Volatility As Investment - Crash Protection with Calendar Spreads of Variance Swaps
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Machine Learning Based Intraday Calibration of End of Day Implied Volatility Surfaces
DEGREE PROJECT IN MATHEMATICS, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2020 Machine Learning Based Intraday Calibration of End of Day Implied Volatility Surfaces CHRISTOPHER HERRON ANDRÉ ZACHRISSON KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES Machine Learning Based Intraday Calibration of End of Day Implied Volatility Surfaces CHRISTOPHER HERRON ANDRÉ ZACHRISSON Degree Projects in Mathematical Statistics (30 ECTS credits) Master's Programme in Applied and Computational Mathematics (120 credits) KTH Royal Institute of Technology year 2020 Supervisor at Nasdaq Technology AB: Sebastian Lindberg Supervisor at KTH: Fredrik Viklund Examiner at KTH: Fredrik Viklund TRITA-SCI-GRU 2020:081 MAT-E 2020:044 Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci Abstract The implied volatility surface plays an important role for Front office and Risk Manage- ment functions at Nasdaq and other financial institutions which require mark-to-market of derivative books intraday in order to properly value their instruments and measure risk in trading activities. Based on the aforementioned business needs, being able to calibrate an end of day implied volatility surface based on new market information is a sought after trait. In this thesis a statistical learning approach is used to calibrate the implied volatility surface intraday. This is done by using OMXS30-2019 implied volatil- ity surface data in combination with market information from close to at the money options and feeding it into 3 Machine Learning models. The models, including Feed For- ward Neural Network, Recurrent Neural Network and Gaussian Process, were compared based on optimal input and data preprocessing steps. -
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. -
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. -
VERTICAL SPREADS: Taking Advantage of Intrinsic Option Value
Advanced Option Trading Strategies: Lecture 1 Vertical Spreads – The simplest spread consists of buying one option and selling another to reduce the cost of the trade and participate in the directional movement of an underlying security. These trades are considered to be the easiest to implement and monitor. A vertical spread is intended to offer an improved opportunity to profit with reduced risk to the options trader. A vertical spread may be one of two basic types: (1) a debit vertical spread or (2) a credit vertical spread. Each of these two basic types can be written as either bullish or bearish positions. A debit spread is written when you expect the stock movement to occur over an intermediate or long- term period [60 to 120 days], whereas a credit spread is typically used when you want to take advantage of a short term stock price movement [60 days or less]. VERTICAL SPREADS: Taking Advantage of Intrinsic Option Value Debit Vertical Spreads Bull Call Spread During March, you decide that PFE is going to make a large up move over the next four months going into the Summer. This position is due to your research on the portfolio of drugs now in the pipeline and recent phase 3 trials that are going through FDA approval. PFE is currently trading at $27.92 [on March 12, 2013] per share, and you believe it will be at least $30 by June 21st, 2013. The following is the option chain listing on March 12th for PFE. View By Expiration: Mar 13 | Apr 13 | May 13 | Jun 13 | Sep 13 | Dec 13 | Jan 14 | Jan 15 Call Options Expire at close Friday, -
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 -
A Glossary of Securities and Financial Terms
A Glossary of Securities and Financial Terms (English to Traditional Chinese) 9-times Restriction Rule 九倍限制規則 24-spread rule 24 個價位規則 1 A AAAC see Academic and Accreditation Advisory Committee【SFC】 ABS see asset-backed securities ACCA see Association of Chartered Certified Accountants, The ACG see Asia-Pacific Central Securities Depository Group ACIHK see ACI-The Financial Markets of Hong Kong ADB see Asian Development Bank ADR see American depositary receipt AFTA see ASEAN Free Trade Area AGM see annual general meeting AIB see Audit Investigation Board AIM see Alternative Investment Market【UK】 AIMR see Association for Investment Management and Research AMCHAM see American Chamber of Commerce AMEX see American Stock Exchange AMS see Automatic Order Matching and Execution System AMS/2 see Automatic Order Matching and Execution System / Second Generation AMS/3 see Automatic Order Matching and Execution System / Third Generation ANNA see Association of National Numbering Agencies AOI see All Ordinaries Index AOSEF see Asian and Oceanian Stock Exchanges Federation APEC see Asia Pacific Economic Cooperation API see Application Programming Interface APRC see Asia Pacific Regional Committee of IOSCO ARM see adjustable rate mortgage ASAC see Asian Securities' Analysts Council ASC see Accounting Society of China 2 ASEAN see Association of South-East Asian Nations ASIC see Australian Securities and Investments Commission AST system see automated screen trading system ASX see Australian Stock Exchange ATI see Account Transfer Instruction ABF Hong -
Spread Scanner Guide Spread Scanner Guide
Spread Scanner Guide Spread Scanner Guide ..................................................................................................................1 Introduction ...................................................................................................................................2 Structure of this document ...........................................................................................................2 The strategies coverage .................................................................................................................2 Spread Scanner interface..............................................................................................................3 Using the Spread Scanner............................................................................................................3 Profile pane..................................................................................................................................3 Position Selection pane................................................................................................................5 Stock Selection pane....................................................................................................................5 Position Criteria pane ..................................................................................................................7 Additional Parameters pane.........................................................................................................7 Results section .............................................................................................................................8 -
Copyrighted Material
Index AA estimate, 68–69 At-the-money (ATM) SPX variance Affine jump diffusion (AJD), 15–16 levels/skews, 39f AJD. See Affine jump diffusion Avellaneda, Marco, 114, 163 Alfonsi, Aurelien,´ 163 American Airlines (AMR), negative book Bakshi, Gurdip, 66, 163 value, 84 Bakshi-Cao-Chen (BCC) parameters, 40, 66, American implied volatilities, 82 67f, 70f, 146, 152, 154f American options, 82 Barrier level, Amortizing options, 135 distribution, 86 Andersen, Leif, 24, 67, 68, 163 equal to strike, 108–109 Andreasen, Jesper, 67, 68, 163 Barrier options, 107, 114. See also Annualized Heston convexity adjustment, Out-of-the-money barrier options 145f applications, 120 Annualized Heston VXB convexity barrier window, 120 adjustment, 160f definitions, 107–108 Ansatz, 32–33 discrete monitoring, adjustment, 117–119 application, 34 knock-in options, 107 Arbitrage, 78–79. See also Capital structure knock-out options, 107, 108 arbitrage limiting cases, 108–109 avoidance, 26 live-out options, 116, 117f calendar spread arbitrage, 26 one-touch options, 110, 111f, 112f, 115 vertical spread arbitrage, 26, 78 out-of-the-money barrier, 114–115 Arrow-Debreu prices, 8–9 Parisian options, 120 Asymptotics, summary, 100 rebate, 108 Benaim, Shalom, 98, 163 At-the-money (ATM) implied volatility (or Berestycki, Henri, 26, 163 variance), 34, 37, 39, 79, 104 Bessel functions, 23, 151. See also Modified structure, computation, 60 Bessel function At-the-money (ATM) lookback (hindsight) weights, 149 option, 119 Bid/offer spread, 26 At-the-money (ATM) option, 70, 78, 126, minimization, -
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. -
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”. -
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 -
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.