Modeling Divergence Swap Rates
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1 Modeling Divergence Swap Raes Pior Or lowski Universiy of Lugano and Swiss Finance Insiue May 20 h, 2016, Chicago Or lowski SFI) Modeling Divergence Swap Raes R in Finance 1 / 8
2 From VIX o power divergence Schneider and Trojani 2015)) CBOE 2000) calculaes he Volailiy Index... VIX 2 = 2 T j K j Q K Kj 2 j ) 2 T 0 [ Q K) dk = 2E Q K 2 ln F T + D py, x) = y p x p pp 1) x p 1 y x) p 1 p R D py, x) = O ln y x ] δ τ 1F τ F τ 1) D 0 = lim p 0 D p ) 2 for ln y x 0 VIX 2 wih he appropriae realized leg rading sraegy is an exac 0-divergence swap. Almos variance swap. Family of power divergence funcions D p defines family of firs-order) quadraic variaion swaps. Compare wih Bondarenko 2014); Marin 2012); Lee 2010). Weighs 2 D K pk, x) = 1. 2 K p HF limis: 1 F p D pf τ, F τ 1) 1 2 Fs 1/F p scaling essenial noaion: D p = σ 2 s ds + DpFs,F ) F p s T D pf s, F s ). Or lowski SFI) Modeling Divergence Swap Raes R in Finance 2 / 8
3 From VIX o power divergence Schneider and Trojani 2015)) CBOE 2000) calculaes he Volailiy Index... VIX 2 = 2 T j K j Q K Kj 2 j ) 2 T 0 [ Q K) dk = 2E Q K 2 ln F T + D py, x) = y p x p pp 1) x p 1 y x) p 1 p R D py, x) = O ln y x ] δ τ 1F τ F τ 1) D 0 = lim p 0 D p ) 2 for ln y x 0 VIX 2 wih he appropriae realized leg rading sraegy is an exac 0-divergence swap. Almos variance swap. Family of power divergence funcions D p defines family of firs-order) quadraic variaion swaps. Compare wih Bondarenko 2014); Marin 2012); Lee 2010). Weighs 2 D K pk, x) = 1. 2 K p HF limis: 1 F p D pf τ, F τ 1) 1 2 Fs 1/F p scaling essenial noaion: D p = σ 2 s ds + DpFs,F ) F p s T D pf s, F s ). Or lowski SFI) Modeling Divergence Swap Raes R in Finance 2 / 8
4 From VIX o power divergence Schneider and Trojani 2015)) CBOE 2000) calculaes he Volailiy Index... VIX 2 = 2 T j K j Q K Kj 2 j ) 2 T 0 [ Q K) dk = 2E Q K 2 ln F T + D py, x) = y p x p pp 1) x p 1 y x) p 1 p R D py, x) = O ln y x ] δ τ 1F τ F τ 1) D 0 = lim p 0 D p ) 2 for ln y x 0 VIX 2 wih he appropriae realized leg rading sraegy is an exac 0-divergence swap. Almos variance swap. Family of power divergence funcions D p defines family of firs-order) quadraic variaion swaps. Compare wih Bondarenko 2014); Marin 2012); Lee 2010). Weighs 2 D K pk, x) = 1. 2 K p HF limis: 1 F p D pf τ, F τ 1) 1 2 Fs 1/F p scaling essenial noaion: D p = σ 2 s ds + DpFs,F ) F p s T D pf s, F s ). Or lowski SFI) Modeling Divergence Swap Raes R in Finance 2 / 8
5 From VIX o power divergence Schneider and Trojani 2015)) CBOE 2000) calculaes he Volailiy Index... VIX 2 = 2E Q [ ln F T + F τ F τ 1 F τ 1 ] E Q σs 2 ds + 2D 0F s, F s ) s T D py, x) = y p x p pp 1) x p 1 y x) p 1 p R D py, x) = O ln y x D 0 = lim D p p 0 ) 2 y for ln x 0 VIX 2 wih he appropriae realized leg rading sraegy is an exac 0-divergence swap. Almos variance swap. Family of power divergence funcions D p defines family of firs-order) quadraic variaion swaps. Compare wih Bondarenko 2014); Marin 2012); Lee 2010). Weighs 2 D K pk, x) = 1. 2 K p HF limis: 1 F p D pf τ, F τ 1) 1 2 Fs 1/F p scaling essenial noaion: Dp = σ 2 s ds + DpFs,F ) F p s T D pf s, F s ). Or lowski SFI) Modeling Divergence Swap Raes R in Finance 2 / 8
6 From VIX o power divergence Schneider and Trojani 2015)) CBOE 2000) calculaes he Volailiy Index... The foundaion of he volailiy / variance swap marke VIX 2 = 2E Q [ ln F T + F τ F τ 1 F τ 1 ] E Q D py, x) = y p x p pp 1) x p 1 y x) p 1 p R D py, x) = O ln y x σ 2 s ds + s T D 0 = lim D p p 0 ) 2 y for ln x 0 2D 0F s, F s ) VIX 2 wih he appropriae realized leg rading sraegy is an exac 0-divergence swap. Almos variance swap. Family of power divergence funcions D p defines family of firs-order) quadraic variaion swaps. Compare wih Bondarenko 2014); Marin 2012); Lee 2010). Weighs 2 D K pk, x) = 1. 2 K p HF limis: 1 F p D pf τ, F τ 1) 1 2 Fs 1/F p scaling essenial noaion: Dp = σ 2 s ds + DpFs,F ) F p s T D pf s, F s ). Or lowski SFI) Modeling Divergence Swap Raes R in Finance 2 / 8
7 From VIX o power divergence Schneider and Trojani 2015)) CBOE 2000) calculaes he Volailiy Index... The foundaion of he volailiy / variance swap marke [ T ] VIX 2 = 2E Q D 0F τ, F τ 1) E Q σs 2 ds + 2D 0F s, F s ) D py, x) = y p x p pp 1) x p 1 y x) p 1 p R D py, x) = O ln y x s T D 0 = lim D p p 0 ) 2 y for ln x 0 VIX 2 wih he appropriae realized leg rading sraegy is an exac 0-divergence swap. Almos variance swap. Family of power divergence funcions D p defines family of firs-order) quadraic variaion swaps. Compare wih Bondarenko 2014); Marin 2012); Lee 2010). Weighs 2 D K pk, x) = 1. 2 K p HF limis: 1 F p D pf τ, F τ 1) 1 2 Fs 1/F p scaling essenial noaion: Dp = σ 2 s ds + DpFs,F ) F p s T D pf s, F s ). Or lowski SFI) Modeling Divergence Swap Raes R in Finance 2 / 8
8 From VIX o power divergence Schneider and Trojani 2015)) CBOE 2000) calculaes he Volailiy Index... The foundaion of he volailiy / variance swap marke [ T ] VIX 2 = 2E Q D 0F τ, F τ 1) E Q σs 2 ds + 2D 0F s, F s ) D py, x) = y p x p pp 1) x p 1 y x) p 1 p R D py, x) = O ln y x s T D 0 = lim D p p 0 ) 2 y for ln x 0 VIX 2 wih he appropriae realized leg rading sraegy is an exac 0-divergence swap. Almos variance swap. Family of power divergence funcions D p defines family of firs-order) quadraic variaion swaps. Compare wih Bondarenko 2014); Marin 2012); Lee 2010). Weighs 2 D K pk, x) = 1. 2 K p HF limis: 1 F p D pf τ, F τ 1) 1 2 Fs 1/F p scaling essenial noaion: Dp = σ 2 s ds + DpFs,F ) F p s T D pf s, F s ). Or lowski SFI) Modeling Divergence Swap Raes R in Finance 2 / 8
9 From VIX o power divergence Schneider and Trojani 2015)) CBOE 2000) calculaes he Volailiy Index... The foundaion of he volailiy / variance swap marke [ T ] VIX 2 = 2E Q D 0F τ, F τ 1) E Q σs 2 ds + 2D 0F s, F s ) D py, x) = y p x p pp 1) x p 1 y x) p 1 p R D py, x) = O ln y x s T D 0 = lim D p p 0 ) 2 y for ln x 0 VIX 2 wih he appropriae realized leg rading sraegy is an exac 0-divergence swap. Almos variance swap. Family of power divergence funcions D p defines family of firs-order) quadraic variaion swaps. Compare wih Bondarenko 2014); Marin 2012); Lee 2010). Weighs 2 D K pk, x) = 1. 2 K p HF limis: 1 F p D pf τ, F τ 1) 1 2 Fs 1/F p scaling essenial noaion: Dp = σ 2 s ds + DpFs,F ) F p s T D pf s, F s ). Or lowski SFI) Modeling Divergence Swap Raes R in Finance 2 / 8
10 From VIX o power divergence Schneider and Trojani 2015)) CBOE 2000) calculaes he Volailiy Index... The foundaion of he volailiy / variance swap marke [ T ] VIX 2 = 2E Q D 0F τ, F τ 1) E Q σs 2 ds + 2D 0F s, F s ) D py, x) = y p x p pp 1) x p 1 y x) p 1 p R D py, x) = O ln y x s T D 0 = lim D p p 0 ) 2 y for ln x 0 VIX 2 wih he appropriae realized leg rading sraegy is an exac 0-divergence swap. Almos variance swap. Family of power divergence funcions D p defines family of firs-order) quadraic variaion swaps. Compare wih Bondarenko 2014); Marin 2012); Lee 2010). Weighs 2 D K pk, x) = 1. 2 K p HF limis: 1 F p D pf τ, F τ 1) 1 2 Fs 1/F p scaling essenial noaion: Dp = σ 2 s ds + DpFs,F ) F p s T D pf s, F s ). Or lowski SFI) Modeling Divergence Swap Raes R in Finance 2 / 8
11 From VIX o power divergence Schneider and Trojani 2015)) CBOE 2000) calculaes he Volailiy Index... The foundaion of he volailiy / variance swap marke [ T ] VIX 2 = 2E Q D 0F τ, F τ 1) E Q σs 2 ds + 2D 0F s, F s ) D py, x) = y p x p pp 1) x p 1 y x) p 1 p R D py, x) = O ln y x s T D 0 = lim D p p 0 ) 2 y for ln x 0 VIX 2 wih he appropriae realized leg rading sraegy is an exac 0-divergence swap. Almos variance swap. Family of power divergence funcions D p defines family of firs-order) quadraic variaion swaps. Compare wih Bondarenko 2014); Marin 2012); Lee 2010). Weighs 2 D K pk, x) = 1. 2 K p HF limis: 1 F p D pf τ, F τ 1) 1 2 Fs 1/F p scaling essenial noaion: Dp = σ 2 s ds + DpFs,F ) F p s T D pf s, F s ). Or lowski SFI) Modeling Divergence Swap Raes R in Finance 2 / 8
12 From VIX o power divergence Schneider and Trojani 2015)) CBOE 2000) calculaes he Volailiy Index... The foundaion of he volailiy / variance swap marke [ T ] VIX 2 = 2E Q D 0F τ, F τ 1) E Q σs 2 ds + 2D 0F s, F s ) D py, x) = y p x p pp 1) x p 1 y x) p 1 p R D py, x) = O ln y x s T D 0 = lim D p p 0 ) 2 y for ln x 0 VIX 2 wih he appropriae realized leg rading sraegy is an exac 0-divergence swap. Almos variance swap. Family of power divergence funcions D p defines family of firs-order) quadraic variaion swaps. Compare wih Bondarenko 2014); Marin 2012); Lee 2010). Weighs 2 D K pk, x) = 1. 2 K p HF limis: 1 F p D pf τ, F τ 1) 1 2 Fs 1/F p scaling essenial noaion: Dp = σ 2 s ds + DpFs,F ) F p s T D pf s, F s ). Or lowski SFI) Modeling Divergence Swap Raes R in Finance 2 / 8
13 Higher-order swaps AJD models no bad a fiing TS of Variance Swaps how many SV facors? Gruber e al. 2015); Andersen e al. 2015). Variaion in p in divergence swaps anoher dimension for fiing. Or lowski SFI) Modeling Divergence Swap Raes R in Finance 3 / 8
14 Higher-order swaps AJD models no bad a fiing TS of Variance Swaps how many SV facors? Gruber e al. 2015); Andersen e al. 2015). Variaion in p in divergence swaps anoher dimension for fiing. Take a difference ε Dp0 +εf T, ) F p 0+ε ) Dp 0 εf T, ) F p = O ln F ) 3 T 0 ε Or lowski SFI) Modeling Divergence Swap Raes R in Finance 3 / 8
15 Higher-order swaps AJD models no bad a fiing TS of Variance Swaps how many SV facors? Gruber e al. 2015); Andersen e al. 2015). Variaion in p in divergence swaps anoher dimension for fiing.... or differeniae. D pf T, ) p F p = O ln F ) 3 T Or lowski SFI) Modeling Divergence Swap Raes R in Finance 3 / 8
16 Higher-order swaps AJD models no bad a fiing TS of Variance Swaps how many SV facors? Gruber e al. 2015); Andersen e al. 2015). Variaion in p in divergence swaps anoher dimension for fiing. Rinse and repea. 2 D pf T, ) p 2 F p = O ln F ) 4 T Or lowski SFI) Modeling Divergence Swap Raes R in Finance 3 / 8
17 Higher-order swaps AJD models no bad a fiing TS of Variance Swaps how many SV facors? Gruber e al. 2015); Andersen e al. 2015). Variaion in p in divergence swaps anoher dimension for fiing. Define S p := D pf T, ) p F p Q p := 2 D pf T, ) p 2 F p Exacly radable wih saic opion porfolio and forward rading under any dynamics. Or lowski SFI) Modeling Divergence Swap Raes R in Finance 3 / 8
18 Higher-order swaps AJD models no bad a fiing TS of Variance Swaps how many SV facors? Gruber e al. 2015); Andersen e al. 2015). Variaion in p in divergence swaps anoher dimension for fiing. Define S p := D pf T, ) p F p Q p := 2 D pf T, ) p 2 F p Exacly radable wih saic opion porfolio and forward rading under any dynamics. Corresponding realized measures: S pf τ, F τ 1) 1 Fs ln Fs σs 2 ds + 2 Q pf τ, F τ 1) 1 2 Fs s T ln 2 F s σ 2 s ds + s T Fs Fs ln Fs D pf s, F s ) ln 2 F s D pf s, F s ) Or lowski SFI) Modeling Divergence Swap Raes R in Finance 3 / 8
19 Higher-order swaps AJD models no bad a fiing TS of Variance Swaps how many SV facors? Gruber e al. 2015); Andersen e al. 2015). Variaion in p in divergence swaps anoher dimension for fiing. Define S p := D pf T, ) p F p Q p := 2 D pf T, ) p 2 F p Exacly radable wih saic opion porfolio and forward rading under any dynamics. Corresponding realized measures: S pf τ, F τ 1) 1 Fs ln Fs σs 2 ds + 2 Q pf τ, F τ 1) 1 2 Fs s T ln 2 F s σ 2 s ds + s T Fs Fs ln Fs D pf s, F s ) ln 2 F s D pf s, F s ) Fully replicable realized measures allow o define higher-order risk premia. Or lowski SFI) Modeling Divergence Swap Raes R in Finance 3 / 8
20 Higher-order swaps AJD models no bad a fiing TS of Variance Swaps how many SV facors? Gruber e al. 2015); Andersen e al. 2015). Variaion in p in divergence swaps anoher dimension for fiing. Define S p := D pf T, ) p F p Q p := 2 D pf T, ) p 2 F p Exacly radable wih saic opion porfolio and forward rading under any dynamics. Corresponding realized measures: S pf τ, F τ 1) 1 Fs ln Fs σs 2 ds + 2 Q pf τ, F τ 1) 1 2 Fs s T ln 2 F s σ 2 s ds + s T Fs Fs ln Fs D pf s, F s ) ln 2 F s D pf s, F s ) Fully replicable realized measures allow o define higher-order risk premia. D p, S p and Q p highly correlaed sandardise: S p := Sp D 3/2 p Q p := Sp D 2 p Or lowski SFI) Modeling Divergence Swap Raes R in Finance 3 / 8
21 Empirical moivaion S&P 500 opions Price of divergence porfolio annualised) p = 0.5 D 1/ τ = 0.08 τ = 0.50 τ = Dae Implied volailiy ATM) IV level Dae Or lowski SFI) Modeling Divergence Swap Raes R in Finance 4 / 8
22 Empirical moivaion S&P 500 opions Price of skewness porfolio scaled) p = 0.5 3/2 S 1/2/D 1/ τ = 0.08 τ = 0.50 τ = Dae Implied volailiy slope a log-srike k = 3/σ AT M τ) IV slope Dae Or lowski SFI) Modeling Divergence Swap Raes R in Finance 4 / 8
23 Empirical moivaion S&P 500 opions Prices of variaion swaps a range of p and τ conain he same informaion as he IV surface; Forming opion porfolios migh alleviae measuremen error, paricularly for deep OTM conracs; Variaion swaps are exacly radable. IV is no. Who knows wha IV is anyway? Or lowski SFI) Modeling Divergence Swap Raes R in Finance 4 / 8
24 Empirical moivaion S&P 500 opions Variaion swaps are exacly radable. IV is no. Who knows wha IV is anyway? Or lowski SFI) Modeling Divergence Swap Raes R in Finance 4 / 8
25 Divergence in Affine Jump Diffusions Affine Jump Diffusion models are a popular and somewha successful) modeling ool. Sae variables: X = [ln S /S s V 1... V M ], V := X [ 1] ] [e u X+s = e αs,u)+βs,u) V, u C M+1 E M Or lowski SFI) Modeling Divergence Swap Raes R in Finance 5 / 8
26 Divergence in Affine Jump Diffusions Affine Jump Diffusion models are a popular and somewha successful) modeling ool. Sae variables: X = [ln S /S s V 1... V M ], V := X [ 1] ] [e u X+s = e αs,u)+βs,u) V, u C M+1 E M Easily calculae D p,,s, S p,,s, Qp,,s in model MGF and derivaives). Or lowski SFI) Modeling Divergence Swap Raes R in Finance 5 / 8
27 Divergence in Affine Jump Diffusions Affine Jump Diffusion models are a popular and somewha successful) modeling ool. Sae variables: X = [ln S /S s V 1... V M ], V := X [ 1] ] [e u X+s = e αs,u)+βs,u) V, u C M+1 E M Easily calculae D p,,s, S p,,s, Qp,,s in model MGF and derivaives). Observable quaniies: S +1 S, D p,+1,s, S p,+1,s, Qp,+1,s: funcions of [V V +1] Or lowski SFI) Modeling Divergence Swap Raes R in Finance 5 / 8
28 Divergence in Affine Jump Diffusions Affine Jump Diffusion models are a popular and somewha successful) modeling ool. Sae variables: X = [ln S /S s V 1... V M ], V := X [ 1] ] [e u X+s = e αs,u)+βs,u) V, u C M+1 E M Easily calculae D p,,s, S p,,s, Qp,,s in model MGF and derivaives). S Observable quaniies: +1 S, D p,+1,s, S p,+1,s, Qp,+1,s: funcions of [V V +1] Laen volailiy facor dynamics: V +1 = µ V +1 + Σ V +1 W +1, µ V +1, Σ V +1 : non-linear funcions of V Or lowski SFI) Modeling Divergence Swap Raes R in Finance 5 / 8
29 Divergence in Affine Jump Diffusions Affine Jump Diffusion models are a popular and somewha successful) modeling ool. Sae variables: X = [ln S /S s V 1... V M ], V := X [ 1] ] [e u X+s = e αs,u)+βs,u) V, u C M+1 E M Easily calculae D p,,s, S p,,s, Qp,,s in model MGF and derivaives). S Observable quaniies: +1 S, D p,+1,s, S p,+1,s, Qp,+1,s: funcions of [V V +1] Laen volailiy facor dynamics: V +1 = µ V +1 + Σ V +1 W +1, µ V +1, Σ V +1 : non-linear funcions of V Unscened Kalman Filer counerfacual assumpion of W N0, I M M ); Specificaion of model under Q Risk Premia specificaion under P; Reasonable performance in simulaed seings. Or lowski SFI) Modeling Divergence Swap Raes R in Finance 5 / 8
30 Preliminary resuls Firs esimaions wih p = 1/2 symmeric swaps), mauriies 1M, 6M; In hree facor specificaion possible o idenify: a variance-level facor, a common facor for S and Q, a erm-srucure facor driving he slopes of he S and Q TS. Exising challenges: Improvemens in fi of S and Q a he price of bad D fi; no enough flexibiliy in he model s pricing properies. Key for improvemen: Careful modeling of jump disribuions Exp-Laplace co-jumps, bu see Bollerslev and Todorov 2014)); Inclusion of purely diffusive vol facor ha does no drive jumps; Furher work: Implemening more flexible jump specificaions; Esimaion wih a greaer number of swap conracs e.g. p = 0, more mauriies); Inference abou risk premia. Or lowski SFI) Modeling Divergence Swap Raes R in Finance 6 / 8
31 R packages affinemodelr Numeric backend for handling P and Q Characerisic Funcions of almos) arbirarily specified Affine Jump-Diffusion models. Semi-closed form[ soluions for up o 3rd ) ] k derivaive wr he sock price argumen. Soluions for E P S+1 s Vj S ) m for m + k 2, m, k > 0. Use wih package ransformopionpricer for vanilla opions. divergencemodelr Model-based pricing of divergence and higher-order swaps. Unscened Kalman Filers for esimaion of AJD models. Builds agains affinemodelr and ukfrcpp. ukfrcpp Rcpp implemenaion of an Unscened Kalman Filer class. Users have o wrie C++ funcions for handling he sae dynamics and observaion equaion, hen wrie an Rcpp funcion o creae an ukfclass objec and filer o reurn saes or likelihood value. Rlibcmaes by András Sali R bindings for he libcmaes opimisaion library. Or lowski SFI) Modeling Divergence Swap Raes R in Finance 7 / 8
32 Secion 1 References Or lowski SFI) Modeling Divergence Swap Raes R in Finance 8 / 8
33 T. Andersen, V. Todorov, and N. Fusari. The risk premia embedded in index opions. Journal of Financial Economics, Forhcoming, Tim Bollerslev and Vikor Todorov. Time-varying jump ails. Journal of Economerics, 1832): , ISSN doi: hp://dx.doi.org/ /j.jeconom URL hp:// Analysis of Financial Daa. Oleg Bondarenko. Variance rading and marke price of variance risk. Journal of Economerics, 1801):81 97, ISSN doi: hp://dx.doi.org/ /j.jeconom URL hp:// CBOE. The cboe volailiy index vix. Technical repor, CBOE, Peer H. Gruber, Claudio Tebaldi, and Fabio Trojani. The price of he smile and variance risk premia. Working Paper, Roger Lee. Weighed Variance Swap. John Wiley & Sons, Ld, ISBN doi: / eqf URL hp://dx.doi.org/ / eqf Ian Marin. Simple variance swaps. Working Paper 16884, Naional Bureau of Economic Research, March URL hp:// Paul G. Schneider and Fabio Trojani. Divergence and he price of uncerainy. Working Paper, Ocober Or lowski SFI) Modeling Divergence Swap Raes R in Finance 8 / 8
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