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Equity Correlation Swaps: a New Approach for Modelling & Pricing Columbia University — Financial Engineering Practitioners Seminar

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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. Standard multi-asset MC engines require a correlation matrix for input which is held constant. Output price will be the average of the correlation matrix, and hedge is meaningless. X A2: The broker market gives you a hedge but not a (non- arbitrage) price. Until a hedging strategy is identified, market prices are purely driven by supply and demand. However, if such a strategy exists, there is a risk that one side of the market leaks value unknowingly... 6 Equity Correlation Swaps: A New Approach For Modelling & Pricing Agenda 1. Fundamentals of index variance, constituent variance and correlation 2. Toy model for derivatives on realised variance 3. Rational pricing of correlation swaps 7 1. Fundamentals of index variance, constituent variance and correlation 8 Equity Correlation Swaps: A New Approach For Modelling & Pricing 1. Fundamentals of index variance, constituent variance and correlation 1.1Realised and Implied Correlation 1.2Correlation Proxy 1.3Application: Variance Dispersion Trading 9 Fundamentals of index variance, constituent variance and correlation Realised and Implied Correlation X Realised Correlation X Pair of stocks: statistical coefficient of correlation between the two time series of daily log-returns X Basket of N stocks: average of the N(N-1)/2 pair-wise correlation coefficients X Implied Correlation X Pair of stocks: usually unobservable X Basket of N stocks: occasionally observable through quotes on basket calls or puts from exotic desks X Liquid indices: observable for listed strikes and maturities 10 Fundamentals of index variance, constituent variance and correlation Realised and Implied Correlation X Realised Correlation Definitions (Equal Weights Assumption) X Average pair-wise (‘naive’) definition: 2 ρPairwise ≡ ∑ ρi, j N(N −1) i< j X Canonical (econometric) definition: 1 N 2 2 σ σ ρ σ Index − 2 ∑σ i ∑ i j i, j N i=1 i< j ρCanonical ≡ N ≈ 2 1 2 < σ σ σ − σ ∑ i j Constituent 2 ∑ i i< j N i=1 N N 2 1 2 ⎛ 1 ⎞ σ i ≈ ⎜ σ i ⎟ ∑ > ∑ N i=1 ⎝ N i=1 ⎠ 11 Fundamentals of index variance, constituent variance and correlation Realised and Implied Correlation X ‘Naive’ Realised Correlation calculation example: DAX 30 Constituent Stocks ADS ALV ALT BAS BAY BMW VOW GY GY GY GY GY GY … GY ADS GY 100% 29% -1% 28% 32% 15% … 22% ALV GY 29% 100% 11% 50% 44% 45% … 36% ALT GY -1% 11% 100% 9% 7% 5% … 5% BAS GY 28% 50% 9% 100% 65% 52% … 33% BAY GY 32% 44% 7% 65% 100% 39% … 32% BMW GY 15% 45% 5% 52% 39% 100% … 53% … … … … … … … 100% … VOW GY 22% 36% 5% 33% 32% 53% … 100% ρPair-wise = 31.26% 12 Fundamentals of index variance, constituent variance and correlation Realised and Implied Correlation X Implied Correlation Definition (Equal Weights Assumption) X No ‘naive’ definition (pair-wise implied correlations not observable) X Canonical (econometric) definition: N *2 1 *2 σ Index − 2 ∑σ i * N i=1 ρCanonical ≡ N *2 1 *2 σ Constituent − 2 ∑σ i N i=1 X Note that the implied volatility surface translates into an implied correlation surface. We use fair variance swap strikes for σ*’s unless mentioned otherwise. 13 Fundamentals of index variance, constituent variance and correlation Realised and Implied Correlation X Implied Correlation calculation example: DAX 1Y ATM Index Vol: 16.88 2 2 * 16.88 − 4.38 1Y ATM Constituent Vol: 23.76 ρCanonical = = 48.73% 23.762 − 4.382 1Y ATM Residual Vol1: 4.38 1Y 90% Index Vol: 18.97 18.972 − 4.522 1Y 90% Constituent Vol: 24.54 ρ * = = 58.31% Canonical 24.542 − 4.522 1Y 90% Residual Vol1: 4.52 N 1 *2 1 Residual Volatility is given as the square root of 2 ∑σ i N i=1 14 Equity Correlation Swaps: A New Approach For Modelling & Pricing 1. Fundamentals of index variance, constituent variance and correlation 1.1Realised and Implied Correlation 1.2Correlation Proxy 1.3Application: Variance Dispersion Trading 15 Fundamentals of index variance, constituent variance and correlation Correlation Proxy X The previous definitions are easily generalised to arbitrary index weights X Proxy Formula: Under certain regularity conditions on the weights, residual volatility becomes negligible and we have: 2 ⎧ ⎛ σ ⎞ ⎪ρ ⎯⎯→⎯ ⎜ Index ⎟ ≡ ρˆ Canonical N →+∞ ⎜ ⎟ ⎪ ⎝σ Constituent ⎠ ⎨ 2 ⎪ ⎛ σ * ⎞ ρ * ⎯⎯→⎯ ⎜ Index ⎟ ≡ ρˆ * ⎪ Canonical N →+∞ ⎜ * ⎟ ⎩ ⎝σ Constituent ⎠ MaxWeight X Condition: = o( N ) MinWeight 16 Fundamentals of index variance, constituent variance and correlation Correlation Proxy X Realised Correlation Proxy calculation example DAX Volatility = 12.79% DAX Constituent Volatility = 20.66% Proxy for DAX Correlation ≈ (12.79/20.66)2 ≈ 38.33% ‘Naive’ DAX Correlation = 31.26% Canonical DAX Correlation = 36.15% X Implied Correlation Proxy calculation example (ATM) DAX Volatility = 16.88% DAX Constituent Volatility = 23.76% Proxy for DAX Correlation ≈ (16.88/23.76)2 ≈ 50.47% Canonical DAX Correlation = 48.73% 17 Fundamentals of index variance, constituent variance and correlation Correlation Proxy X Correlation (realised and implied) is thus close to the ratio of index variance to average constituent variance Index Variance Correlation ≈ Average Constituent Variance X This is interesting because index variance and average constituent variance can be traded on the OTC variance swap market 18 Equity Correlation Swaps: A New Approach For Modelling & Pricing 1. Fundamentals of index variance, constituent variance and correlation 1.1Realised and Implied Correlation 1.2Correlation Proxy 1.3Application: Variance Dispersion Trading 19 Fundamentals of index variance, constituent variance and correlation Application: Variance Dispersion Trading X Variance Dispersion Trades Spread of variance swap positions between an index and its constituents, usually: X Long Average Constituent Variance X Short Index Variance 2 2 2 X - Payoff = σ Constituent −σ Index = σ Constituent ×[1− ρˆ]≥ 0 *2 *2 *2 * X - Cost = σ Constituent −σ Index = σ Constituent ×[]1− ρˆ ≥ 0 X Exposure: long volatility, short correlation 20 Fundamentals of index variance, constituent variance and correlation Application: Variance Dispersion Trading X By underweighting the constituents’ leg with a factor β = ρ* < 1, several benefits are obtained: X Vega-Neutrality On trade date, if constituent variance goes up 1 point and implied correlation is unchanged, index variance would go up by ρ* points and the P&L is: β x 1pt – ρ* pts = 0 X Zero cost *2 *2 Cost = βσ Constituent −σ Index = 0 X Straightforward p&l decomposition 2 * P & L = Payoff − Cost = σ Constituent ×[ρˆ − ρˆ] { { Zero β 21 Fundamentals of index variance, constituent variance and correlation Application: Variance Dispersion Trading X Note: X In practice, constituent variance is often ‘packaged’ using a common ‘Vega Notional’: 1 N σ 2 Vega Notional i × ∑ * N i=1 2σ i X In this case the aggregate vega is: N 1 * * Vega Not.× ∑σ i = Vega Not.×σ Constituent N i=1 X and the Beta coefficient for a Vega-Neutral Dispersion trade would be β = √ρ*. 22 2. Toy Model for Derivatives on Realised Variance 23 Equity Correlation Swaps: A New Approach For Modelling & Pricing 2. Toy Model for Derivatives on Realised Variance 2.1Realised Variance: A Tradable Asset 2.2Toy Model for Realised Variance 2.3Application: Volatility Swap 2.4Parameter Estimation 2.5Model Limitations 24 Toy Model for Derivatives on Realised Variance Realised Variance: A Tradable Asset X Variance Swap At expiry two parties exchange the realised variance of e.g. DJ EuroStoxx 50 daily log-returns, against a strike (‘implied variance’) X OTC market has become very liquid on S&P 500 and DJ EuroStoxx 50, with bid-offer spreads sometimes as tight as ¼ vega. X CBOE introduced Three-Month Variance Futures on the S&P 500 in 2004. 25 Equity Correlation Swaps: A New Approach For Modelling & Pricing 2. Toy Model for Derivatives on Realised Variance 2.1Realised Variance: A Tradable Asset 2.2Toy Model for Realised Variance 2.3Application: Volatility Swap 2.4Parameter Estimation 2.5Model Limitations 26 Toy Model for Derivatives on Realised Variance Which Model for Realised Variance? Fischer Black: ‘I start with the view that nothing is really constant.
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