DOCSLIB.ORG

A New Approach for Modelling and Pricing Correlation Swaps

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A New Approach for Modelling and Pricing Correlation Swaps A New Approach For Modelling and Pricing Correlation Swaps First draft: 20 February 2007 This version: 9 May 2007 Sebastien Bossu [email protected] Equity Structuring — ECD London – Disclaimer This document only reflects the views of the author and not necessarily those of Dresdner Kleinwort research, sales or trading departments. This document is for research or educational purposes only and is not intended to promote any financial investment or security. THIS DOCUMENT IS A COMMUNICATION MADE, OR APPROVED FOR COMMUNICATION IN THE UK, BY DRESDNER BANK AG LONDON BRANCH, AUTHORISED BY THE GERMAN FEDERAL FINANCIAL SUPERVISORY AUTHORITY AND BY THE FINANCIAL SERVICES AUTHORITY; REGULATED BY THE FINANCIAL SERVICES AUTHORITY FOR THE CONDUCT OF DESIGNATED INVESTMENT BUSINESS IN THE UK AND INCORPORATED IN GERMANY AS A STOCK CORPORATION WITH LIMITED LIABILITY. DRESDNER BANK AG LONDON BRANCH DOES NOT DEAL FOR, OR ADVISE OR OTHERWISE OFFER ANY INVESTMENT SERVICES TO PRIVATE CUSTOMERS. IN THE US, ANY BANKING SERVICES ARE PROVIDED BY DRESDNER BANK AG NEW YORK BRANCH AND/OR DRESDNER BANK AG GRAND CAYMAN BRANCH. ANY SECURITIES RELATED BUSINESS IS PROVIDED BY DRESDNER KLEINWORT SECURITIES LLC, A REGISTERED BROKER DEALER. This document is directed exclusively to market counterparties and intermediate customers who have professional experience in matters relating to investments. It is not directed at private customers and any investments or services to which the document may relate are not available to private customers. No persons other than a market counterparty or an intermediate customer should read or rely on any information in this document. This document has been prepared by Dresdner Kleinwort and is intended for discussion purposes only. This document is an indicative summary of the terms and conditions of the securities and may be subsequently changed without notice. The final terms and conditions of the securities will be set out in full in the offering documentation relating to the securities. Neither this document nor any other statement (oral or otherwise) made at any time in connection herewith is an offer, invitation or recommendation to acquire or dispose of any securities or to enter into any transaction. Dresdner Kleinwort is acting solely as principal and not as advisor or fiduciary. Dresdner Kleinwort may have positions or deal in transactions or securities (or related derivatives) identical or similar to those described herein. In any such circumstance, if conflicts of interest were to arise, Dresdner Kleinwort would conduct its business without regard to the consequences of such conduct (adverse or otherwise to you). Potential counterparties are advised to independently review and/or obtain independent professional advice and draw their own conclusions regarding the economic benefit and risks of this transaction and legal, regulatory, credit, tax and accounting aspects in relation to their particular circumstances. Distribution of this document does not oblige Dresdner Kleinwort to enter into any transaction. Any offer would be made at a later date and subject to contract, satisfactory documentation, prior transaction and market conditions. Dresdner Kleinwort makes no representations as to any matter or as to the accuracy or completeness of any statements made herein or made at any time orally or otherwise in connection herewith and all liability (in negligence or otherwise) in respect of any such matters or statements is expressly excluded, except only in the case of fraud or wilful default. Any forecasts and projections provided herein are indicative only as at the dates indicated and do not purport to be anything else and may in particular (but without limitation) be affected by changes in market conditions. Past performance is not indicative of future results. As used herein, “Dresdner Kleinwort” means Dresdner Bank AG (whether or not acting by its London Branch) and any of its associated or affiliated companies. NO ACTION HAS BEEN MADE OR WILL BE TAKEN THAT WOULD PERMIT A PUBLIC OFFERING OF THE SECURITIES DESCRIBED HEREIN IN ANY JURISDICTION IN WHICH ACTION FOR THAT PURPOSE IS REQUIRED. NO OFFERS, SALES, RESALES OR DELIVERY OF THE SECURITIES DESCRIBED HEREIN OR DISTRIBUTION OF ANY OFFERING MATERIAL RELATING TO SUCH SECURITIES MAY BE MADE IN OR FROM ANY JURISDICTION EXCEPT IN CIRCUMSTANCES WHICH WILL RESULT IN COMPLIANCE WITH ANY APPLICABLE LAWS AND REGULATIONS AND WHICH WILL NOT IMPOSE ANY OBLIGATION ON DRESDNER KLEINWORT. This document is confidential, and no part of it may be reproduced, distributed or transmitted without the prior written permission of Dresdner Kleinwort, except that the recipient may make any disclosure required by law or requested by a regulator having jurisdiction over the recipient. Dresdner Bank AG London Branch. Registered in England and Wales No FC007638. Located at: 30 Gresham Street, London, EC2P 2XY. Incorporated in Germany as a stock corporation with limited liability. A member of the Allianz Group. Contents Section Page: Introduction 1 1. Fundamentals of index volatility, constituent volatility, correlation and dispersion 2 1.1. Definitions 2 1.2. Proxy formulas for realised and implied correlation 5 1.3. Variance dispersion trades 7 2. Toy model for derivatives on realised variance 10 2.1. Model framework 10 2.2. Fair value of derivatives on realised variance. Volatility claims. 11 2.3. Parameter estimation 12 2.4. Model limitations 12 3. Rational pricing of equity correlation swaps 13 3.1. Two-factor toy model 14 3.2. Fair value of the correlation claim 14 3.3. Parameter estimation 15 3.4. Hedging strategy 16 3.5. Model limitations 16 4. Further research 18 5. References 19 6. Appendices 20 A New Approach For Modelling and Pricing Correlation Swaps Fundamentals of index volatility, constituent volatility, correlation and dispersion Introduction This report carries forward an earlier work (2005) on the arbitrage pricing of correlation swaps on a stock index. The theoretical derivations have been made more rigorous, and we also include tentative parameter estimates based on a break-even historical analysis. Our aim here is to provide some elementary fundamental and practical results, which may serve as guiding principles and rules of thumbs for a new category of derivatives on realised volatility and correlation. These ‘statistical derivatives’ have grown in popularity over the past few years, giving sophisticated investors the opportunity to take advantage of specific market structural imbalances. Correlation swaps are over-the-counter derivative instruments allowing to trade the observed correlation between the returns of several assets, against a pre-agreed price. In the equity derivatives sphere, these contracts appeared in the early 2000’s as a means to hedge the parametric risk exposure of exotic desks to changes in correlation. Exotic derivatives indeed frequently involve multiple assets, and their valuation requires a correlation matrix as input parameter. Unlike volatility, whose implied levels have become observable due to the development of listed option markets, implied correlation coefficients are unobservable, which makes the pricing of correlation swaps a perfect example of ‘chicken-egg problem.’ We show how a correlation swap on the constituent stocks of an index can be viewed as a simple derivative on two types of tradable variance – the square of volatility –, and derive an analytical formula for its fair value relying upon dynamic trading of these instruments. The report is organised as follows. Section 1 gives precise definitions of the concepts of realised and implied volatility of an index and its constituent stocks, realised and implied dispersion as well as realised and implied correlation; some key mathematical properties and practical applications are then introduced. Section 2 proposes a one-factor ‘toy model’ for derivatives on realised variance which is a straightforward modification of the Black- Scholes (1973) model; an analytical formula for the fair value of volatility (as opposed to variance) is then derived and used for parameter estimation. Section 3 extends the toy model to two factors in order to derive an analytical formula for the fair value of realised correlation; our numerical results suggest that the fair value of a correlation swap should be close to implied correlation; finally, a formal link between dispersion trading and the hedging strategy for correlation is established. 1. Fundamentals of index volatility, constituent volatility, correlation and dispersion 1.1. Definitions 1 Consider a universe of N stocks S = (Si)i=1..N, and a vector of positive real numbers w = (wi)i=1..N such that Σi=1..N wi = 1. Denote Si(t) the price of stock Si at time t, with convention S(0) = 1, and define their geometric average as: N w (110) i I(t) ≡ ∏()Si (t) i=1 From an econometric point of view, (S, w, I) is a simplified system2 for the calculation of a stock index I with constituent stocks S and weights w. We complete the quantitative setup by considering a probability space (Ω, E, P) with a P-filtration F, and assuming that the vector S of stock prices is an F-adapted, positive Ito process. 1 The positivity and unit sum conditions imply N ≥ 2. 2 In practice, most stock indexes are defined as an arithmetic weighted average, with weights corresponding to market capitalisations; as such, weights change continuously with stock prices. Additionally the constituent stocks are typically reviewed on a quarterly or annual basis. 2 A New Approach For Modelling and Pricing Correlation Swaps Fundamentals of index volatility, constituent volatility, correlation and dispersion Given a time period3 τ and a positive Ito process X, define: (111) ||τ ≡ ds ∫τ −1 (112) στX ()≡ τ (dX ln )2 ∫τ s N 2 S S (113) i σ ()τστ≡ ∑ wi () () i=1 N 2 2 S (114) i ε ()τστ≡ ∑ wi () () i=1 From an econometric point of view, (111) is the length of a time period, (112) is the continuously sampled realised volatility of any positive Ito process X (in particular that of the constituent stock prices S, and the index value I), (113) is the continuously sampled average realised volatility of the constituent stocks4, and (114) corresponds to a realised residual quantity useful below.
Load more

Recommended publications
  • Updated Trading Participants' Trading Manual
    Annexure 3 TRADING MANUAL BURSA MALAYSIA DERIVATIVES BHD TRADING MANUAL (Version 4.10) This manual is the intellectual property of BURSA MALAYSIA. No part of the manual is to be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system, without permission in writing from Head of BMD Exchange Operations. TRADING MANUAL Version History Version Date Author Comments V 1.0 9 Aug 2010 BMDB Initial Version V 1.1 27 Aug 2010 BMDB Updated # 6.6.3 Review of Trades – Price Adjustments and Cancellations V1.2 6 Sep 2010 BMDB Inserted 15. Operator ID (“Tag 50 ID”) Required for All BMD orders traded on CME Globex V1.3 9 Sep 2010 BMDB Update 1. Introduction – 1.6 TPs’ compliance in relation to access, connectivity, specification or use of CME Globex V1.4 13 Sep 2010 BMDB Updated 13. Messaging And Market Performance Protection Policy V1.5 9 Nov 2011 BMDB Inserted 16 Negotiated Large Trade V1.6 18 Nov 2011 BMDB Amended section 16 for typo errors, consistency and clarity. V1.7 24 Nov 2011 BMDB Amended section 16 - Extended NLT cut-off time for FKLI, FKB3 and FMG5 to 4.00pm and for FCPO to 5.00pm. -Amended the NLT Facility Trade Registration form. V1.8 10 Feb 2012 BMDB Updated section 11 EFP to EFRP V1.9 23 Mar 2012 BMDB Amended sections 11 and 16 (forms and processes) V2.0 5 Apr 2012 BMDB Renamed to “Trading Manual” V2.1 14 May 2012 BMDB Updated Section 6 for OKLI and to align with CME practice V2.2 29 May 2012 BMDB i) Updated for OCPO ii) Change of terminology to be consistent with CME iii) Updated Sections 7.7 & 14.1 for consistency with Rules iv) Updated Sections 12.3 & 12.4 for accuracy v) Updated Section 3.1 on options naming convention V2.3 18 Feb 2013 BMDB Updated Section 16 NLT V2.4 3 Apr 2013 BMDB Updated Section 9 Circuit Breaker on timing V2.5 2 Jul 2013 BMDB Updated FGLD.
    [Show full text]
  • Three Essays on Variance Risk and Correlation Risk
    Imperial College London Imperial College Business School Three Essays on Variance Risk and Correlation Risk Xianghe Kong Submitted in part fulfilment of the requirements for the degree of Doctor of Philosophy in Finance of Imperial College and the Diploma of Imperial College October 2010 1 Abstract This thesis focuses on variance risk and correlation risk in the equity market, and consists of three essays. The first essay demonstrates that the variance risk, mea- sured as the difference between the realized return variance and its risk-neutral expectation, is an important determinant of the cross-sectional variation of hedge fund returns. Empirical evidence shows that funds with significantly higher loadings on variance risk outperform lower-loading funds on average. However, they incur severe losses during market downturns. Failure to account for variance risk results in overestimation of funds' absolute returns and underestimation of risk. The results provide important implications for hedge fund risk management and performance evaluations. The second essay examines the empirical properties of a widely-used correlation risk proxy, namely the dispersion trade between the index and individual stock options. I find that discrete hedging errors in such trading strategy can result in incorrect inferences on the magnitude of correlation risk premium and render the proxy unreliable as a measure of pure exposure to correlation risk. I implement a dynamic hedging scheme for the dispersion trade, which significantly improves the estimation accuracy of correlation risk and enhances the risk-return profile of the trading strategy. Finally, the third essay aims to forecast the average pair-wise correlations between stocks in the market portfolio.
    [Show full text]
  • Fast Monte Carlo Simulation for Pricing Covariance Swap Under Correlated Stochastic Volatility Models Junmei Ma, Ping He
    IAENG International Journal of Applied Mathematics, 46:3, IJAM_46_3_09 ______________________________________________________________________________________ Fast Monte Carlo Simulation for Pricing Covariance Swap under Correlated Stochastic Volatility Models Junmei Ma, Ping He Abstract—The modeling and pricing of covariance swap with probability 1 − δ, where σn is the standard deviation derivatives under correlated stochastic volatility models are of estimation of V , δ is significance level and Z δ is the 2 studied. The pricing problem of the derivative under the case δ of discrete sampling covariance is mainly discussed, by efficient quantile of standard normal distribution under 2 . The error pσn is Z δ , which just rely on the deviation σ and sample Control Variate Monte Carlo simulation. Based on the closed n 2 form solutions derived for approximate models with correlated storage n.However, although Monte Carlo method is easy deterministic volatility by partial differential equation method, to implement and effective to solve the multi-dimensional two kinds of acceleration methods are therefore proposed when problem, it’s disadvantage is its slow convergence. The the volatility processes obey the Hull-White stochastic processes. − 1 Through analyzing the moments for the underlying processes, convergence rate of Monte Carlo method is O(n 2 ).We the efficient control volatility under the approximate model have to spend 100 times as much as computer time in order is constructed to make sure the high correlation between the to reduce the error by a factor of 10 [8]. control variate and the problem. The numerical results illustrate Many financial practitioners and researchers concentrate the high efficiency of the control variate Monte Carlo method; on the problem of computational efficiency, several ap- the results coincide with the theoretical results.
    [Show full text]
  • Variance Risk: a Bird's Eye View
    Variance Risk: A Bird’s Eye View∗ Fabian Hollstein† and Chardin Wese Simen‡ January 11, 2018 Abstract Prior research documents a significant variance risk premium (VRP) for the S&P 500 index but only for few equities. Using high-frequency data, we show that these results are affected by measurement errors in the realized variance estimates. We decompose the index VRP into factors related to the VRP of equities and the correlation risk pre- mium. The former mostly drives the variations in the index VRP while the latter mainly captures the level of the index VRP. The two factors predict excess stock returns in the time-series and cross-section, but at different horizons. Together, they improve the return predictability. JEL classification: G11, G12 Keywords: Correlation Swaps, Realized Variance, Return Predictability, Variance Swaps ∗We are grateful to Arndt Claußen, Maik Dierkes, Binh Nguyen, Marcel Prokopczuk, Roméo Tédon- gap and seminar participants at Leibniz University Hannover for helpful comments and discussions. Part of this research project was completed when Chardin Wese Simen visited Leibniz University Hannover. †School of Economics and Management, Leibniz University Hannover, Koenigsworther Platz 1, 30167 Hannover, Germany. Contact: [email protected]. ‡ICMA Centre, Henley Business School, University of Reading, Reading, RG6 6BA, UK. Contact: [email protected]. 1 Introduction Carr & Wu(2009) and Driessen et al.(2009) document a significantly negative av- erage variance swap payoff (VSP) for the U.S. stock market index.1 However, they find significant VSPs only for a small number of individual equities that make up the stock index.
    [Show full text]
  • Correlation Risk and the Cross Section of Hedge Fund Returns
    ‘When There Is No Place to Hide’: Correlation Risk and the Cross-Section of Hedge Fund Returns ANDREA BURASCHI, ROBERT KOSOWSKI and FABIO TROJANI ABSTRACT This paper analyzes the relation between correlation risk and the cross-section of hedge fund re- turns. Legal framework and investment mandate a¤ect the nature of risks that hedge funds are exposed to: Hedge funds’ability to enter long-short positions can be useful to reduce market beta, but it can severely expose the funds to unexpected changes in correlations. We use a novel dataset on correlation swaps and investigate this link. We …nd a number of interesting results. First, the dy- namics of hedge funds’absolute returns are explained to a statistically and economically signi…cant percentage by exposure to correlation risk. Second, di¤erent exposures to correlation risk explain cross-sectional di¤erences in hedge fund excess returns. Third, Fama-Macbeth regressions highlight that correlation risk carries the largest and most signi…cant risk premium in the cross-section of hedge fund returns. Fourth, exposure to correlation risk is linked to an asymmetric risk pro…le: Funds selling protection against correlation increases have maximum drawdowns much higher than funds buying protection against correlation risk. Fifth, failure to account for correlation risk expo- sures leads to a strongly biased estimation of funds’risk-adjusted performance. These …ndings have implications for hedge fund risk management, the categorization of hedge funds according to their risk pro…le and recent legislation that allows mutual funds to follow so-called 130/30 long-short strategies. JEL classi…cation: D9, E3, E4, G11, G14, G23 Keywords: Stochastic correlation and volatility, hedge fund performance, optimal portfolio choice.
    [Show full text]
  • Options Trading
    OPTIONS TRADING: THE HIDDEN REALITY RI$K DOCTOR GUIDE TO POSITION ADJUSTMENT AND HEDGING Charles M. Cottle ● OPTIONS: PERCEPTION AND DECEPTION and ● COULDA WOULDA SHOULDA revised and expanded www.RiskDoctor.com www.RiskIllustrated.com Chicago © Charles M. Cottle, 1996-2006 All rights reserved. No part of this publication may be printed, reproduced, stored in a retrieval system, or transmitted, emailed, uploaded in any form or by any means, electronic, mechanical photocopying, recording, or otherwise, without the prior written permission of the publisher. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold with the understanding that neither the author or the publisher is engaged in rendering legal, accounting, or other professional service. If legal advice or other expert assistance is required, the services of a competent professional person should be sought. From a Declaration of Principles jointly adopted by a Committee of the American Bar Association and a Committee of Publishers. Published by RiskDoctor, Inc. Library of Congress Cataloging-in-Publication Data Cottle, Charles M. Adapted from: Options: Perception and Deception Position Dissection, Risk Analysis and Defensive Trading Strategies / Charles M. Cottle p. cm. ISBN 1-55738-907-1 ©1996 1. Options (Finance) 2. Risk Management 1. Title HG6024.A3C68 1996 332.63’228__dc20 96-11870 and Coulda Woulda Shoulda ©2001 Printed in the United States of America ISBN 0-9778691-72 First Edition: January 2006 To Sarah, JoJo, Austin and Mom Thanks again to Scott Snyder, Shelly Brown, Brian Schaer for the OptionVantage Software Graphics, Allan Wolff, Adam Frank, Tharma Rajenthiran, Ravindra Ramlakhan, Victor Brancale, Rudi Prenzlin, Roger Kilgore, PJ Scardino, Morgan Parker, Carl Knox and Sarah Williams the angel who revived the Appendix and Chapter 10.
    [Show full text]
  • October 4, 2017 Extent, the Dividend Acts Like an Anchor, Slowing the Stock Down
    In many time periods, dividends accounted for much more than 50% of the S&P 500’s total return. Non- dividend-paying stocks only have one way to recognize value… their share prices must rise. Dividend-paying stocks have two ways to recognize value… regular cash dividend payments, and a rise in their share prices. Howard Silverblatt, a Senior Index Analyst with S&P Dow Jones Indices said: “Dividend stocks have several advantages. Since 1926, dividends have accounted for about 42% of investor re- turns, while being less volatile than the market. To some October 4, 2017 extent, the dividend acts like an anchor, slowing the stock down. The beauty of dividends is that you get paid, In Today’s Issue whether or not the market is up.” • The Dividend Issue We know dividends are a crucial component of long- term investment returns. So, how do you scour the • DIVY—The lone ETF that invests in pure dividend growth. smorgasbord of dividend funds available? • REGL and SMDV—Finding the next Dividend Aristocrats through Mid and small-cap dividend growth powerhouses. The general answer is pick a reputable fund family; a fund with a satisfactory asset level (volume too, if it’s an ETF), decent track record and low expenses and a divi- Finding a Needle(s) in the Dividend Haystack dend strategy that makes sense. Last year, Morningstar reported there were 469 US- One way to shorten the list is to strictly use ETFs. Many listed dividend strategies across the open-end, closed- advisors are doing this. According to FPA’s Trends in In- end and ETF universes.
    [Show full text]
  • Equity Correlation
    Equity correlation – explaining the investment opportunity Barclays Capital provides an overview of the equity derivatives flows and products that contribute to dealers’ correlation exposures, presenting several strategies that allow investors, of all types, to take advantage of the resultant opportunities One of the most talked-about metrics in the world of equity options on higher volatility underlyings (for example, stocks for derivatives currently is correlation. While the basic intuition behind single underlying reverse convertibles) and purchase options the metric is quite straightforward, correlation risks and exposures can on lower volatility underlyings (for example, indices for optically be significant. Indeed, dealers have focused much time and effort into higher participation rates in capital guaranteed notes). managing correlation risk. Moreover, many of these efforts have been While many of the above flows are over the counter and aimed at isolating this risk to make it more efficient to trade. evidence is anecdotal at best, we are confident that these flows are responsible for a bulk of dealers’ short correlation exposures. Correlation is in demand We now consider why and how we think investors should take We believe there are two primary flows that contribute to dealers’ advantage of opportunities arising from this phenomenon. overall short correlation positioning. The first flow comes from structured products and is the most important, in our view. As Opportunities for investors investors generally like diversification and limited losses, they prefer On the back of the above flows and recent sell-offs in the equity products with exposures to multiple underlyings and some sort markets, implied correlation (i.e.
    [Show full text]
  • Equity Correlation Swaps: a New Approach for Modelling & Pricing Columbia University — Financial Engineering Practitioners Seminar
    26 February 2007 Equity Correlation Swaps: A New Approach For Modelling & Pricing Columbia University — Financial Engineering Practitioners Seminar Sebastien Bossu Equity Derivatives Structuring — Product Development Equity Correlation Swaps: A New Approach For Modelling & Pricing Disclaimer This document only reflects the views of the author and not necessarily those of Dresdner Kleinwort research, sales or trading departments. This document is for research or educational purposes only and is not intended to promote any financial investment or security. 2 Equity Correlation Swaps: A New Approach For Modelling & Pricing Introduction X The daily life of an Equity Exotic Structurer (1): X Q: ‘Can you price this payoff: ⎡ ⎤ ⎛ ST − S0 ⎞ min⎢20%,max⎜5%, ⎟⎥ ⎣ ⎝ S0 ⎠⎦ ?’ X A: Hedge is a fixed cash flow and a static call spread: 1 1 5% + max[]0, ST −105%× S0 − max[]0, ST −125%× S0 S0 S0 Price: PV(5%) + 1/S0 x (Call Spread 105-125) 3 Equity Correlation Swaps: A New Approach For Modelling & Pricing Introduction X The daily life of an Equity Exotic Structurer (2): X Q: ‘Can you price this payoff: 2 252 ⎛ St ⎞ ∑⎜ln ⎟ t=1 ⎝ St−1 ⎠ ?’ X A: 1-year variance. Hedge is a static portfolio of puts and calls approximating ln(ST/S0), delta-hedged daily. ⎡ 1 1 +∞ 1 ⎤ Price: 2 Put(k)dk + Call(k)dk ⎢ 2 2 ⎥ ⎣∫0 k ∫1 k ⎦ 4 Equity Correlation Swaps: A New Approach For Modelling & Pricing Introduction X The daily life of an Equity Exotic Structurer (3): X Q: ‘Can you price this payoff: 2 ∑ ρi, j N(N −1) i< j where ρi,j is the coefficient of correlation between stocks i and j observed between t = 0 and t = T?’ X A0: ‘Go home’ X A1: ‘Let me program this into our Monte-Carlo engine’ X A2: ‘Let me ask my broker’ 5 Equity Correlation Swaps: A New Approach For Modelling & Pricing Introduction X Pros and cons X A1: Monte-Carlo gives you a price but not a hedge.
    [Show full text]
  • Introduction to Variance Swaps
    Introduction to Variance Swaps Sebastien Bossu, Dresdner Kleinwort Wasserstein, Equity Derivatives Structuring Abstract Bank AG (whether or not acting by its London Branch) and any of its associated or affiliated companies and their directors, representatives or employees. Dresdner The purpose of this article is to introduce the properties of variance swaps, and give in- Kleinwort Wasserstein does not deal for, or advise or otherwise offer any investment sights into the hedging and valuation of these instruments from the particular lens of services to private customers. an option trader. Dresdner Bank AG London Branch, authorised by the German Federal Financial • Section 1 gives general details about variance swaps and their applications. Supervisory Authority and by the Financial Services Authority, regulated by the • Section 2 explains in ‘intuitive’ financial mathematics terms how variance swaps Financial Services Authority for the conduct of designated investment business in the are hedged and priced. UK. Registered in England and Wales No FC007638. Located at: Riverbank House, 2 Swan Lane, London, EC4R 3UX. Incorporated in Germany with limited liability. A Keywords member of Allianz. Variance swap, volatility, path-dependent, gamma risk, static hedge. Disclaimer This document has been prepared by Dresdner Kleinwort Wasserstein and is intend- ed for discussion purposes only. “Dresdner Kleinwort Wasserstein” means Dresdner Payoff 4. The notional is specified in volatility terms (here h50,000 per ‘vega’ or volatility point.) The true notional of the trade, called variance no- A variance swap is a derivative contract which allows investors to trade fu- tional or variance units, is given as: ture realized (or historical) volatility against current implied volatility.
    [Show full text]
  • Correlation Trading Strategies – Opportunities and Limitations
    Correlation Trading Strategies – Opportunities and Limitations Gunter Meissner1 Abstract: Correlation trading has become popular in the investment bank and hedge fund community in the recent past. This paper discusses six types of correlation trading strategies and analysis their opportunities and limitations. The correlation strategies, roughly in chronological order of their occurrence are 1) Empirical Correlation Trading, 2) Pairs Trading, 3) Multi-asset Options, 4) Structured Products, 5) Correlation Swaps, and 6) Dispersion trading. While traders can apply correlation trading strategies to enhance returns, correlation products are also a convenient tool to hedge correlation risk and systemic risk. Keywords: Correlation Trading, Pairs Trading, Multi-asset options, Correlation Swaps, Dispersion Trading Introduction This paper gives an overview of the most popular correlation trading strategies and analysis their opportunities and limitations with respect to enhancing returns. Six correlation strategies are discussed: 1) Empirical Correlation Trading, 2) Pairs Trading, 3) Multi-asset Options, 4) Structured Products, 5) Correlation Swaps, and 6) Dispersion trading. This paper focuses on trading correlation, however, briefly in point 7, the risk managing properties of correlation products are outlined. So without further ado, let’s analyze the correlation trading strategies. 1) Empirical Correlation Trading Empirical Correlation Trading attempts to exploit historically significant correlations within or between financial markets. Numerous financial correlations can be investigated. One area of interest is the autocorrelation between stocks or indices. Figure 1 shows the autocorrelation of the Dow Jones Industrial Index (Dow) from 1920 to 2014: 1 Gunter Meissner is President of Derivatives Software, www.dersoft.com, CEO of Cassandra Capital Management, www.cassandracm.com and Adjunct Professor of MathFinance at NYU-Courant.
    [Show full text]
  • A Proposal for Multi-Asset Generalised Variance Swaps
    A Proposal for Multi-asset Generalised Variance Swaps Subhojit Biswas Indian Statistical Institute, Kolkata [email protected] Diganta Mukherjee Sampling and Official Statistics Unit Indian Statistical Institute, Kolkata [email protected] ∗ August 13, 2019 Abstract This paper proposes swaps on two important new measures of generalized variance, namely the maximum eigen- value and trace of the covariance matrix of the assets involved. We price these generalized variance swaps for financial markets with Markov-modulated volatilities. We consider multiple assets in the portfolio for theoretical purpose and demonstrate our approach with numerical examples taking three stocks in the portfolio. The results obtained in this paper have important implications for the commodity sector where such swaps would be useful for hedging risk. Keywords: Generalized Variance, Swaps, Trace, Maximum Eigenvalue AMS Classification: 91G10, 91G80 arXiv:1908.03899v1 [q-fin.MF] 11 Aug 2019 1 Introduction Covariance and correlation swaps are among recent financial products which are useful for volatility hedging and speculation using two different financial underlying assets. For example, option dependent on exchange rate move- ments, such as those paying in a currency different from the underlying currency, have an exposure to movements of the correlation between the asset and the exchange rate, this risk may be eliminated by using a covariance swap. The literature devoted to the volatility derivatives is growing. The Non-Gaussian Ornstein-Uhlenbeck stochastic volatility model was used by Benth et al. (2007) to study volatility and variance swaps. Broadie and Jain (2008a) evaluated price and hedging strategy for volatility derivatives in the Heston square root stochastic volatility model ∗We thank Indranil Sengupta for helpful comments and suggestions which has helped improve the exposition considerably.
    [Show full text]