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. 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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.