pioneered by SchÄonbucher [S99] and has recently been put into a more general framework by Schweizer/Wissel [SW05] who also consider power-type payo®s. This approach is attractive for pricing strike-dependent options such as compound options. However, the dependency on a ¯xed cash strike also implies that if the market moves, the model's ¯xed strike may drift too far out of the money to be suitable for hedging purposes. This is of particular concern if we want to price and hedge mainly volatility- dependent products such as options on realized variance or cliquets.
Load more

Recommended publications
  • Arithmetic Variance Swaps
    Arithmetic variance swaps Article (Accepted Version) Leontsinis, Stamatis and Alexander, Carol (2017) Arithmetic variance swaps. Quantitative Finance, 17 (4). pp. 551-569. ISSN 1469-7688 This version is available from Sussex Research Online: http://sro.sussex.ac.uk/id/eprint/62303/ This document is made available in accordance with publisher policies and may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher’s version. Please see the URL above for details on accessing the published version. Copyright and reuse: Sussex Research Online is a digital repository of the research output of the University. Copyright and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable, the material made available in SRO has been checked for eligibility before being made available. Copies of full text items generally can be reproduced, displayed or performed and given to third parties in any format or medium for personal research or study, educational, or not-for-profit purposes without prior permission or charge, provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way. http://sro.sussex.ac.uk Arithmetic Variance Swaps Stamatis Leontsinisa and Carol Alexanderb a RwC Asset Management, London b School of Business, Management and Economics, University of Sussex To appear in Quantitative Finance, 2016 (in press) Abstract Biases in standard variance swap rates can induce substantial deviations below market rates.
    [Show full text]
  • Pricing Financial Derivatives Under the Regime-Switching Models Mengzhe Zhang
    Pricing Financial Derivatives Under the Regime-Switching Models Mengzhe Zhang A thesis in fulfilment of the requirements the degree of Doctor of Philosophy School of Mathematics and Statistics Faculty of Science, University of New South of Wales, Australia 19th April 2017 Acknowledgement First of all, I would like to thank my supervisor Dr Leung Lung Chan for his valuable guidance, helpful comments and enlightening advice throughout the preparation of this thesis. I also want to thank my parents for their love, support and encouragement throughout my graduate study time at the University of NSW. Last but not the least, I would like to thank my friends- Dr Xin Gao, Dr Xin Zhang, Dr Wanchuang Zhu, Dr Philip Chen and Dr Jinghao Huang from the Math school of UNSW and Dr Zhuo Chen, Dr Xueting Zhang, Dr Tyler Kwong, Dr Henry Rui, Mr Tianyu Cai, Mr Zhongyuan Liu, Mr Yue Peng and Mr Huaizhou Li from the Business school of UNSW. i Contents 1 Introduction1 1.1 BS-Type Model with Regime-Switching.................5 1.2 Heston's Model with Regime-Switching.................6 2 Asymptotics for Option Prices with Regime-Switching Model: Short- Time and Large-Time9 2.1 Introduction................................9 2.2 BS Model with Regime-Switching.................... 13 2.3 Small Time Asymptotics for Option Prices............... 13 2.3.1 Out-of-the-Money and In-the-Money.............. 13 2.3.2 At-the-Money........................... 25 2.3.3 Numerical Results........................ 35 2.4 Large-Maturity Asymptotics for Option Prices............. 39 ii 2.5 Calibration................................ 53 2.6 Conclusion................................. 57 3 Saddlepoint Approximations to Option Price in A Regime-Switching Model 59 3.1 Introduction...............................
    [Show full text]
  • Further Developments in Volatility Derivatives Modeling
    Introduction Prior work Parameter estimation Model vs market prices Conclusion Further Developments in Volatility Derivatives Modeling Jim Gatheral Global Derivatives Trading & Risk Management, Paris May 21, 2008 Introduction Prior work Parameter estimation Model vs market prices Conclusion Disclaimer The opinions expressed in this presentation are those of the author alone, and do not necessarily reflect the views of of Merrill Lynch, its subsidiaries or affiliates. Introduction Prior work Parameter estimation Model vs market prices Conclusion Motivation and context We would like to have a model that prices consistently 1 options on SPX 2 options on VIX 3 options on realized variance We believe there may be such a model because we can identify relationships between options on SPX, VIX and variance. For example: 1 Puts on SPX and calls on VIX both protect against market dislocations. 2 Bruno Dupire constructs an upper bound on the price of options on variance from the prices of index options. 3 The underlying of VIX options is the square-root of a forward-starting variance swap. The aim is not necessarily to find new relationships; the aim is to devise a tool for efficient determination of relative value. Introduction Prior work Parameter estimation Model vs market prices Conclusion Outline 1 Review of prior work Double Lognormal vs Double Heston 2 Parameter estimation Time series analysis of variance curves Time series analysis of SABR fits to SPX options 3 Model vs market prices Numerical techniques Pricing of options on VIX and SPX Options on realized variance 4 Conclusion Is the model right? Introduction Prior work Parameter estimation Model vs market prices Conclusion Review of prior work We found that double-mean reverting lognormal dynamics for SPX instantaneous variance gave reasonable fits to both SPX and VIX option volatility smiles.
    [Show full text]
  • Credit Derivatives Handbook
    08 February 2007 Fixed Income Research http://www.credit-suisse.com/researchandanalytics Credit Derivatives Handbook Credit Strategy Contributors Ira Jersey +1 212 325 4674 [email protected] Alex Makedon +1 212 538 8340 [email protected] David Lee +1 212 325 6693 [email protected] This is the second edition of our Credit Derivatives Handbook. With the continuous growth of the derivatives market and new participants entering daily, the Handbook has become one of our most requested publications. Our goal is to make this publication as useful and as user friendly as possible, with information to analyze instruments and unique situations arising from market action. Since we first published the Handbook, new innovations have been developed in the credit derivatives market that have gone hand in hand with its exponential growth. New information included in this edition includes CDS Orphaning, Cash Settlement of Single-Name CDS, Variance Swaps, and more. We have broken the information into several convenient sections entitled "Credit Default Swap Products and Evaluation”, “Credit Default Swaptions and Instruments with Optionality”, “Capital Structure Arbitrage”, and “Structure Products: Baskets and Index Tranches.” We hope this publication is useful for those with various levels of experience ranging from novices to long-time practitioners, and we welcome feedback on any topics of interest. FOR IMPORTANT DISCLOSURE INFORMATION relating to analyst certification, the Firm’s rating system, and potential conflicts
    [Show full text]
  • White Paper Volatility: Instruments and Strategies
    White Paper Volatility: Instruments and Strategies Clemens H. Glaffig Panathea Capital Partners GmbH & Co. KG, Freiburg, Germany July 30, 2019 This paper is for informational purposes only and is not intended to constitute an offer, advice, or recommendation in any way. Panathea assumes no responsibility for the content or completeness of the information contained in this document. Table of Contents 0. Introduction ......................................................................................................................................... 1 1. Ihe VIX Index .................................................................................................................................... 2 1.1 General Comments and Performance ......................................................................................... 2 What Does it mean to have a VIX of 20% .......................................................................... 2 A nerdy side note ................................................................................................................. 2 1.2 The Calculation of the VIX Index ............................................................................................. 4 1.3 Mathematical formalism: How to derive the VIX valuation formula ...................................... 5 1.4 VIX Futures .............................................................................................................................. 6 The Pricing of VIX Futures ................................................................................................
    [Show full text]
  • Volatility As Investment - Crash Protection with Calendar Spreads of Variance Swaps
    Journal of Applied Operational Research (2014) 6(4), 243–254 © Tadbir Operational Research Group Ltd. All rights reserved. www.tadbir.ca ISSN 1735-8523 (Print), ISSN 1927-0089 (Online) Volatility as investment - crash protection with calendar spreads of variance swaps Uwe Wystup 1,* and Qixiang Zhou 2 1 MathFinance AG, Frankfurt, Germany 2 Frankfurt School of Finance & Management, Germany Abstract. Nowadays, volatility is not only a risk measure but can be also considered an individual asset class. Variance swaps, one of the main investment vehicles, can obtain pure exposure on realized volatility. In normal market phases, implied volatility is often higher than the realized volatility will turn out to be. We present a volatility investment strategy that can benefit from both negative risk premium and correlation of variance swaps to the underlying stock index. The empirical evidence demonstrates a significant diversification effect during the financial crisis by adding this strategy to an existing portfolio consisting of 70% stocks and 30% bonds. The back-testing analysis includes the last ten years of history of the S&P500 and the EUROSTOXX50. Keywords: volatility; investment strategy; stock index; crash protection; variance swap * Received July 2014. Accepted November 2014 Introduction Volatility is an important variable of the financial market. The accurate measurement of volatility is important not only for investment but also as an integral part of risk management. Generally, there are two methods used to assess volatility: The first one entails the application of time series methods in order to make statistical conclusions based on historical data. Examples of this method are ARCH/GARCH models.
    [Show full text]
  • Nonparametric Pricing and Hedging of Volatility Swaps in Stochastic
    Nonparametric Pricing and Hedging of Volatility Swaps in Stochastic Volatility Models Frido Rolloos∗ March 28, 2020 Abstract In this paper the zero vanna implied volatility approximation for the price of freshly minted volatility swaps is generalised to seasoned volatility swaps. We also derive how volatility swaps can be hedged using a strip of vanilla options with weights that are directly related to trading intuition. Additionally, we derive first and second order hedges for volatil- ity swaps using only variance swaps. As dynamically trading variance swaps is in general cheaper and operationally less cumbersome compared to dynamically rebalancing a continu- ous strip of options, our result makes the hedging of volatility swaps both practically feasible and robust. Within the class of stochastic volatility models our pricing and hedging results are model-independent and can be implemented at almost no computational cost. arXiv:2001.02404v4 [q-fin.PR] 3 Apr 2020 ∗[email protected] 1 1 Assumptions and notations We will work under the premise that the market implied volatility surface is generated by the following general stochastic volatility (SV) model dS = σS ρ dW + ρdZ¯ (1.1) [ ] dσ = a σ, t dt + b σ, t dW (1.2) ( ) ( ) where ρ¯ = 1 ρ2, dW and dZ are independent standard Brownian motions, and the func- tions a and b −are deterministic functions of time and volatility. The results derived in this p paper are valid for any SV model satisfying (1.1) and (1.2), which includes among others the Heston model, the lognormal SABR model, and the 3 2 model. / The SV process is assumed to be well-behaved in the sense that vanilla options prices are risk-neutral expectations of the payoff function: C S, K = Et S T K + (1.3) ( ) [( ( )− ) ] The option price C can always be expressed in terms of the Black-Merton-Scholes (BS) price CBS with an implied volatility parameter I: BS C S, K = C S, K, I (1.4) ( ) ( ) It is assumed that the implied volatility parameter I = I S, t, K,T, σ, ρ .
    [Show full text]
  • The Term Structure of Variance Swap Rates and Optimal Variance Swap Investments
    JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSIS Vol. 45, No. 5, Oct. 2010, pp. 1279–1310 COPYRIGHT 2010, MICHAEL G. FOSTER SCHOOL OF BUSINESS, UNIVERSITY OF WASHINGTON, SEATTLE, WA 98195 doi:10.1017/S0022109010000463 The Term Structure of Variance Swap Rates and Optimal Variance Swap Investments Daniel Egloff, Markus Leippold, and Liuren Wu∗ Abstract This paper performs specification analysis on the term structure of variance swap rates on the S&P 500 index and studies the optimal investment decision on the variance swaps and the stock index. The analysis identifies 2 stochastic variance risk factors, which govern the short and long end of the variance swap term structure variation, respectively. The highly negative estimate for the market price of variance risk makes it optimal for an investor to take short positions in a short-term variance swap contract, long positions in a long-term variance swap contract, and short positions in the stock index. I. Introduction The financial market is becoming increasingly aware of the fact that the re- turn variance on stock indexes is stochastic (Engle (2004)) and the variance risk is heavily priced (Carr and Wu (2009)). Associated with this recognition is the development of a large number of variance-related derivative products. The most actively traded is the variance swap contract. The contract has 0 value at inception. At maturity, the long side of the variance swap contract receives the difference be- tween a standard measure of the realized variance and a fixed rate, called the vari- ance swap rate, determined at the inception of the contract. Traditional derivative contracts such as calls, puts, and straddles also have variance risk exposure, ∗Egloff, [email protected], QuantCatalyst, Hardturmstrasse 101, 8005 Zurich, Switzerland; Leippold, [email protected], University of Zurich, Swiss Banking Institute, Plattenstrasse 14, 8032 Zurich, Switzerland; and Wu, [email protected], Baruch College, Zicklin School of Business, One Bernard Baruch Way, Box B10-225, New York, NY 10010.
    [Show full text]
  • The Price of Interest Rate Variance Risk and Optimal Investments in Interest Rate Derivatives
    The price of interest rate variance risk and optimal investments in interest rate derivatives Anders B. Trolle Copenhagen Business School Abstract Recent research on unspanned stochastic variance raises the possibility that interest rate derivatives constitute an important component of optimal fixed income portfo- lios. In this paper, I estimate a flexible dynamic term structure model that allows for unspanned stochastic variance on an extensive data set of swaps and swaptions. I find that variance risk is predominantly unspanned by bonds, and that the price of risk on the unspanned variance factor is significantly larger in absolute value than the prices of risk on the term structure factors. Consequently, Sharpe ratios on variance sensitive derivatives are about three times larger than Sharpe ratios on bonds or short-term bond futures. These findings are corroborated by an analysis of the Treasury futures market, where the variance risk premium is estimated with a model independent approach. I then solve the dynamic portfolio choice problem for a long-term fixed income investor with and without access to interest rate derivatives and find substantial utility gains from participating in the derivatives market. JEL Classification: G11 Keywords: Portfolio choice, derivatives, stochastic variance, swaps, Treasury futures This version: December 2008 ——————————— I have benefitted from discussions with Giovanni Barone-Adesi, Michael Brennan, Wolfgang B¨uhler, Pierre Collin-Dufresne, David Lando, Francis Longstaff, Claus Munk, Carsten Sørensen and seminar participants at Copenhagen Business School. I am particularly grateful to Eduardo Schwartz for extensive comments. Some of the results in this paper were previously circulated under the title “Dynamic interest rate derivative strategies in the presence of unspanned stochastic volatility”.
    [Show full text]
  • Volatility Swaps Valuation Under Stochastic Volatility with Jumps and Stochastic Intensity
    Volatility swaps valuation under stochastic volatility with jumps and stochastic intensity∗ Ben-zhang Yanga, Jia Yueb, Ming-hui Wanga and Nan-jing Huanga† a. Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P.R. China b. Department of Economic Mathematics, South Western University of Finance and Economics, Chengdu, Sichuan 610074, P.R. China Abstract. In this paper, a pricing formula for volatility swaps is delivered when the underlying asset follows the stochastic volatility model with jumps and stochastic intensity. By using Feynman-Kac theorem, a partial integral differential equation is obtained to derive the joint moment generating function of the previous model. Moreover, discrete and continuous sampled volatility swap pricing formulas are given by employing transform techniques and the relationship between two pricing formulas is discussed. Finally, some numerical simulations are reported to support the results presented in this paper. Key Words and Phrases: Stochastic volatility model with jumps; Stochastic intensity; Volatility derivatives; Pricing. 2010 AMS Subject Classification: 91G20, 91G80, 60H10. 1 Introduction Since the finance volatility has always been considered as a key measure, the development and growth of the financial market have changed the role of the volatility over last century. Volatility derivatives in general are important tools to display the market fluctuation and manage volatility risk for investors. arXiv:1805.06226v2 [q-fin.PR] 18 May 2018 More precisely, volatility derivatives are traded for decision-making between long or short positions, trading spreads between realized and implied volatility, and hedging against volatility risks. The utmost advantage of volatility derivatives is their capability in providing direct exposure towards the assets volatility without being burdened with the hassles of continuous delta-hedging.
    [Show full text]
  • Variance Swap Payoffs, Risk Premia and Extreme Market Conditions
    Variance swap payoffs, risk premia and extreme market conditions Jeroen V.K. Rombouts1, Lars Stentoft2 and Francesco Violante3 June 5, 2017 Abstract This paper estimates the Variance Risk Premium (VRP) directly from synthetic variance swap payoffs. Since variance swap payoffs are highly volatile, we extract the VRP by using signal extraction techniques based on a state-space representation of our model in combination with a simple economic constraint. Our approach, only requiring option implied volatilities and daily returns for the underlying, provides measurement error free estimates of the part of the VRP related to normal market conditions, and allows constructing variables indicating agents' expectations under extreme market conditions. The latter variables and the VRP generate different re- turn predictability on the major US indices. A factor model is proposed to extract a market VRP which turns out to be priced when considering Fama and French port- folios. Keywords: Variance risk premium; Variance swaps; Return predictability; Factor Model, Kalman filter, CAPM. JEL Classification: C12, C22, G12, G13 1ESSEC Business School, Av. B. Hirsch, Cergy Pontoise, France 95021. Phone: +33 1 3443 3049 - E-mail: [email protected] 2Department of Economics and Department of Statistical and Actuarial Sciences, University of Western Ontario, Social Science Centre, London, ON, Canada N6A 5C2, Phone: +1 519 661 2111 ext. 85311 - E-mail: [email protected] 3MEMOTEF - Sapienza Universit´adi Roma, Via del Castro Laurenziano 9, Roma, Italy 00161, and
    [Show full text]
  • Local Volatility, Stochastic Volatility and Jump-Diffusion Models
    IEOR E4707: Financial Engineering: Continuous-Time Models Fall 2013 ⃝c 2013 by Martin Haugh Local Volatility, Stochastic Volatility and Jump-Diffusion Models These notes provide a brief introduction to local and stochastic volatility models as well as jump-diffusion models. These models extend the geometric Brownian motion model and are often used in practice to price exotic derivative securities. It is worth emphasizing that the prices of exotics and other non-liquid securities are generally not available in the market-place and so models are needed in order to both price them and calculate their Greeks. This is in contrast to vanilla options where prices are available and easily seen in the market. For these more liquid options, we only need a model, i.e. Black-Scholes, and the volatility surface to calculate the Greeks and perform other risk-management tasks. In addition to describing some of these models, we will also provide an introduction to a commonly used fourier transform method for pricing vanilla options when analytic solutions are not available. This transform method is quite general and can also be used in any model where the characteristic function of the log-stock price is available. Transform methods now play a key role in the numerical pricing of derivative securities. 1 Local Volatility Models The GBM model for stock prices states that dSt = µSt dt + σSt dWt where µ and σ are constants. Moreover, when pricing derivative securities with the cash account as numeraire, we know that µ = r − q where r is the risk-free interest rate and q is the dividend yield.
    [Show full text]