Quadratic Variance Swap Models Loriano Mancini Swiss Finance - PowerPoint PPT Presentation

Quadratic Variance Swap Models Loriano Mancini Swiss Finance Institute and EPFL joint with Damir Filipovi´ c , EPFL and Elise Gourier , Princeton University Workshop on Stochastic and Quantitative Finance Imperial College London, 28–29 November 2014

Variance Swap (time 0) 30 20 10 Payoff 0 −10 −20 −30 0 50 100 150 200 250 Time (days) 1200 1150 Stock Price 1100 1050 1000 0 50 100 150 200 250 Time (days) 2/30

Variance Swap (time T ) 30 20 10 Payoff 0 −10 −20 −30 0 50 100 150 200 250 Time (days) 1200 1150 Stock Price 1100 1050 1000 0 50 100 150 200 250 Time (days) 3/30

Outline Motivation Variance Swap Quadratic Model Optimal Investment 4/30

Outline Motivation Variance Swap Quadratic Model Optimal Investment 5/30 Motivation

Volatility Derivatives Over last two decades, large demand of volatility derivatives: variance swaps , volatility swaps, corridor integrated variance,. . . Variance swaps: traded over-the-counter ◮ on various underlying assets (equity indices, exchange/interest rates, commodities, etc.) ◮ at many different maturities ( ⇒ term structure) 6/30 Motivation

Why Variance Swaps? Variance swaps have two distinctive features: 1. Easier to hedge than other volatility derivatives: static position in options and dynamic trading of futures; Dupire (1993), Neuberger (1994), Carr and Madan (1998), a.o. 2. Direct exposure to variance risk over a fixed time horizon. CBOE futures and options on VIX not equally direct exposure ◮ VIX index (30-day S&P 500 volatility index) ◮ introduced in 1993 (back-calculated to 1990, revised in 2003) ◮ 3/2004 futures on VIX ◮ 2/2006 European options on VIX ◮ 12/2012 “S&P 500 Variance Futures” 7/30 Motivation

Wall Street Journal, 22 October 2008 8/30 Motivation

Contribution 1. Novel and flexible model for term structure of variance swaps 2. Dynamic optimal investment in variance swaps, S&P 500, index option, and risk-free bond 9/30 Motivation

Outline Motivation Variance Swap Quadratic Model Optimal Investment Variance Swap 10/30

Setup ◮ S t index price (S&P 500), under Q : dS t � = r t dt + σ t dB t + ξ ( χ ( dt , d ξ ) − ν t ( d ξ ) dt ) S t − R ◮ Quadratic variation over horizon [ t , t + τ ]: �� t + τ � t + τ QV ( t , t + τ ) = 1 � � σ 2 (log(1 + ξ )) 2 χ ( ds , d ξ ) s ds + τ t t R ◮ Variance Swap payoff: QV ( t , t + τ ) − VS ( t , t + τ ) ◮ Variance Swap rate (depends on τ ⇒ term structure): VS ( t , t + τ ) = E Q t [ QV ( t , t + τ )] Variance Swap 11/30

Variance Swap Data set 80 70 60 Variance Swap Rate % 50 40 30 24 20 10 12 97 98 99 00 01 02 03 04 05 06 07 08 09 10 6 23 Maturity months Year � Figure: Variance swap rates, VS ( t , t + τ ) × 100, on the S&P 500 index from 4-Jan-1996 to 2-Sep-2010, daily quotes. Source: Bloomberg. Variance Swap 12/30

Summary Statistics Variance Swap rates τ Mean Std Skew Kurt 2 22.14 8.18 1.53 7.08 3 22.32 7.81 1.32 6.05 6 22.87 7.40 1.10 4.97 12 23.44 6.88 0.80 3.77 24 23.93 6.48 0.57 2.92 Realized Variances 2 18.90 12.40 4.31 28.40 3 19.06 12.04 3.80 21.81 6 19.46 11.33 2.93 13.17 12 20.13 10.47 1.97 6.86 24 20.60 8.81 1.09 3.48 Table: Daily data from 4-Jan-1996 to 2-Sep-2010. Volatility percentage units. Variance Swap 13/30

Outline Motivation Variance Swap Quadratic Model Optimal Investment 14/30 Quadratic Model

Quadratic Variance Swap Model ◮ X is m -dimensional diffusion state process: dX t = µ ( X t ) dt + Σ( X t ) dW t ◮ X is quadratic if µ ( x ) = b + β x m m Σ( x )Σ( x ) ⊤ = a + � � α k x k + A kl x k x l k =1 k , l =1 ◮ Define spot variance � v t = VS ( t , t ) = σ 2 (log(1 + ξ )) 2 ν t ( d ξ ) t + R ◮ A quadratic variance swap model is obtained when v t = φ + ψ ⊤ X t + X ⊤ t π X t 15/30 Quadratic Model

Term Structure of Variance Swaps Quadratic variance swap model admits a quadratic term structure: 1 VS ( t , T ) = E Q t [ QV ( t , T )] = T − t G ( T − t , X t ) with G ( τ, x ) = Φ( τ ) + Ψ( τ ) ⊤ x + x ⊤ Π( τ ) x and Φ, Ψ and Π satisfy a linear system of ODEs. 16/30 Quadratic Model

Model Selection Do we need the quadratic feature? Data: Daily variance swap rates, and quadratic variation from intraday futures returns ◮ In-sample (pre-crisis): Jan 4, 1996 to Apr 2, 2007 ◮ Out-of-sample: Apr 3, 2007 to Jun 7, 2010 Method: Maximum Likelihood with Unscented Kalman filter Estimation results: ◮ Good fit of the bivariate quadratic model (likelihood tests, AIC and BIC criteria, pricing errors, forecasting power) ◮ Somewhat better than affine model with jumps 17/30 Quadratic Model

Fitting Variance Swap Rates T = 2 months 80 Model−based VS rates 70 Actual VS rates 60 50 40 30 20 10 1998 2000 2002 2004 2006 2008 2010 T = 24 months 50 Model−based VS rates 45 Actual VS rates 40 35 30 25 20 15 10 1998 2000 2002 2004 2006 2008 2010 18/30 Quadratic Model

Outline Motivation Variance Swap Quadratic Model Optimal Investment Optimal Investment 19/30

Optimal Portfolio Problem Maximize expected utility from terminal wealth V T of a power utility investor with constant relative risk aversion (CRRA) η � V 1 − η � n t , w t ,φ t , 0 ≤ t ≤ T E P T max 1 − η By dynamically and optimally investing: ◮ n t = ( n 1 t , . . . , n nt ) ⊤ relative notional exposures to each on-the-run τ i - variance swap , i = 1 , . . . , n ◮ w t fraction of wealth invested in stock index ◮ φ t fraction of wealth invested in index option ◮ and risk-free bond Optimal Investment 20/30

Investing in a Variance Swap ◮ Variance swap issued at t ∗ with maturity T ∗ = t ∗ + τ ◮ Spot value Γ t at date t ∈ [ t ∗ , T ∗ ] of a one dollar notional long position in this variance swap: �� T ∗ � �� e − r ( T ∗ − t ) 1 Γ t = E Q v s ds − τ VS ( t ∗ , T ∗ ) t τ t ∗ �� t = e − r ( T ∗ − t ) � t ∗ v s ds + ( T ∗ − t ) VS ( t , T ∗ ) − τ VS ( t ∗ , T ∗ ) τ ◮ Extends to τ -variance swaps issued at a sequence of inception dates 0 = t ∗ 0 < t ∗ 1 < · · · , with t ∗ k +1 − t ∗ k ≤ τ . At any date t ∈ [ t ∗ k , t ∗ k +1 ) the investor takes a position in the respective on-the-run τ -variance swap with maturity T ∗ ( t ) = t ∗ k + τ . Optimal Investment 21/30

Investing in an Index Option ◮ Assume: index price jumps by a deterministic size ξ > − 1 ◮ One index option needed to complete the market, with price O t = O ( S t , X t ). The Q -dynamics of O t � ∂ s O t S t σ ( X t ) R ( X t ) ⊤ + ∇ x O ⊤ � dO t = r O t dt + t Σ( X t ) dW t + ∆ O t ( dN t − ν Q ( X t ) dt ) ◮ Index put option in our empirical analysis Optimal Investment 22/30

Wealth Dynamics ◮ Resulting wealth process has Q -dynamics dV t dS t dO t = n ⊤ + (1 − n ⊤ t d Γ t + w t + φ t t Γ t − w t − φ t ) r dt V t − S t − O t − = r dt + θ W ⊤ dW t + θ N t ξ ( dN t − ν Q ( X t ) dt ) t   n t � θ W � ◮ θ W and θ N t  with t are defined by = G t w t t θ N  t φ t ∇ x O t   D t 0 m × 1 O t − � Σ( X t ) ⊤ � σ ( X t ) R ( X t ) 0 d × 1 ∂ s O t S t 0 1 × n 1 G t =   O t − 0 1 0 1 × m   ∆ O t 0 1 × n 1 ξ O t − and D t is the m × n matrix whose i th column is given by e − r ( T ∗ � i ( t ) − t ) /τ i � ∇ x G ( T ∗ i ( t ) − t , X t ) Optimal Investment 23/30

Optimal Portfolio Problem ◮ Maximize expected utility from terminal wealth V T of a power utility investor with constant relative risk aversion (CRRA) η � V 1 − η � n t , w t ,φ t , 0 ≤ t ≤ T E P T max 1 − η ◮ Pricing kernel: � ν Q ( X t ) d π t � = − r dt − Λ( X t ) ⊤ dW P ( dN t − ν P ( X t ) dt ) t + ν P ( X t ) − 1 π t − ◮ Assumption: The market is complete with respect to stock index , index option , and n on-the-run τ i - variance swaps . Thus, n = m = d − 1, and the ( d + 1) × ( d + 1) matrix G t is invertible dt ⊗ d Q -a.s. Optimal Investment 24/30

Optimal Portfolio Problem: Solution via HJB ∂ 2 J � ∂ J ∂ t + ∂ J + 1 � � r + θ W ⊤ Λ( x ) − θ N ξν Q ( x ) ∂ v 2 v 2 θ W ⊤ θ W 0 = max ∂ v v 2 θ W , θ N m ∂ 2 J + ∇ x J ⊤ ( µ ( x ) + Σ( x )Λ( x )) + 1 � Σ( x )Σ( x ) ⊤ � � 2 ∂ x i ∂ x j ij i , j =1 � ∂ J � � � � + θ W ⊤ v Σ( x ) ⊤ ∇ x J ( t , v (1 + θ N ξ ) , x ) − J ( t , v , x ) ν P ( x ) + ∂ v Optimal Allocation : There exists an optimal strategy n ∗ t , w ∗ t , φ ∗ t recovered from: = 1 θ W ∗ η Λ( X t ) + Σ( X t ) ⊤ ∇ x h ( T − t , X t ) t �� ν P ( X t ) � 1 /η � = 1 θ N ∗ − 1 t ξ ν Q ( X t ) where h is such that e h satisfies a known PDE Optimal Investment 25/30

Optimal Investment in VS: Short-Long Strategy 2 n 1t n 2t 1 0 −1 −2 −3 −4 −5 1998 2000 2002 2004 2006 2008 2010 ◮ Short position in 2-year VS (earn variance risk premium), long position in 3-month VS (hedge volatility risk) ◮ Periodic patterns in n t ◮ Based on bivariate quadratic model Optimal Investment 26/30